Research ArticleCaristi Type Selections of Multivalued Mappings

Calogero Vetro1 and Francesca Vetro2

1Dipartimento di Matematica e Informatica Universita degli Studi di Palermo Via Archirafi 34 90123 Palermo Italy2Dipartimento Energia Ingegneria dellrsquoInformazione e Modelli Matematici (DEIM) Universita degli Studi di PalermoViale delle Scienze 90128 Palermo Italy

Correspondence should be addressed to Francesca Vetro francescavetrounipait

Received 29 October 2014 Accepted 16 December 2014

Academic Editor Sompong Dhompongsa

Copyright copy 2015 C Vetro and F VetroThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Multivaluedmappings and related selection theorems are fundamental tools inmany branches ofmathematics and applied sciencesIn this paper we continue this theory and prove the existence of Caristi type selections for generalized multivalued contractions oncomplete metric spaces by using some classes of functions Also we prove fixed point and quasi-fixed point theorems

1 Introduction and Preliminaries

In 1998 Repovs and Semenov [1] furnished a comprehensivestudy of the theory of continuous selections for multivaluedmappings They point out that ldquothis interesting branch ofmodern topology was started by Michael [2] and sincethen has received a great amount of interest with variousapplications outside topology for instance approximationtheory control theory convex sets differential inclusionseconomics fixed point theory and vector measuresrdquo Thusan interesting matter is to obtain existence conditions forselections under different regularity hypotheses for instanceLipschitz-continuity and measurability see also [3ndash5] Inparticular we are interested in developing this theory forfixed point theorems by using Caristirsquos mappings Someprecise results concerning existence of fixed points forCaristirsquos single-valued and multivalued mappings and datadependence of fixed points set are proved in [6] see also thereferences therein We recall that Browder [7] was the firstauthor to use continuous selections to prove a fixed pointresult but the first result of Caristi type selection was provedby Jachymski [8] for Nadlerrsquos multivalued contraction withclosed values

Definition 1 Let (119883 119889) be ametric space A function120601 119883 rarr

[0 +infin) is lower semicontinuous at 119909 isin 119883 if and only iffor every sequence 119909

119899 in 119883 with 119909

119899rarr 119909 as 119899 rarr +infin

lim inf119899rarr+infin

120601(119909119899) ge 120601(119909) Also 120601 is lower semicontinuous

if and only if it is lower semicontinuous at every 119909 isin 119883

Also 119871(119910) = 119909 isin 119883 120601(119909) le 119910 is called thelower counter set defined by a point 119910 isin [0 +infin) Then thefollowing results hold true

Proposition 2 Let (119883 119889) be a metric space Let 120601 119883 rarr

[0 +infin) be a function Then 120601 is lower semicontinuous if andonly if 119871(119910) is closed for every 119910 isin [0 +infin)

Theorem 3 (Caristi [9]) Let (119883 119889) be a complete metric spaceand let 119891 119883 rarr 119883 be a mapping not necessarily continuousAssume that there exists a function 120601 119883 rarr [0 +infin) whichis lower semicontinuous such that

119889 (119909 119891119909) le 120601 (119909) minus 120601 (119891119909) forall119909 isin 119883 (1)

Then 119891 has a fixed point 119911 that is 119911 = 119891119911

Also 119891 is called Caristirsquos mapping on (119883 119889) On the otherhand Nadler [10 11] established the following result

Theorem 4 (Nadler [10 11]) Let (119883 119889) be a complete metricspace and let 119865 119883 rarr 119862119897(119883) be a multivalued mapping suchthat for some 119896 isin (0 1) one has

119867(119865119909 119865119910) le 119896119889 (119909 119910) forall119909 119910 isin 119883 (2)

where 119862119897(119883) denotes the class of all nonempty closed subsetsof 119883 and 119867 denotes a generalized Hausdorff metric on 119862119897(119883)Then 119865 has a fixed point 119911 that is 119911 isin 119865119911

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 941856 6 pageshttpdxdoiorg1011552015941856

2 Journal of Function Spaces

For119860 119861 isin 119862119897(119883) we recall that119867(119860 119861) = maxsup119863(119909

119861) 119909 isin 119860 sup119863(119910 119860) 119910 isin 119861 where 119863(119909 119861) = inf119889(119909

119910) 119910 isin 119861 Also a multivalued mapping 119865 satisfyingthe assumption of Theorem 4 is called Nadlerrsquos multivaluedcontraction

Definition 5 Let 119865 119883 rarr 119862119897(119883) be a multivalued mappingand let 119891 119883 rarr 119883 be a (single-valued) mapping Then 119891 issaid to be a selection for 119865 if

119891119909 isin 119865119909 119909 isin 119883 (3)

Also 119891 is called Caristi type selection if it is Caristirsquos map-ping As mentioned above Jachymski established existencetheorems stating that certain multivalued mappings admitselections that are Caristirsquos mappings which do not need tobe continuous (see for instance Example 1 in [8])

Theorem 6 (Jachymski [8]) If 119865 is Nadlerrsquos multivaluedcontraction on a complete metric space (119883 119889) then 119865 admitsa selection 119891 119883 rarr 119883 which is Caristirsquos mapping on (119883 119889)

generated by a Lipschitz function 120601

Clearly Theorem 3 yields Theorem 4 that is everyNadlerrsquos multivalued mapping admits a fixed point butthe converse does not hold in general Obviously if themultivalued mapping does not admit a fixed point then aCaristi type selection cannot exist The following exampleillustrates the case of a multivalued mapping which does notadmit a Caristi type selection even if it has a fixed point

Example 7 (Xu [12]) Consider the complete metric space([0 +infin) 119889) where 119889 denotes the standard metric Define 119865

[0 +infin) rarr 119862119897([0 +infin)) as119865119909 = [2119909 3119909] for all 119909 isin [0 +infin)

Trivially 0 is a unique fixed point of 119865 Now assumethat there exists a Caristi type selection for 119865 say 119891 Thenreferring to notions and notations of Theorem 3 we write119889(119909 119891119909) le 120601(119909) minus 120601(119891119909) for all 119909 isin [0 +infin) By definitionof 119865 we have 2119909 le 119891119909 le 3119909 and so 119909 le 120601(119909) minus 120601(119891119909) for all119909 isin (0 +infin) By iteration we can get easily that

119891119899

119909 le 120601 (119891119899

119909) minus 120601 (119891119899+1

119909) forall119899 isin N cup 0 (4)

This implies that the sequence 120601(119891119899119909) is nonincreasing andso being bounded below convergent to some 119903 ge 0 Alsofrom (4) as 119899 rarr +infin we get 119891

119899119909 rarr 0 On the otherhand the reader can immediately prove that 119891119899119909 is a strictlyincreasing sequence and hence we get a contradiction withthe above limit Then we conclude that 119891 is not a Caristi typeselection

Definition 8 Given a function 120583 [0 +infin) rarr [0 +infin) with120583(119905) lt 119905 for 119905 gt 0 a multivalued mapping 119865 119883 rarr 119862119897(119883) issaid to be a multivalued 120583-contraction if

119867(119865119909 119865119910) le 120583 (119889 (119909 119910)) forall119909 119910 isin 119883 (5)

Definition 9 A function 120583 [0 +infin) rarr [0 +infin) is said to besubadditive if

120583 (119904 + 119905) le 120583 (119904) + 120583 (119905) forall119904 119905 isin [0 +infin) (6)

Also 120583 is said to be superadditive if the reverse inequalityholds true

Theorem 10 (Jachymski [8]) Let 119865 119883 rarr 119862119897(119883) be amultivalued 120583-contraction on a complete metric space (119883 119889)

such that 120583 is superadditive and the function 119905 rarr 119905minus120583(119905) (119905 isin

[0 +infin)) is nondecreasing Then there exist a selection 119891 of119865 and a function ℎ [0 +infin) rarr [0 +infin) which isnondecreasing and subadditive and continuous at 119905 = 0 suchthat ℎ

minus1(0) = 0 Moreover there is an equivalent metric 984858

such that (119883 984858) is complete and f is Caristirsquos mapping on (119883 984858)

In this paper we continue this study and prove the exis-tence of Caristi type selections for generalized multivaluedcontractions on complete metric spaces Our results fit intothe theory of selections for multivalued mappings showingcertain ways to establish selection theorems by using someclasses of functions Alsowe prove fixed point and quasi-fixedpoint theoremsWe remark that the existence of a Caristi typeselection for a multivalued mapping ensures the existence ofa fixed point

2 Caristi Type Selection Theorems

We start to develop our theory by using the concept of lowersemicontinuity which is one of the most important conceptsin multivalued analysis

Theorem 11 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping and let 119902 gt 1 be a real

number Consider 119878119902(119909) = 119910 isin 119865119909 119889(119909 119910) le 119902119863(119909 119865119909)

and suppose that 119865 satisfies the following conditions(i) there exist two nonnegative real numbers 119886 119887with 119886119902+

119887 lt 1 such that for each 119909 isin 119883 there is 119910 isin 119878119902(119909)

having the property

119863(119910 119865119910) le 119886119889 (119909 119910) + 119887119863 (119909 119865119909) (7)

(ii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuousThen 119865 has a selection 119891 that is Caristirsquos mapping

Proof By the axiom of choice and condition (i) there is amapping 119891 119883 rarr 119883 with 119891119909 isin 119878

119902(119909) for all 119909 isin 119883 such

that

119863(119891119909 119865119891119909) le 119886119889 (119909 119891119909) + 119887119863 (119909 119865119909) (8)

and so

119863(119891119909 119865119891119909) minus 119887119863 (119909 119865119909) le 119886119889 (119909 119891119909) (9)

Then(1 minus 119887 minus 119886119902) 119889 (119909 119891119909) le (1 minus 119887) 119902119863 (119909 119865119909) minus 119886119902119889 (119909 119891119909)

le (1 minus 119887) 119902119863 (119909 119865119909) minus 119902119863 (119891119909 119865119891119909)

+ 119887119902119863 (119909 119865119909)

= 119902119863 (119909 119865119909) minus 119902119863 (119891119909 119865119891119909)

(10)

Journal of Function Spaces 3

By condition (ii) the function 120601 119883 rarr [0 +infin) definedby 120601(119905) = (119902(1 minus 119887 minus 119886119902))119863(119905 119865119905) for all 119905 isin 119883 is lowersemicontinuous and hence 119891 is Caristirsquos mapping that is aselection of 119865

Example 12 Let119883 = [0 1] be endowed with the usual metric119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) is a completemetric space Also let 119865 119883 rarr 2119883 0 be defined by

119865119909 =

[0119909

2] if 119909 isin [0 1[

[1

2 1] if 119909 = 1

(11)

Consider 119902 = 43 119886 = 12 and 119887 = 16 such that 119886119902 + 119887 =

56 lt 1 Then for 119909 = 1 and 119910 = 1199092 we have 119889(119909 119910) =

1199092 le (43) sdot (1199092) = (43)119863(119909 119865119909) that is 119910 isin 119878119902(119909)

Moreover

119863(119910 119865119910) = 119863(119909

2 [0

119909

4]) =

119909

4le

1

2sdot119909

2+

1

6sdot119909

2

= 119886119889 (119909 119910) + 119887119863 (119909 119865119909)

(12)

Also for 119909 = 1 and 119910 = 1 we have 119889(119909 119910) = 0 = 119902 sdot 0 =

119902119863(119909 119865119909) that is 1 isin 119878119902(119909) Moreover

119863(119910 119865119910) = 119863(1 [1

2 1]) = 0 le 119886119889 (119909 119910) + 119887119863 (119909 119865119909)

(13)

Finally the function 119901 119883 rarr [0 +infin) defined by

119901119909 = 119863 (119909 119865119909) =

119909

2if 119909 isin [0 1[

0 if 119909 = 1(14)

is lower semicontinuous in [0 1] Thus all the hypotheses ofTheorem 11 are satisfied and so 119865 has a selection 119891 that isCaristirsquos mapping In fact 119891 119883 rarr 119883 defined by 119891119909 = 1199092

for all 119909 isin 119883 is such that 119891119909 isin 119865119909 and also 119889(119909 119891119909) = 1199092 =

120601(119909) minus 120601(119891119909) where 120601 119883 rarr [0 +infin) is given by 120601(119909) = 119909

for all 119909 isin 119883Notice that

119867(1198650 1198651) = 119867(0 [1

2 1]) = max

1

2 1 = 119889 (0 1)

(15)

and hence 119865 is not Nadlerrsquos multivalued contraction

Analogous results toTheorem 11 can be established underdifferent hypotheses For instance in the next theoremthe multivalued mapping 119865 satisfies another contractivecondition

Theorem 13 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping and let 119902 gt 1 be a real

number Consider 119878119902(119909) = 119910 isin 119865119909 119889(119909 119910) le 119902119863119889(119909 119865119909)

and suppose that 119865 satisfies the following conditions(i) there exist nonnegative real numbers 120572 120573 120574 with (120572 +

120574)119902 + 120573 + 120574 lt 1 such that for each 119909 isin 119883 there is119910 isin 119878119902(119909) having the property

119863(119910 119865119910) le 120572119889 (119909 119910) + 120573119863 (119909 119865119909) + 120574119863 (119909 119865119910) (16)

(ii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous

Then 119865 has a selection 119891 that is Caristirsquos mapping

Proof By the axiom of choice and condition (i) there is amapping 119891 119883 rarr 119883 with 119891119909 isin 119878

119902(119909) for all 119909 isin 119883 such

that

119863(119891119909 119865119891119909) le 120572119889 (119909 119891119909) + 120573119863 (119909 119865119909) + 120574119863 (119909 119865119891119909)

(17)

and so by using the triangular inequality for 119863(119909 119865119891119909)

(1 minus 120574)119863 (119891119909 119865119891119909) minus 120573119863 (119909 119865119909) le (120572 + 120574) 119889 (119909 119891119909)

(18)

Then

(1 minus 120573 minus 120574 minus (120572 + 120574) 119902) 119889 (119909 119891119909)

le (1 minus 120573 minus 120574) 119902119863 (119909 119865119909) minus (120572 + 120574) 119902119889 (119909 119891119909)

le (1 minus 120573 minus 120574) 119902119863 (119909 119865119909) minus (1 minus 120574) 119902119863 (119891119909 119865119891119909)

+ 120573119902119863 (119909 119865119909)

= (1 minus 120574) 119902119863 (119909 119865119909) minus (1 minus 120574) 119902119863 (119891119909 119865119891119909)

(19)

By condition (ii) the function 120601 119883 rarr [0 +infin) definedby 120601(119905) = ((1minus120574)119902(1minus120573minus120574minus(120572+120574)119902))119863(119905 119865119905) for all 119905 isin 119883is lower semicontinuous and hence119891 is Caristirsquosmapping thatis a selection of 119865

We would like to remark that other results can be statedby involving upper semicontinuousmultivaluedmappings inview of the following situation

Definition 14 Let (119883 119889) be a metric space Then a multi-valued mapping 119865 119883 rarr 2119883 0 is said to be ℎ-uppersemicontinuous at 119909

0isin 119883 if the function

ℎ (119865119909 1198651199090) = sup 119863 (119910 119865119909

0) 119910 isin 119865119909 (20)

is continuous at 1199090 Clearly 119865 is said to be ℎ-upper semicon-

tinuous whenever ℎ(119865119909 1198651199090) is continuous at every 119909

0isin 119883

Now we present a class of multivalued mappings suchthat the function 120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 is lowersemicontinuous

Proposition 15 Let (119883 119889) be a metric space If 119865 119883 rarr

2119883 0 is ℎ-upper semicontinuous then the function 120601(119909) =

119863(119909 119865119909) is lower semicontinuous

Proof Given 119909 isin 119883 for all 119910 isin 119883 we get

120601 (119909) = 119863 (119909 119865119909) le 119889 (119909 119910) + 119863 (119910 119865119910) + ℎ (119865119910 119865119909)

le 119889 (119909 119910) + 120601 (119910) + ℎ (119865119910 119865119909) (21)

From above inequalities we deduce that 120601(119909) le

lim inf119910rarr119909

120601(119910) and so 120601 is a lower semicontinuousfunction

4 Journal of Function Spaces

For instance fromTheorem 13 and Proposition 15 we getthe following corollary

Corollary 16 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883

0 be an ℎ-upper semicontinuous multivaluedmapping and let 119902 gt 1 be a real number Consider 119878

119902(119909) =

119910 isin 119865119909 119889(119909 119910) le 119902119863(119909 119865119909) and suppose that 119865 satisfiesthe condition (119894) of Theorem 13 Then 119865 has a selection 119891 thatis Caristirsquos mapping

3 Extension to Quasi-Fixed Point Theorems

Let (119883 119889) be a metric space We recall that a multivaluedmapping 119865 119883 rarr 2119883 0 has a quasi-fixed point ifthere exists a point 119911 isin 119883 such that 119863(119911 119865119911) = 0 Thenwe extend our theory by considering functions instead ofconstant values Therefore let (119883 119889) be a metric space andlet 119886 119887 119883 rarr [0 +infin) and 119902 119883 rarr (0 +infin) be functionssuch that

119886 (119909) 119902minus1

(119909) + 119887 (119909) lt 1 forall119909 isin 119883 (22)

Remark 17 Notice that in (22) we do not need that 119902(119909) lt 1Wewill return on this fact to derive a particular situation fromthe following theorem

Theorem 18 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (23)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(24)

Then 119865 has a quasi-fixed point that is there exists 119911 isin 119883 suchthat 119863(119911 119865119911) = 0

Proof Assume that 119863(119909 119865119909) gt 0 for all 119909 isin 119883 By the axiomof choice and condition (ii) there is a mapping 119891 119883 rarr 119883

with 119891119909 = 119909 such that

119902 (119909) 119889 (119909 119891119909) le 119863 (119909 119865119909)

119863 (119891119909 119865119891119909) le 119886 (119909) 119889 (119909 119891119909) + 119887 (119909)119863 (119909 119865119909) (25)

Then for each 119909 isin 119883

119863 (119891119909 119865119891119909) minus 119887 (119909)119863 (119909 119865119909) le 119886 (119909) 119889 (119909 119891119909) (26)

This implies

(1 minus 119887 (119909) minus 119886 (119909) 119902minus1

(119909)) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119886 (119909) 119902minus1

(119909) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

+ 119887 (119909) 119902minus1

(119909)119863 (119909 119865119909)

= 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

(27)

Consequently we have

119889 (119909 119891119909) le1

119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909)

times [119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)]

(28)

Now let 119884 = 119909 isin 119883 119863(119909 119865119909) le inf119911isin119883

119863(119911 119865119911) + 120578 Sinceby (iii)119884 is a closed subset of119883 we deduce that119884 is completeDenote by 120574 = inf119902(119909) minus 119887(119909)119902(119909) minus 119886(119909) 119909 isin 119884 gt 0 Forall 119909 isin 119884 we get

119889 (119909 119891119909) le1

120574[119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)] = 120601 (119909) minus 120601 (119891119909)

(29)

where the function 120601 119883 rarr [0 +infin) is defined by 120601(119905) =

120574minus1119863(119905 119865119905) for all 119905 isin 119883 Clearly by condition (iii) thefunction 120601 is lower semicontinuous From (29) we get that119891119909 isin 119884 whenever 119909 isin 119884 and hence 119891 119884 rarr 119884 isCaristirsquos mapping This implies that 119891 has a fixed point in 119884a contradiction since 119891119909 = 119909 for all 119909 isin 119883 Hence there is119911 isin 119883 such that 119863(119911 119865119911) = 0

As a consequence of Theorem 18 in the case that 119865119909 isalso closed we obtain the following corollary

Corollary 19 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping such that119865119909 is closedSuppose that 119865 satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (30)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous

Journal of Function Spaces 5

(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(31)

Then 119865 has a fixed point

In view of Remark 17 by assuming 119902(119909) lt 1 for all 119909 isin 119883on the same lines of the proof of Theorem 18 one can provethe following result

Theorem 20 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions(i) there exist three functions 119886 119887 119883 rarr [0 1) and 119902

119883 rarr (0 1) such that (22) holds(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin 119865119909

such that119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (32)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(33)

Then 119865 has a selection 119891 that is Caristirsquos mapping on a closedsubset of 119883

Remark 21 If in Theorems 18 and 20 and Corollary 19we assume that the multivalued mapping 119865 is ℎ-uppersemicontinuous then (iii) holds true In this case we canreformulate the statements of these results requiring that 119865

satisfies only conditions (i) (ii) and (iv)

4 Generalization of Caristirsquos Theorem

We denote by Φ the set of all functions 120577 [0 +infin) rarr

[0 +infin) such that there exist 120576 gt 0 and 119888 isin (0 1) satisfying120575120576

= sup 120577minus1

([0 120576]) lt +infin 120577(119905) ge 119888119905 for all 119905 isin [0 120575120576] and

120577(119905) gt 120576 for all 119905 gt 120575120576

Remark 22 Given a nondecreasing function 120577 [0 +infin) rarr

[0 +infin) continuous at 119905 = 0 with 120577(0) = 0 consider the rightlower Dini derivative of 120577 at 119905 isin [0 +infin) that is

[119863+120577] (119905) = lim inf

119904rarr 119905+

120577 (119904) minus 120577 (119905)

119904 minus 119905 (34)

Then 120577 isin Φ provided that [119863+120577](0) gt 0 see [8] Also each

function 120577 [0 +infin) rarr [0 +infin) that is nondecreasingsubadditive and continuous at 119905 = 0 with 120577(0) = 0 belongs toΦ

Inspired by Khamsi [13] and Jachymski [8] we givetwo fixed point theorems In particular our first theoremfurnishes an alternative proof to Theorem 3 of [13] and therelated Kirkrsquos problem without using order relations (seeSection 3 in [13] for more details)

Theorem 23 Let (119883 119889) be a complete metric space Let 119891

119883 rarr 119883 be a mapping Suppose that there exist a lowersemicontinuous function 120601 119883 rarr [0 +infin) and a function120577 isin Φ such that

120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) forall119909 isin 119883 (35)

Then 119891 has a fixed point in 119883

Proof Let 120576 gt 0 119888 isin (0 1) and let 120575120576be as stated above Let

119884 = 119909 isin 119883 120601 (119909) le inf119911isin119883

120601 (119911) + 120576 (36)

The set119884 is closed since 120601 is lower semicontinuous and hencecomplete Now from (35) we get that 119891119909 isin 119884 whenever 119909 isin

119884 Also120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) le 120601 (119909) minus inf

119911isin119883

120601 (119911) le 120576

(37)

for all 119909 isin 119884 we obtain that 119889(119909 119891119909) isin [0 120575120576] whenever 119909 isin

119884 Hence

119888119889 (119909 119891119909) le 120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) (38)

that is

119889 (119909 119891119909) le1

119888120601 (119909) minus

1

119888120601 (119891119909) forall119909 isin 119884 (39)

Since the function (1119888)120601 is lower semicontinuous byTheorem 3 the mapping 119891 119884 rarr 119884 has a fixed point in119884 and so in 119883

Example 24 Let119883 = minus3 minus1cup[0 +infin) be endowed with theusual metric 119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) isa complete metric space Also let 119891 119883 rarr 119883 be defined by

119891119909 = 0 if 119909 notin [1 3]

1 if 119909 isin [1 3] (40)

It follows that

119889 (119909 119891119909) = |119909| if 119909 notin [1 3]

119909 minus 1 if 119909 isin [1 3] (41)

Notice that 120601 119883 rarr [0 +infin) defined by 120601(119909) = |119909| forall 119909 isin 119883 is a lower semicontinuous function such that120577(119889(119909 119891119909)) le 120601(119909) minus 120601(119891119909) where 120577 [0 +infin) rarr [0 +infin) isgiven by 120577(119905) = 119888119905 for all 119905 ge 0 where 119888 isin (0 1) Thus we canapplyTheorem 23 to conclude that119891 has a fixed point clearly0 and 1 are fixed points of 119891

The inspiration of our next theorem is Theorem 10 Inparticular our result does not use a monotonic conditionFor a comprehensive discussion we refer the reader to thefundamental paper of Jachymski [8]

6 Journal of Function Spaces

Theorem 25 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 119862119897(119883) be a multivalued mapping Suppose that 119865 is a120583-contraction with 120583 right upper semicontinuous such that thefunction 120577(119905) = (119905 minus 120583(119905))2 for all 119905 ge 0 belongs to Φ Then 119865

has a fixed point

Proof Let ] [0 +infin) rarr [0 +infin) be the function definedby ](119905) = (119905 + 120583(119905))2 for all 119905 ge 0 Clearly ] is right uppersemicontinuous and ](119905) lt 119905 for all 119905 gt 0 Therefore the set

119910 isin 119865119909 ] (119889 (119909 119910)) le 119863 (119909 119865119909) = 0 (42)

for all 119909 isin 119883 In fact if 119910119899 sub 119865119909 is a sequence such that

119889(119909 119910119899) darr 119863(119909 119865119909) the right upper semicontinuity of the

function ] ensures that

lim sup119899rarr+infin

] (119889 (119909 119910119899)) le ] (119863 (119909 119865119909)) (43)

and hence there exists 119910119899such that

] (119889 (119909 119910119899)) lt 119863 (119909 119865119909) (44)

The axiom of choice ensures that there is a mapping 119891 119883 rarr

119883 such that

119891119909 isin 119865119909 ] (119889 (119909 119891119909)) le 119863 (119909 119865119909) for 119909 isin 119883 (45)

Since 119863(119891119909 119865119891119909) le 119867(119865119909 119865119891119909) le 120583(119889(119909 119891119909)) we get

120577 (119889 (119909 119891119909)) = ] (119889 (119909 119891119909)) minus 120583 (119889 (119909 119891119909))

le 119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)

= 120601 (119909) minus 120601 (119891119909)

(46)

where the function 120601 119883 rarr [0 +infin) is defined by120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 Since the function 120601 is lowersemicontinuous then we get that 119891 has a fixed point whichis a fixed point for 119865

5 Conclusion

Under suitable hypotheses for multivalued mappings weestablished the existence of Caristi type selections Also weproved fixed point and quasi-fixed point theorems by usingweaker andmodified hypotheses on some classes of functionspresent in the literature Our results extend and complementmany theorems in the literature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

The first author is a Member of the Gruppo Nazionale perlrsquoAnalisi Matematica la Probabilita e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM) The second author is a Member of the GruppoNazionale per le Strutture Algebriche Geometriche e le loroApplicazioni (GNSAGA) of the Istituto Nazionale di AltaMatematica (INdAM)

References

[1] D Repovs and P V Semenov Continuous Selections of Multi-valued Mappings vol 455 of Mathematics and Its Applications1998

[2] E Michael ldquoContinuous selections Irdquo Annals of MathematicsSecond Series vol 63 pp 361ndash382 1956

[3] A Petrusel ldquoMultivalued operators and continuous selectionsrdquoPure Mathematics and Applications vol 9 no 1-2 pp 165ndash1701998

[4] A Petrusel andG Petrusel ldquoSelection theorems formultivaluedgeneralized contractionsrdquo Mathematica Moravica vol 9 pp43ndash52 2005

[5] A Sıntamarian ldquoSelections and common fixed points for somegeneralized multivalued contractionsrdquo Demonstratio Mathe-matica vol 39 no 3 pp 609ndash617 2006

[6] A Petrusel and A Sıntamarian ldquoSingle-valued and multi-valued Caristi type operatorsrdquo Publicationes MathematicaeDebrecen vol 60 no 1-2 pp 167ndash177 2002

[7] F E Browder ldquoThefixed point theory ofmulti-valuedmappingsin topological vector spacesrdquo Mathematische Annalen vol 177pp 283ndash301 1968

[8] J R Jachymski ldquoCaristirsquos fixed point theorem and selections ofset-valued contractionsrdquo Journal of Mathematical Analysis andApplications vol 227 no 1 pp 55ndash67 1998

[9] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[10] H Covitz and S B Nadler Jr ldquoMulti-valued contractionmappings in generalized metric spacesrdquo Israel Journal of Math-ematics vol 8 pp 5ndash11 1970

[11] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969

[12] H-K Xu ldquoSome recent results and problems for set-valuedmappingsrdquo in Advances in Mathematics Research vol 1 pp 31ndash49 2002

[13] M A Khamsi ldquoRemarks on Caristirsquos fixed point theoremrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 227ndash231 2009

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

For119860 119861 isin 119862119897(119883) we recall that119867(119860 119861) = maxsup119863(119909

119861) 119909 isin 119860 sup119863(119910 119860) 119910 isin 119861 where 119863(119909 119861) = inf119889(119909

119910) 119910 isin 119861 Also a multivalued mapping 119865 satisfyingthe assumption of Theorem 4 is called Nadlerrsquos multivaluedcontraction

Definition 5 Let 119865 119883 rarr 119862119897(119883) be a multivalued mappingand let 119891 119883 rarr 119883 be a (single-valued) mapping Then 119891 issaid to be a selection for 119865 if

119891119909 isin 119865119909 119909 isin 119883 (3)

Also 119891 is called Caristi type selection if it is Caristirsquos map-ping As mentioned above Jachymski established existencetheorems stating that certain multivalued mappings admitselections that are Caristirsquos mappings which do not need tobe continuous (see for instance Example 1 in [8])

Theorem 6 (Jachymski [8]) If 119865 is Nadlerrsquos multivaluedcontraction on a complete metric space (119883 119889) then 119865 admitsa selection 119891 119883 rarr 119883 which is Caristirsquos mapping on (119883 119889)

generated by a Lipschitz function 120601

Clearly Theorem 3 yields Theorem 4 that is everyNadlerrsquos multivalued mapping admits a fixed point butthe converse does not hold in general Obviously if themultivalued mapping does not admit a fixed point then aCaristi type selection cannot exist The following exampleillustrates the case of a multivalued mapping which does notadmit a Caristi type selection even if it has a fixed point

Example 7 (Xu [12]) Consider the complete metric space([0 +infin) 119889) where 119889 denotes the standard metric Define 119865

[0 +infin) rarr 119862119897([0 +infin)) as119865119909 = [2119909 3119909] for all 119909 isin [0 +infin)

Trivially 0 is a unique fixed point of 119865 Now assumethat there exists a Caristi type selection for 119865 say 119891 Thenreferring to notions and notations of Theorem 3 we write119889(119909 119891119909) le 120601(119909) minus 120601(119891119909) for all 119909 isin [0 +infin) By definitionof 119865 we have 2119909 le 119891119909 le 3119909 and so 119909 le 120601(119909) minus 120601(119891119909) for all119909 isin (0 +infin) By iteration we can get easily that

119891119899

119909 le 120601 (119891119899

119909) minus 120601 (119891119899+1

119909) forall119899 isin N cup 0 (4)

This implies that the sequence 120601(119891119899119909) is nonincreasing andso being bounded below convergent to some 119903 ge 0 Alsofrom (4) as 119899 rarr +infin we get 119891

119899119909 rarr 0 On the otherhand the reader can immediately prove that 119891119899119909 is a strictlyincreasing sequence and hence we get a contradiction withthe above limit Then we conclude that 119891 is not a Caristi typeselection

Definition 8 Given a function 120583 [0 +infin) rarr [0 +infin) with120583(119905) lt 119905 for 119905 gt 0 a multivalued mapping 119865 119883 rarr 119862119897(119883) issaid to be a multivalued 120583-contraction if

119867(119865119909 119865119910) le 120583 (119889 (119909 119910)) forall119909 119910 isin 119883 (5)

Definition 9 A function 120583 [0 +infin) rarr [0 +infin) is said to besubadditive if

120583 (119904 + 119905) le 120583 (119904) + 120583 (119905) forall119904 119905 isin [0 +infin) (6)

Also 120583 is said to be superadditive if the reverse inequalityholds true

Theorem 10 (Jachymski [8]) Let 119865 119883 rarr 119862119897(119883) be amultivalued 120583-contraction on a complete metric space (119883 119889)

such that 120583 is superadditive and the function 119905 rarr 119905minus120583(119905) (119905 isin

[0 +infin)) is nondecreasing Then there exist a selection 119891 of119865 and a function ℎ [0 +infin) rarr [0 +infin) which isnondecreasing and subadditive and continuous at 119905 = 0 suchthat ℎ

minus1(0) = 0 Moreover there is an equivalent metric 984858

such that (119883 984858) is complete and f is Caristirsquos mapping on (119883 984858)

In this paper we continue this study and prove the exis-tence of Caristi type selections for generalized multivaluedcontractions on complete metric spaces Our results fit intothe theory of selections for multivalued mappings showingcertain ways to establish selection theorems by using someclasses of functions Alsowe prove fixed point and quasi-fixedpoint theoremsWe remark that the existence of a Caristi typeselection for a multivalued mapping ensures the existence ofa fixed point

2 Caristi Type Selection Theorems

We start to develop our theory by using the concept of lowersemicontinuity which is one of the most important conceptsin multivalued analysis

Theorem 11 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping and let 119902 gt 1 be a real

number Consider 119878119902(119909) = 119910 isin 119865119909 119889(119909 119910) le 119902119863(119909 119865119909)

and suppose that 119865 satisfies the following conditions(i) there exist two nonnegative real numbers 119886 119887with 119886119902+

119887 lt 1 such that for each 119909 isin 119883 there is 119910 isin 119878119902(119909)

having the property

119863(119910 119865119910) le 119886119889 (119909 119910) + 119887119863 (119909 119865119909) (7)

(ii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuousThen 119865 has a selection 119891 that is Caristirsquos mapping

Proof By the axiom of choice and condition (i) there is amapping 119891 119883 rarr 119883 with 119891119909 isin 119878

119902(119909) for all 119909 isin 119883 such

that

119863(119891119909 119865119891119909) le 119886119889 (119909 119891119909) + 119887119863 (119909 119865119909) (8)

and so

119863(119891119909 119865119891119909) minus 119887119863 (119909 119865119909) le 119886119889 (119909 119891119909) (9)

Then(1 minus 119887 minus 119886119902) 119889 (119909 119891119909) le (1 minus 119887) 119902119863 (119909 119865119909) minus 119886119902119889 (119909 119891119909)

le (1 minus 119887) 119902119863 (119909 119865119909) minus 119902119863 (119891119909 119865119891119909)

+ 119887119902119863 (119909 119865119909)

= 119902119863 (119909 119865119909) minus 119902119863 (119891119909 119865119891119909)

(10)

Journal of Function Spaces 3

By condition (ii) the function 120601 119883 rarr [0 +infin) definedby 120601(119905) = (119902(1 minus 119887 minus 119886119902))119863(119905 119865119905) for all 119905 isin 119883 is lowersemicontinuous and hence 119891 is Caristirsquos mapping that is aselection of 119865

Example 12 Let119883 = [0 1] be endowed with the usual metric119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) is a completemetric space Also let 119865 119883 rarr 2119883 0 be defined by

119865119909 =

[0119909

2] if 119909 isin [0 1[

[1

2 1] if 119909 = 1

(11)

Consider 119902 = 43 119886 = 12 and 119887 = 16 such that 119886119902 + 119887 =

56 lt 1 Then for 119909 = 1 and 119910 = 1199092 we have 119889(119909 119910) =

1199092 le (43) sdot (1199092) = (43)119863(119909 119865119909) that is 119910 isin 119878119902(119909)

Moreover

119863(119910 119865119910) = 119863(119909

2 [0

119909

4]) =

119909

4le

1

2sdot119909

2+

1

6sdot119909

2

= 119886119889 (119909 119910) + 119887119863 (119909 119865119909)

(12)

Also for 119909 = 1 and 119910 = 1 we have 119889(119909 119910) = 0 = 119902 sdot 0 =

119902119863(119909 119865119909) that is 1 isin 119878119902(119909) Moreover

119863(119910 119865119910) = 119863(1 [1

2 1]) = 0 le 119886119889 (119909 119910) + 119887119863 (119909 119865119909)

(13)

Finally the function 119901 119883 rarr [0 +infin) defined by

119901119909 = 119863 (119909 119865119909) =

119909

2if 119909 isin [0 1[

0 if 119909 = 1(14)

is lower semicontinuous in [0 1] Thus all the hypotheses ofTheorem 11 are satisfied and so 119865 has a selection 119891 that isCaristirsquos mapping In fact 119891 119883 rarr 119883 defined by 119891119909 = 1199092

for all 119909 isin 119883 is such that 119891119909 isin 119865119909 and also 119889(119909 119891119909) = 1199092 =

120601(119909) minus 120601(119891119909) where 120601 119883 rarr [0 +infin) is given by 120601(119909) = 119909

for all 119909 isin 119883Notice that

119867(1198650 1198651) = 119867(0 [1

2 1]) = max

1

2 1 = 119889 (0 1)

(15)

and hence 119865 is not Nadlerrsquos multivalued contraction

Analogous results toTheorem 11 can be established underdifferent hypotheses For instance in the next theoremthe multivalued mapping 119865 satisfies another contractivecondition

Theorem 13 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping and let 119902 gt 1 be a real

number Consider 119878119902(119909) = 119910 isin 119865119909 119889(119909 119910) le 119902119863119889(119909 119865119909)

and suppose that 119865 satisfies the following conditions(i) there exist nonnegative real numbers 120572 120573 120574 with (120572 +

120574)119902 + 120573 + 120574 lt 1 such that for each 119909 isin 119883 there is119910 isin 119878119902(119909) having the property

119863(119910 119865119910) le 120572119889 (119909 119910) + 120573119863 (119909 119865119909) + 120574119863 (119909 119865119910) (16)

(ii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous

Then 119865 has a selection 119891 that is Caristirsquos mapping

Proof By the axiom of choice and condition (i) there is amapping 119891 119883 rarr 119883 with 119891119909 isin 119878

119902(119909) for all 119909 isin 119883 such

that

119863(119891119909 119865119891119909) le 120572119889 (119909 119891119909) + 120573119863 (119909 119865119909) + 120574119863 (119909 119865119891119909)

(17)

and so by using the triangular inequality for 119863(119909 119865119891119909)

(1 minus 120574)119863 (119891119909 119865119891119909) minus 120573119863 (119909 119865119909) le (120572 + 120574) 119889 (119909 119891119909)

(18)

Then

(1 minus 120573 minus 120574 minus (120572 + 120574) 119902) 119889 (119909 119891119909)

le (1 minus 120573 minus 120574) 119902119863 (119909 119865119909) minus (120572 + 120574) 119902119889 (119909 119891119909)

le (1 minus 120573 minus 120574) 119902119863 (119909 119865119909) minus (1 minus 120574) 119902119863 (119891119909 119865119891119909)

+ 120573119902119863 (119909 119865119909)

= (1 minus 120574) 119902119863 (119909 119865119909) minus (1 minus 120574) 119902119863 (119891119909 119865119891119909)

(19)

By condition (ii) the function 120601 119883 rarr [0 +infin) definedby 120601(119905) = ((1minus120574)119902(1minus120573minus120574minus(120572+120574)119902))119863(119905 119865119905) for all 119905 isin 119883is lower semicontinuous and hence119891 is Caristirsquosmapping thatis a selection of 119865

We would like to remark that other results can be statedby involving upper semicontinuousmultivaluedmappings inview of the following situation

Definition 14 Let (119883 119889) be a metric space Then a multi-valued mapping 119865 119883 rarr 2119883 0 is said to be ℎ-uppersemicontinuous at 119909

0isin 119883 if the function

ℎ (119865119909 1198651199090) = sup 119863 (119910 119865119909

0) 119910 isin 119865119909 (20)

is continuous at 1199090 Clearly 119865 is said to be ℎ-upper semicon-

tinuous whenever ℎ(119865119909 1198651199090) is continuous at every 119909

0isin 119883

Now we present a class of multivalued mappings suchthat the function 120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 is lowersemicontinuous

Proposition 15 Let (119883 119889) be a metric space If 119865 119883 rarr

2119883 0 is ℎ-upper semicontinuous then the function 120601(119909) =

119863(119909 119865119909) is lower semicontinuous

Proof Given 119909 isin 119883 for all 119910 isin 119883 we get

120601 (119909) = 119863 (119909 119865119909) le 119889 (119909 119910) + 119863 (119910 119865119910) + ℎ (119865119910 119865119909)

le 119889 (119909 119910) + 120601 (119910) + ℎ (119865119910 119865119909) (21)

From above inequalities we deduce that 120601(119909) le

lim inf119910rarr119909

120601(119910) and so 120601 is a lower semicontinuousfunction

4 Journal of Function Spaces

For instance fromTheorem 13 and Proposition 15 we getthe following corollary

Corollary 16 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883

0 be an ℎ-upper semicontinuous multivaluedmapping and let 119902 gt 1 be a real number Consider 119878

119902(119909) =

119910 isin 119865119909 119889(119909 119910) le 119902119863(119909 119865119909) and suppose that 119865 satisfiesthe condition (119894) of Theorem 13 Then 119865 has a selection 119891 thatis Caristirsquos mapping

3 Extension to Quasi-Fixed Point Theorems

Let (119883 119889) be a metric space We recall that a multivaluedmapping 119865 119883 rarr 2119883 0 has a quasi-fixed point ifthere exists a point 119911 isin 119883 such that 119863(119911 119865119911) = 0 Thenwe extend our theory by considering functions instead ofconstant values Therefore let (119883 119889) be a metric space andlet 119886 119887 119883 rarr [0 +infin) and 119902 119883 rarr (0 +infin) be functionssuch that

119886 (119909) 119902minus1

(119909) + 119887 (119909) lt 1 forall119909 isin 119883 (22)

Remark 17 Notice that in (22) we do not need that 119902(119909) lt 1Wewill return on this fact to derive a particular situation fromthe following theorem

Theorem 18 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (23)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(24)

Then 119865 has a quasi-fixed point that is there exists 119911 isin 119883 suchthat 119863(119911 119865119911) = 0

Proof Assume that 119863(119909 119865119909) gt 0 for all 119909 isin 119883 By the axiomof choice and condition (ii) there is a mapping 119891 119883 rarr 119883

with 119891119909 = 119909 such that

119902 (119909) 119889 (119909 119891119909) le 119863 (119909 119865119909)

119863 (119891119909 119865119891119909) le 119886 (119909) 119889 (119909 119891119909) + 119887 (119909)119863 (119909 119865119909) (25)

Then for each 119909 isin 119883

119863 (119891119909 119865119891119909) minus 119887 (119909)119863 (119909 119865119909) le 119886 (119909) 119889 (119909 119891119909) (26)

This implies

(1 minus 119887 (119909) minus 119886 (119909) 119902minus1

(119909)) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119886 (119909) 119902minus1

(119909) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

+ 119887 (119909) 119902minus1

(119909)119863 (119909 119865119909)

= 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

(27)

Consequently we have

119889 (119909 119891119909) le1

119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909)

times [119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)]

(28)

Now let 119884 = 119909 isin 119883 119863(119909 119865119909) le inf119911isin119883

119863(119911 119865119911) + 120578 Sinceby (iii)119884 is a closed subset of119883 we deduce that119884 is completeDenote by 120574 = inf119902(119909) minus 119887(119909)119902(119909) minus 119886(119909) 119909 isin 119884 gt 0 Forall 119909 isin 119884 we get

119889 (119909 119891119909) le1

120574[119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)] = 120601 (119909) minus 120601 (119891119909)

(29)

where the function 120601 119883 rarr [0 +infin) is defined by 120601(119905) =

120574minus1119863(119905 119865119905) for all 119905 isin 119883 Clearly by condition (iii) thefunction 120601 is lower semicontinuous From (29) we get that119891119909 isin 119884 whenever 119909 isin 119884 and hence 119891 119884 rarr 119884 isCaristirsquos mapping This implies that 119891 has a fixed point in 119884a contradiction since 119891119909 = 119909 for all 119909 isin 119883 Hence there is119911 isin 119883 such that 119863(119911 119865119911) = 0

As a consequence of Theorem 18 in the case that 119865119909 isalso closed we obtain the following corollary

Corollary 19 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping such that119865119909 is closedSuppose that 119865 satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (30)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous

Journal of Function Spaces 5

(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(31)

Then 119865 has a fixed point

In view of Remark 17 by assuming 119902(119909) lt 1 for all 119909 isin 119883on the same lines of the proof of Theorem 18 one can provethe following result

Theorem 20 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions(i) there exist three functions 119886 119887 119883 rarr [0 1) and 119902

119883 rarr (0 1) such that (22) holds(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin 119865119909

such that119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (32)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(33)

Then 119865 has a selection 119891 that is Caristirsquos mapping on a closedsubset of 119883

Remark 21 If in Theorems 18 and 20 and Corollary 19we assume that the multivalued mapping 119865 is ℎ-uppersemicontinuous then (iii) holds true In this case we canreformulate the statements of these results requiring that 119865

satisfies only conditions (i) (ii) and (iv)

4 Generalization of Caristirsquos Theorem

We denote by Φ the set of all functions 120577 [0 +infin) rarr

[0 +infin) such that there exist 120576 gt 0 and 119888 isin (0 1) satisfying120575120576

= sup 120577minus1

([0 120576]) lt +infin 120577(119905) ge 119888119905 for all 119905 isin [0 120575120576] and

120577(119905) gt 120576 for all 119905 gt 120575120576

Remark 22 Given a nondecreasing function 120577 [0 +infin) rarr

[0 +infin) continuous at 119905 = 0 with 120577(0) = 0 consider the rightlower Dini derivative of 120577 at 119905 isin [0 +infin) that is

[119863+120577] (119905) = lim inf

119904rarr 119905+

120577 (119904) minus 120577 (119905)

119904 minus 119905 (34)

Then 120577 isin Φ provided that [119863+120577](0) gt 0 see [8] Also each

function 120577 [0 +infin) rarr [0 +infin) that is nondecreasingsubadditive and continuous at 119905 = 0 with 120577(0) = 0 belongs toΦ

Inspired by Khamsi [13] and Jachymski [8] we givetwo fixed point theorems In particular our first theoremfurnishes an alternative proof to Theorem 3 of [13] and therelated Kirkrsquos problem without using order relations (seeSection 3 in [13] for more details)

Theorem 23 Let (119883 119889) be a complete metric space Let 119891

119883 rarr 119883 be a mapping Suppose that there exist a lowersemicontinuous function 120601 119883 rarr [0 +infin) and a function120577 isin Φ such that

120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) forall119909 isin 119883 (35)

Then 119891 has a fixed point in 119883

Proof Let 120576 gt 0 119888 isin (0 1) and let 120575120576be as stated above Let

119884 = 119909 isin 119883 120601 (119909) le inf119911isin119883

120601 (119911) + 120576 (36)

The set119884 is closed since 120601 is lower semicontinuous and hencecomplete Now from (35) we get that 119891119909 isin 119884 whenever 119909 isin

119884 Also120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) le 120601 (119909) minus inf

119911isin119883

120601 (119911) le 120576

(37)

for all 119909 isin 119884 we obtain that 119889(119909 119891119909) isin [0 120575120576] whenever 119909 isin

119884 Hence

119888119889 (119909 119891119909) le 120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) (38)

that is

119889 (119909 119891119909) le1

119888120601 (119909) minus

1

119888120601 (119891119909) forall119909 isin 119884 (39)

Since the function (1119888)120601 is lower semicontinuous byTheorem 3 the mapping 119891 119884 rarr 119884 has a fixed point in119884 and so in 119883

Example 24 Let119883 = minus3 minus1cup[0 +infin) be endowed with theusual metric 119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) isa complete metric space Also let 119891 119883 rarr 119883 be defined by

119891119909 = 0 if 119909 notin [1 3]

1 if 119909 isin [1 3] (40)

It follows that

119889 (119909 119891119909) = |119909| if 119909 notin [1 3]

119909 minus 1 if 119909 isin [1 3] (41)

Notice that 120601 119883 rarr [0 +infin) defined by 120601(119909) = |119909| forall 119909 isin 119883 is a lower semicontinuous function such that120577(119889(119909 119891119909)) le 120601(119909) minus 120601(119891119909) where 120577 [0 +infin) rarr [0 +infin) isgiven by 120577(119905) = 119888119905 for all 119905 ge 0 where 119888 isin (0 1) Thus we canapplyTheorem 23 to conclude that119891 has a fixed point clearly0 and 1 are fixed points of 119891

The inspiration of our next theorem is Theorem 10 Inparticular our result does not use a monotonic conditionFor a comprehensive discussion we refer the reader to thefundamental paper of Jachymski [8]

6 Journal of Function Spaces

Theorem 25 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 119862119897(119883) be a multivalued mapping Suppose that 119865 is a120583-contraction with 120583 right upper semicontinuous such that thefunction 120577(119905) = (119905 minus 120583(119905))2 for all 119905 ge 0 belongs to Φ Then 119865

has a fixed point

Proof Let ] [0 +infin) rarr [0 +infin) be the function definedby ](119905) = (119905 + 120583(119905))2 for all 119905 ge 0 Clearly ] is right uppersemicontinuous and ](119905) lt 119905 for all 119905 gt 0 Therefore the set

119910 isin 119865119909 ] (119889 (119909 119910)) le 119863 (119909 119865119909) = 0 (42)

for all 119909 isin 119883 In fact if 119910119899 sub 119865119909 is a sequence such that

119889(119909 119910119899) darr 119863(119909 119865119909) the right upper semicontinuity of the

function ] ensures that

lim sup119899rarr+infin

] (119889 (119909 119910119899)) le ] (119863 (119909 119865119909)) (43)

and hence there exists 119910119899such that

] (119889 (119909 119910119899)) lt 119863 (119909 119865119909) (44)

The axiom of choice ensures that there is a mapping 119891 119883 rarr

119883 such that

119891119909 isin 119865119909 ] (119889 (119909 119891119909)) le 119863 (119909 119865119909) for 119909 isin 119883 (45)

Since 119863(119891119909 119865119891119909) le 119867(119865119909 119865119891119909) le 120583(119889(119909 119891119909)) we get

120577 (119889 (119909 119891119909)) = ] (119889 (119909 119891119909)) minus 120583 (119889 (119909 119891119909))

le 119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)

= 120601 (119909) minus 120601 (119891119909)

(46)

where the function 120601 119883 rarr [0 +infin) is defined by120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 Since the function 120601 is lowersemicontinuous then we get that 119891 has a fixed point whichis a fixed point for 119865

5 Conclusion

Under suitable hypotheses for multivalued mappings weestablished the existence of Caristi type selections Also weproved fixed point and quasi-fixed point theorems by usingweaker andmodified hypotheses on some classes of functionspresent in the literature Our results extend and complementmany theorems in the literature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

The first author is a Member of the Gruppo Nazionale perlrsquoAnalisi Matematica la Probabilita e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM) The second author is a Member of the GruppoNazionale per le Strutture Algebriche Geometriche e le loroApplicazioni (GNSAGA) of the Istituto Nazionale di AltaMatematica (INdAM)

References

[1] D Repovs and P V Semenov Continuous Selections of Multi-valued Mappings vol 455 of Mathematics and Its Applications1998

[2] E Michael ldquoContinuous selections Irdquo Annals of MathematicsSecond Series vol 63 pp 361ndash382 1956

[3] A Petrusel ldquoMultivalued operators and continuous selectionsrdquoPure Mathematics and Applications vol 9 no 1-2 pp 165ndash1701998

[4] A Petrusel andG Petrusel ldquoSelection theorems formultivaluedgeneralized contractionsrdquo Mathematica Moravica vol 9 pp43ndash52 2005

[5] A Sıntamarian ldquoSelections and common fixed points for somegeneralized multivalued contractionsrdquo Demonstratio Mathe-matica vol 39 no 3 pp 609ndash617 2006

[6] A Petrusel and A Sıntamarian ldquoSingle-valued and multi-valued Caristi type operatorsrdquo Publicationes MathematicaeDebrecen vol 60 no 1-2 pp 167ndash177 2002

[7] F E Browder ldquoThefixed point theory ofmulti-valuedmappingsin topological vector spacesrdquo Mathematische Annalen vol 177pp 283ndash301 1968

[8] J R Jachymski ldquoCaristirsquos fixed point theorem and selections ofset-valued contractionsrdquo Journal of Mathematical Analysis andApplications vol 227 no 1 pp 55ndash67 1998

[9] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[10] H Covitz and S B Nadler Jr ldquoMulti-valued contractionmappings in generalized metric spacesrdquo Israel Journal of Math-ematics vol 8 pp 5ndash11 1970

[11] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969

[12] H-K Xu ldquoSome recent results and problems for set-valuedmappingsrdquo in Advances in Mathematics Research vol 1 pp 31ndash49 2002

[13] M A Khamsi ldquoRemarks on Caristirsquos fixed point theoremrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 227ndash231 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

By condition (ii) the function 120601 119883 rarr [0 +infin) definedby 120601(119905) = (119902(1 minus 119887 minus 119886119902))119863(119905 119865119905) for all 119905 isin 119883 is lowersemicontinuous and hence 119891 is Caristirsquos mapping that is aselection of 119865

Example 12 Let119883 = [0 1] be endowed with the usual metric119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) is a completemetric space Also let 119865 119883 rarr 2119883 0 be defined by

119865119909 =

[0119909

2] if 119909 isin [0 1[

[1

2 1] if 119909 = 1

(11)

Consider 119902 = 43 119886 = 12 and 119887 = 16 such that 119886119902 + 119887 =

56 lt 1 Then for 119909 = 1 and 119910 = 1199092 we have 119889(119909 119910) =

1199092 le (43) sdot (1199092) = (43)119863(119909 119865119909) that is 119910 isin 119878119902(119909)

Moreover

119863(119910 119865119910) = 119863(119909

2 [0

119909

4]) =

119909

4le

1

2sdot119909

2+

1

6sdot119909

2

= 119886119889 (119909 119910) + 119887119863 (119909 119865119909)

(12)

Also for 119909 = 1 and 119910 = 1 we have 119889(119909 119910) = 0 = 119902 sdot 0 =

119902119863(119909 119865119909) that is 1 isin 119878119902(119909) Moreover

119863(119910 119865119910) = 119863(1 [1

2 1]) = 0 le 119886119889 (119909 119910) + 119887119863 (119909 119865119909)

(13)

Finally the function 119901 119883 rarr [0 +infin) defined by

119901119909 = 119863 (119909 119865119909) =

119909

2if 119909 isin [0 1[

0 if 119909 = 1(14)

is lower semicontinuous in [0 1] Thus all the hypotheses ofTheorem 11 are satisfied and so 119865 has a selection 119891 that isCaristirsquos mapping In fact 119891 119883 rarr 119883 defined by 119891119909 = 1199092

for all 119909 isin 119883 is such that 119891119909 isin 119865119909 and also 119889(119909 119891119909) = 1199092 =

120601(119909) minus 120601(119891119909) where 120601 119883 rarr [0 +infin) is given by 120601(119909) = 119909

for all 119909 isin 119883Notice that

119867(1198650 1198651) = 119867(0 [1

2 1]) = max

1

2 1 = 119889 (0 1)

(15)

and hence 119865 is not Nadlerrsquos multivalued contraction

Analogous results toTheorem 11 can be established underdifferent hypotheses For instance in the next theoremthe multivalued mapping 119865 satisfies another contractivecondition

Theorem 13 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping and let 119902 gt 1 be a real

number Consider 119878119902(119909) = 119910 isin 119865119909 119889(119909 119910) le 119902119863119889(119909 119865119909)

and suppose that 119865 satisfies the following conditions(i) there exist nonnegative real numbers 120572 120573 120574 with (120572 +

120574)119902 + 120573 + 120574 lt 1 such that for each 119909 isin 119883 there is119910 isin 119878119902(119909) having the property

119863(119910 119865119910) le 120572119889 (119909 119910) + 120573119863 (119909 119865119909) + 120574119863 (119909 119865119910) (16)

(ii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous

Then 119865 has a selection 119891 that is Caristirsquos mapping

Proof By the axiom of choice and condition (i) there is amapping 119891 119883 rarr 119883 with 119891119909 isin 119878

119902(119909) for all 119909 isin 119883 such

that

119863(119891119909 119865119891119909) le 120572119889 (119909 119891119909) + 120573119863 (119909 119865119909) + 120574119863 (119909 119865119891119909)

(17)

and so by using the triangular inequality for 119863(119909 119865119891119909)

(1 minus 120574)119863 (119891119909 119865119891119909) minus 120573119863 (119909 119865119909) le (120572 + 120574) 119889 (119909 119891119909)

(18)

Then

(1 minus 120573 minus 120574 minus (120572 + 120574) 119902) 119889 (119909 119891119909)

le (1 minus 120573 minus 120574) 119902119863 (119909 119865119909) minus (120572 + 120574) 119902119889 (119909 119891119909)

le (1 minus 120573 minus 120574) 119902119863 (119909 119865119909) minus (1 minus 120574) 119902119863 (119891119909 119865119891119909)

+ 120573119902119863 (119909 119865119909)

= (1 minus 120574) 119902119863 (119909 119865119909) minus (1 minus 120574) 119902119863 (119891119909 119865119891119909)

(19)

By condition (ii) the function 120601 119883 rarr [0 +infin) definedby 120601(119905) = ((1minus120574)119902(1minus120573minus120574minus(120572+120574)119902))119863(119905 119865119905) for all 119905 isin 119883is lower semicontinuous and hence119891 is Caristirsquosmapping thatis a selection of 119865

We would like to remark that other results can be statedby involving upper semicontinuousmultivaluedmappings inview of the following situation

Definition 14 Let (119883 119889) be a metric space Then a multi-valued mapping 119865 119883 rarr 2119883 0 is said to be ℎ-uppersemicontinuous at 119909

0isin 119883 if the function

ℎ (119865119909 1198651199090) = sup 119863 (119910 119865119909

0) 119910 isin 119865119909 (20)

is continuous at 1199090 Clearly 119865 is said to be ℎ-upper semicon-

tinuous whenever ℎ(119865119909 1198651199090) is continuous at every 119909

0isin 119883

Now we present a class of multivalued mappings suchthat the function 120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 is lowersemicontinuous

Proposition 15 Let (119883 119889) be a metric space If 119865 119883 rarr

2119883 0 is ℎ-upper semicontinuous then the function 120601(119909) =

119863(119909 119865119909) is lower semicontinuous

Proof Given 119909 isin 119883 for all 119910 isin 119883 we get

120601 (119909) = 119863 (119909 119865119909) le 119889 (119909 119910) + 119863 (119910 119865119910) + ℎ (119865119910 119865119909)

le 119889 (119909 119910) + 120601 (119910) + ℎ (119865119910 119865119909) (21)

From above inequalities we deduce that 120601(119909) le

lim inf119910rarr119909

120601(119910) and so 120601 is a lower semicontinuousfunction

4 Journal of Function Spaces

For instance fromTheorem 13 and Proposition 15 we getthe following corollary

Corollary 16 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883

0 be an ℎ-upper semicontinuous multivaluedmapping and let 119902 gt 1 be a real number Consider 119878

119902(119909) =

119910 isin 119865119909 119889(119909 119910) le 119902119863(119909 119865119909) and suppose that 119865 satisfiesthe condition (119894) of Theorem 13 Then 119865 has a selection 119891 thatis Caristirsquos mapping

3 Extension to Quasi-Fixed Point Theorems

Let (119883 119889) be a metric space We recall that a multivaluedmapping 119865 119883 rarr 2119883 0 has a quasi-fixed point ifthere exists a point 119911 isin 119883 such that 119863(119911 119865119911) = 0 Thenwe extend our theory by considering functions instead ofconstant values Therefore let (119883 119889) be a metric space andlet 119886 119887 119883 rarr [0 +infin) and 119902 119883 rarr (0 +infin) be functionssuch that

119886 (119909) 119902minus1

(119909) + 119887 (119909) lt 1 forall119909 isin 119883 (22)

Remark 17 Notice that in (22) we do not need that 119902(119909) lt 1Wewill return on this fact to derive a particular situation fromthe following theorem

Theorem 18 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (23)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(24)

Then 119865 has a quasi-fixed point that is there exists 119911 isin 119883 suchthat 119863(119911 119865119911) = 0

Proof Assume that 119863(119909 119865119909) gt 0 for all 119909 isin 119883 By the axiomof choice and condition (ii) there is a mapping 119891 119883 rarr 119883

with 119891119909 = 119909 such that

119902 (119909) 119889 (119909 119891119909) le 119863 (119909 119865119909)

119863 (119891119909 119865119891119909) le 119886 (119909) 119889 (119909 119891119909) + 119887 (119909)119863 (119909 119865119909) (25)

Then for each 119909 isin 119883

119863 (119891119909 119865119891119909) minus 119887 (119909)119863 (119909 119865119909) le 119886 (119909) 119889 (119909 119891119909) (26)

This implies

(1 minus 119887 (119909) minus 119886 (119909) 119902minus1

(119909)) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119886 (119909) 119902minus1

(119909) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

+ 119887 (119909) 119902minus1

(119909)119863 (119909 119865119909)

= 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

(27)

Consequently we have

119889 (119909 119891119909) le1

119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909)

times [119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)]

(28)

Now let 119884 = 119909 isin 119883 119863(119909 119865119909) le inf119911isin119883

119863(119911 119865119911) + 120578 Sinceby (iii)119884 is a closed subset of119883 we deduce that119884 is completeDenote by 120574 = inf119902(119909) minus 119887(119909)119902(119909) minus 119886(119909) 119909 isin 119884 gt 0 Forall 119909 isin 119884 we get

119889 (119909 119891119909) le1

120574[119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)] = 120601 (119909) minus 120601 (119891119909)

(29)

where the function 120601 119883 rarr [0 +infin) is defined by 120601(119905) =

120574minus1119863(119905 119865119905) for all 119905 isin 119883 Clearly by condition (iii) thefunction 120601 is lower semicontinuous From (29) we get that119891119909 isin 119884 whenever 119909 isin 119884 and hence 119891 119884 rarr 119884 isCaristirsquos mapping This implies that 119891 has a fixed point in 119884a contradiction since 119891119909 = 119909 for all 119909 isin 119883 Hence there is119911 isin 119883 such that 119863(119911 119865119911) = 0

As a consequence of Theorem 18 in the case that 119865119909 isalso closed we obtain the following corollary

Corollary 19 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping such that119865119909 is closedSuppose that 119865 satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (30)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous

Journal of Function Spaces 5

(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(31)

Then 119865 has a fixed point

In view of Remark 17 by assuming 119902(119909) lt 1 for all 119909 isin 119883on the same lines of the proof of Theorem 18 one can provethe following result

Theorem 20 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions(i) there exist three functions 119886 119887 119883 rarr [0 1) and 119902

119883 rarr (0 1) such that (22) holds(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin 119865119909

such that119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (32)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(33)

Then 119865 has a selection 119891 that is Caristirsquos mapping on a closedsubset of 119883

Remark 21 If in Theorems 18 and 20 and Corollary 19we assume that the multivalued mapping 119865 is ℎ-uppersemicontinuous then (iii) holds true In this case we canreformulate the statements of these results requiring that 119865

satisfies only conditions (i) (ii) and (iv)

4 Generalization of Caristirsquos Theorem

We denote by Φ the set of all functions 120577 [0 +infin) rarr

[0 +infin) such that there exist 120576 gt 0 and 119888 isin (0 1) satisfying120575120576

= sup 120577minus1

([0 120576]) lt +infin 120577(119905) ge 119888119905 for all 119905 isin [0 120575120576] and

120577(119905) gt 120576 for all 119905 gt 120575120576

Remark 22 Given a nondecreasing function 120577 [0 +infin) rarr

[0 +infin) continuous at 119905 = 0 with 120577(0) = 0 consider the rightlower Dini derivative of 120577 at 119905 isin [0 +infin) that is

[119863+120577] (119905) = lim inf

119904rarr 119905+

120577 (119904) minus 120577 (119905)

119904 minus 119905 (34)

Then 120577 isin Φ provided that [119863+120577](0) gt 0 see [8] Also each

function 120577 [0 +infin) rarr [0 +infin) that is nondecreasingsubadditive and continuous at 119905 = 0 with 120577(0) = 0 belongs toΦ

Inspired by Khamsi [13] and Jachymski [8] we givetwo fixed point theorems In particular our first theoremfurnishes an alternative proof to Theorem 3 of [13] and therelated Kirkrsquos problem without using order relations (seeSection 3 in [13] for more details)

Theorem 23 Let (119883 119889) be a complete metric space Let 119891

119883 rarr 119883 be a mapping Suppose that there exist a lowersemicontinuous function 120601 119883 rarr [0 +infin) and a function120577 isin Φ such that

120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) forall119909 isin 119883 (35)

Then 119891 has a fixed point in 119883

Proof Let 120576 gt 0 119888 isin (0 1) and let 120575120576be as stated above Let

119884 = 119909 isin 119883 120601 (119909) le inf119911isin119883

120601 (119911) + 120576 (36)

The set119884 is closed since 120601 is lower semicontinuous and hencecomplete Now from (35) we get that 119891119909 isin 119884 whenever 119909 isin

119884 Also120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) le 120601 (119909) minus inf

119911isin119883

120601 (119911) le 120576

(37)

for all 119909 isin 119884 we obtain that 119889(119909 119891119909) isin [0 120575120576] whenever 119909 isin

119884 Hence

119888119889 (119909 119891119909) le 120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) (38)

that is

119889 (119909 119891119909) le1

119888120601 (119909) minus

1

119888120601 (119891119909) forall119909 isin 119884 (39)

Since the function (1119888)120601 is lower semicontinuous byTheorem 3 the mapping 119891 119884 rarr 119884 has a fixed point in119884 and so in 119883

Example 24 Let119883 = minus3 minus1cup[0 +infin) be endowed with theusual metric 119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) isa complete metric space Also let 119891 119883 rarr 119883 be defined by

119891119909 = 0 if 119909 notin [1 3]

1 if 119909 isin [1 3] (40)

It follows that

119889 (119909 119891119909) = |119909| if 119909 notin [1 3]

119909 minus 1 if 119909 isin [1 3] (41)

Notice that 120601 119883 rarr [0 +infin) defined by 120601(119909) = |119909| forall 119909 isin 119883 is a lower semicontinuous function such that120577(119889(119909 119891119909)) le 120601(119909) minus 120601(119891119909) where 120577 [0 +infin) rarr [0 +infin) isgiven by 120577(119905) = 119888119905 for all 119905 ge 0 where 119888 isin (0 1) Thus we canapplyTheorem 23 to conclude that119891 has a fixed point clearly0 and 1 are fixed points of 119891

The inspiration of our next theorem is Theorem 10 Inparticular our result does not use a monotonic conditionFor a comprehensive discussion we refer the reader to thefundamental paper of Jachymski [8]

6 Journal of Function Spaces

Theorem 25 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 119862119897(119883) be a multivalued mapping Suppose that 119865 is a120583-contraction with 120583 right upper semicontinuous such that thefunction 120577(119905) = (119905 minus 120583(119905))2 for all 119905 ge 0 belongs to Φ Then 119865

has a fixed point

Proof Let ] [0 +infin) rarr [0 +infin) be the function definedby ](119905) = (119905 + 120583(119905))2 for all 119905 ge 0 Clearly ] is right uppersemicontinuous and ](119905) lt 119905 for all 119905 gt 0 Therefore the set

119910 isin 119865119909 ] (119889 (119909 119910)) le 119863 (119909 119865119909) = 0 (42)

for all 119909 isin 119883 In fact if 119910119899 sub 119865119909 is a sequence such that

119889(119909 119910119899) darr 119863(119909 119865119909) the right upper semicontinuity of the

function ] ensures that

lim sup119899rarr+infin

] (119889 (119909 119910119899)) le ] (119863 (119909 119865119909)) (43)

and hence there exists 119910119899such that

] (119889 (119909 119910119899)) lt 119863 (119909 119865119909) (44)

The axiom of choice ensures that there is a mapping 119891 119883 rarr

119883 such that

119891119909 isin 119865119909 ] (119889 (119909 119891119909)) le 119863 (119909 119865119909) for 119909 isin 119883 (45)

Since 119863(119891119909 119865119891119909) le 119867(119865119909 119865119891119909) le 120583(119889(119909 119891119909)) we get

120577 (119889 (119909 119891119909)) = ] (119889 (119909 119891119909)) minus 120583 (119889 (119909 119891119909))

le 119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)

= 120601 (119909) minus 120601 (119891119909)

(46)

where the function 120601 119883 rarr [0 +infin) is defined by120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 Since the function 120601 is lowersemicontinuous then we get that 119891 has a fixed point whichis a fixed point for 119865

5 Conclusion

Under suitable hypotheses for multivalued mappings weestablished the existence of Caristi type selections Also weproved fixed point and quasi-fixed point theorems by usingweaker andmodified hypotheses on some classes of functionspresent in the literature Our results extend and complementmany theorems in the literature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

The first author is a Member of the Gruppo Nazionale perlrsquoAnalisi Matematica la Probabilita e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM) The second author is a Member of the GruppoNazionale per le Strutture Algebriche Geometriche e le loroApplicazioni (GNSAGA) of the Istituto Nazionale di AltaMatematica (INdAM)

References

[1] D Repovs and P V Semenov Continuous Selections of Multi-valued Mappings vol 455 of Mathematics and Its Applications1998

[2] E Michael ldquoContinuous selections Irdquo Annals of MathematicsSecond Series vol 63 pp 361ndash382 1956

[3] A Petrusel ldquoMultivalued operators and continuous selectionsrdquoPure Mathematics and Applications vol 9 no 1-2 pp 165ndash1701998

[4] A Petrusel andG Petrusel ldquoSelection theorems formultivaluedgeneralized contractionsrdquo Mathematica Moravica vol 9 pp43ndash52 2005

[5] A Sıntamarian ldquoSelections and common fixed points for somegeneralized multivalued contractionsrdquo Demonstratio Mathe-matica vol 39 no 3 pp 609ndash617 2006

[6] A Petrusel and A Sıntamarian ldquoSingle-valued and multi-valued Caristi type operatorsrdquo Publicationes MathematicaeDebrecen vol 60 no 1-2 pp 167ndash177 2002

[7] F E Browder ldquoThefixed point theory ofmulti-valuedmappingsin topological vector spacesrdquo Mathematische Annalen vol 177pp 283ndash301 1968

[8] J R Jachymski ldquoCaristirsquos fixed point theorem and selections ofset-valued contractionsrdquo Journal of Mathematical Analysis andApplications vol 227 no 1 pp 55ndash67 1998

[9] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[10] H Covitz and S B Nadler Jr ldquoMulti-valued contractionmappings in generalized metric spacesrdquo Israel Journal of Math-ematics vol 8 pp 5ndash11 1970

[11] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969

[12] H-K Xu ldquoSome recent results and problems for set-valuedmappingsrdquo in Advances in Mathematics Research vol 1 pp 31ndash49 2002

[13] M A Khamsi ldquoRemarks on Caristirsquos fixed point theoremrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 227ndash231 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

For instance fromTheorem 13 and Proposition 15 we getthe following corollary

Corollary 16 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883

0 be an ℎ-upper semicontinuous multivaluedmapping and let 119902 gt 1 be a real number Consider 119878

119902(119909) =

119910 isin 119865119909 119889(119909 119910) le 119902119863(119909 119865119909) and suppose that 119865 satisfiesthe condition (119894) of Theorem 13 Then 119865 has a selection 119891 thatis Caristirsquos mapping

3 Extension to Quasi-Fixed Point Theorems

Let (119883 119889) be a metric space We recall that a multivaluedmapping 119865 119883 rarr 2119883 0 has a quasi-fixed point ifthere exists a point 119911 isin 119883 such that 119863(119911 119865119911) = 0 Thenwe extend our theory by considering functions instead ofconstant values Therefore let (119883 119889) be a metric space andlet 119886 119887 119883 rarr [0 +infin) and 119902 119883 rarr (0 +infin) be functionssuch that

119886 (119909) 119902minus1

(119909) + 119887 (119909) lt 1 forall119909 isin 119883 (22)

Remark 17 Notice that in (22) we do not need that 119902(119909) lt 1Wewill return on this fact to derive a particular situation fromthe following theorem

Theorem 18 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (23)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(24)

Then 119865 has a quasi-fixed point that is there exists 119911 isin 119883 suchthat 119863(119911 119865119911) = 0

Proof Assume that 119863(119909 119865119909) gt 0 for all 119909 isin 119883 By the axiomof choice and condition (ii) there is a mapping 119891 119883 rarr 119883

with 119891119909 = 119909 such that

119902 (119909) 119889 (119909 119891119909) le 119863 (119909 119865119909)

119863 (119891119909 119865119891119909) le 119886 (119909) 119889 (119909 119891119909) + 119887 (119909)119863 (119909 119865119909) (25)

Then for each 119909 isin 119883

119863 (119891119909 119865119891119909) minus 119887 (119909)119863 (119909 119865119909) le 119886 (119909) 119889 (119909 119891119909) (26)

This implies

(1 minus 119887 (119909) minus 119886 (119909) 119902minus1

(119909)) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119886 (119909) 119902minus1

(119909) 119889 (119909 119891119909)

le (1 minus 119887 (119909)) 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

+ 119887 (119909) 119902minus1

(119909)119863 (119909 119865119909)

= 119902minus1

(119909)119863 (119909 119865119909) minus 119902minus1

(119909)119863 (119891119909 119865119891119909)

(27)

Consequently we have

119889 (119909 119891119909) le1

119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909)

times [119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)]

(28)

Now let 119884 = 119909 isin 119883 119863(119909 119865119909) le inf119911isin119883

119863(119911 119865119911) + 120578 Sinceby (iii)119884 is a closed subset of119883 we deduce that119884 is completeDenote by 120574 = inf119902(119909) minus 119887(119909)119902(119909) minus 119886(119909) 119909 isin 119884 gt 0 Forall 119909 isin 119884 we get

119889 (119909 119891119909) le1

120574[119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)] = 120601 (119909) minus 120601 (119891119909)

(29)

where the function 120601 119883 rarr [0 +infin) is defined by 120601(119905) =

120574minus1119863(119905 119865119905) for all 119905 isin 119883 Clearly by condition (iii) thefunction 120601 is lower semicontinuous From (29) we get that119891119909 isin 119884 whenever 119909 isin 119884 and hence 119891 119884 rarr 119884 isCaristirsquos mapping This implies that 119891 has a fixed point in 119884a contradiction since 119891119909 = 119909 for all 119909 isin 119883 Hence there is119911 isin 119883 such that 119863(119911 119865119911) = 0

As a consequence of Theorem 18 in the case that 119865119909 isalso closed we obtain the following corollary

Corollary 19 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 21198830 be amultivaluedmapping such that119865119909 is closedSuppose that 119865 satisfies the following conditions

(i) there exist three functions 119886 119887 119883 rarr [0 +infin) and119902 119883 rarr (0 +infin) such that (22) holds

(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin

119883 119909 such that

119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (30)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous

Journal of Function Spaces 5

(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(31)

Then 119865 has a fixed point

In view of Remark 17 by assuming 119902(119909) lt 1 for all 119909 isin 119883on the same lines of the proof of Theorem 18 one can provethe following result

Theorem 20 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions(i) there exist three functions 119886 119887 119883 rarr [0 1) and 119902

119883 rarr (0 1) such that (22) holds(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin 119865119909

such that119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (32)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(33)

Then 119865 has a selection 119891 that is Caristirsquos mapping on a closedsubset of 119883

Remark 21 If in Theorems 18 and 20 and Corollary 19we assume that the multivalued mapping 119865 is ℎ-uppersemicontinuous then (iii) holds true In this case we canreformulate the statements of these results requiring that 119865

satisfies only conditions (i) (ii) and (iv)

4 Generalization of Caristirsquos Theorem

We denote by Φ the set of all functions 120577 [0 +infin) rarr

[0 +infin) such that there exist 120576 gt 0 and 119888 isin (0 1) satisfying120575120576

= sup 120577minus1

([0 120576]) lt +infin 120577(119905) ge 119888119905 for all 119905 isin [0 120575120576] and

120577(119905) gt 120576 for all 119905 gt 120575120576

Remark 22 Given a nondecreasing function 120577 [0 +infin) rarr

[0 +infin) continuous at 119905 = 0 with 120577(0) = 0 consider the rightlower Dini derivative of 120577 at 119905 isin [0 +infin) that is

[119863+120577] (119905) = lim inf

119904rarr 119905+

120577 (119904) minus 120577 (119905)

119904 minus 119905 (34)

Then 120577 isin Φ provided that [119863+120577](0) gt 0 see [8] Also each

function 120577 [0 +infin) rarr [0 +infin) that is nondecreasingsubadditive and continuous at 119905 = 0 with 120577(0) = 0 belongs toΦ

Inspired by Khamsi [13] and Jachymski [8] we givetwo fixed point theorems In particular our first theoremfurnishes an alternative proof to Theorem 3 of [13] and therelated Kirkrsquos problem without using order relations (seeSection 3 in [13] for more details)

Theorem 23 Let (119883 119889) be a complete metric space Let 119891

119883 rarr 119883 be a mapping Suppose that there exist a lowersemicontinuous function 120601 119883 rarr [0 +infin) and a function120577 isin Φ such that

120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) forall119909 isin 119883 (35)

Then 119891 has a fixed point in 119883

Proof Let 120576 gt 0 119888 isin (0 1) and let 120575120576be as stated above Let

119884 = 119909 isin 119883 120601 (119909) le inf119911isin119883

120601 (119911) + 120576 (36)

The set119884 is closed since 120601 is lower semicontinuous and hencecomplete Now from (35) we get that 119891119909 isin 119884 whenever 119909 isin

119884 Also120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) le 120601 (119909) minus inf

119911isin119883

120601 (119911) le 120576

(37)

for all 119909 isin 119884 we obtain that 119889(119909 119891119909) isin [0 120575120576] whenever 119909 isin

119884 Hence

119888119889 (119909 119891119909) le 120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) (38)

that is

119889 (119909 119891119909) le1

119888120601 (119909) minus

1

119888120601 (119891119909) forall119909 isin 119884 (39)

Since the function (1119888)120601 is lower semicontinuous byTheorem 3 the mapping 119891 119884 rarr 119884 has a fixed point in119884 and so in 119883

Example 24 Let119883 = minus3 minus1cup[0 +infin) be endowed with theusual metric 119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) isa complete metric space Also let 119891 119883 rarr 119883 be defined by

119891119909 = 0 if 119909 notin [1 3]

1 if 119909 isin [1 3] (40)

It follows that

119889 (119909 119891119909) = |119909| if 119909 notin [1 3]

119909 minus 1 if 119909 isin [1 3] (41)

Notice that 120601 119883 rarr [0 +infin) defined by 120601(119909) = |119909| forall 119909 isin 119883 is a lower semicontinuous function such that120577(119889(119909 119891119909)) le 120601(119909) minus 120601(119891119909) where 120577 [0 +infin) rarr [0 +infin) isgiven by 120577(119905) = 119888119905 for all 119905 ge 0 where 119888 isin (0 1) Thus we canapplyTheorem 23 to conclude that119891 has a fixed point clearly0 and 1 are fixed points of 119891

The inspiration of our next theorem is Theorem 10 Inparticular our result does not use a monotonic conditionFor a comprehensive discussion we refer the reader to thefundamental paper of Jachymski [8]

6 Journal of Function Spaces

Theorem 25 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 119862119897(119883) be a multivalued mapping Suppose that 119865 is a120583-contraction with 120583 right upper semicontinuous such that thefunction 120577(119905) = (119905 minus 120583(119905))2 for all 119905 ge 0 belongs to Φ Then 119865

has a fixed point

Proof Let ] [0 +infin) rarr [0 +infin) be the function definedby ](119905) = (119905 + 120583(119905))2 for all 119905 ge 0 Clearly ] is right uppersemicontinuous and ](119905) lt 119905 for all 119905 gt 0 Therefore the set

119910 isin 119865119909 ] (119889 (119909 119910)) le 119863 (119909 119865119909) = 0 (42)

for all 119909 isin 119883 In fact if 119910119899 sub 119865119909 is a sequence such that

119889(119909 119910119899) darr 119863(119909 119865119909) the right upper semicontinuity of the

function ] ensures that

lim sup119899rarr+infin

] (119889 (119909 119910119899)) le ] (119863 (119909 119865119909)) (43)

and hence there exists 119910119899such that

] (119889 (119909 119910119899)) lt 119863 (119909 119865119909) (44)

The axiom of choice ensures that there is a mapping 119891 119883 rarr

119883 such that

119891119909 isin 119865119909 ] (119889 (119909 119891119909)) le 119863 (119909 119865119909) for 119909 isin 119883 (45)

Since 119863(119891119909 119865119891119909) le 119867(119865119909 119865119891119909) le 120583(119889(119909 119891119909)) we get

120577 (119889 (119909 119891119909)) = ] (119889 (119909 119891119909)) minus 120583 (119889 (119909 119891119909))

le 119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)

= 120601 (119909) minus 120601 (119891119909)

(46)

where the function 120601 119883 rarr [0 +infin) is defined by120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 Since the function 120601 is lowersemicontinuous then we get that 119891 has a fixed point whichis a fixed point for 119865

5 Conclusion

Under suitable hypotheses for multivalued mappings weestablished the existence of Caristi type selections Also weproved fixed point and quasi-fixed point theorems by usingweaker andmodified hypotheses on some classes of functionspresent in the literature Our results extend and complementmany theorems in the literature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

The first author is a Member of the Gruppo Nazionale perlrsquoAnalisi Matematica la Probabilita e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM) The second author is a Member of the GruppoNazionale per le Strutture Algebriche Geometriche e le loroApplicazioni (GNSAGA) of the Istituto Nazionale di AltaMatematica (INdAM)

References

[1] D Repovs and P V Semenov Continuous Selections of Multi-valued Mappings vol 455 of Mathematics and Its Applications1998

[2] E Michael ldquoContinuous selections Irdquo Annals of MathematicsSecond Series vol 63 pp 361ndash382 1956

[3] A Petrusel ldquoMultivalued operators and continuous selectionsrdquoPure Mathematics and Applications vol 9 no 1-2 pp 165ndash1701998

[4] A Petrusel andG Petrusel ldquoSelection theorems formultivaluedgeneralized contractionsrdquo Mathematica Moravica vol 9 pp43ndash52 2005

[5] A Sıntamarian ldquoSelections and common fixed points for somegeneralized multivalued contractionsrdquo Demonstratio Mathe-matica vol 39 no 3 pp 609ndash617 2006

[6] A Petrusel and A Sıntamarian ldquoSingle-valued and multi-valued Caristi type operatorsrdquo Publicationes MathematicaeDebrecen vol 60 no 1-2 pp 167ndash177 2002

[7] F E Browder ldquoThefixed point theory ofmulti-valuedmappingsin topological vector spacesrdquo Mathematische Annalen vol 177pp 283ndash301 1968

[8] J R Jachymski ldquoCaristirsquos fixed point theorem and selections ofset-valued contractionsrdquo Journal of Mathematical Analysis andApplications vol 227 no 1 pp 55ndash67 1998

[9] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[10] H Covitz and S B Nadler Jr ldquoMulti-valued contractionmappings in generalized metric spacesrdquo Israel Journal of Math-ematics vol 8 pp 5ndash11 1970

[11] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969

[12] H-K Xu ldquoSome recent results and problems for set-valuedmappingsrdquo in Advances in Mathematics Research vol 1 pp 31ndash49 2002

[13] M A Khamsi ldquoRemarks on Caristirsquos fixed point theoremrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 227ndash231 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(31)

Then 119865 has a fixed point

In view of Remark 17 by assuming 119902(119909) lt 1 for all 119909 isin 119883on the same lines of the proof of Theorem 18 one can provethe following result

Theorem 20 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 2119883 0 be a multivalued mapping Suppose that 119865

satisfies the following conditions(i) there exist three functions 119886 119887 119883 rarr [0 1) and 119902

119883 rarr (0 1) such that (22) holds(ii) for each 119909 isin 119883 with 119863(119909 119865119909) gt 0 there exists 119910 isin 119865119909

such that119902 (119909) 119889 (119909 119910) le 119863 (119909 119865119909)

119863 (119910 119865119910) le 119886 (119909) 119889 (119909 119910) + 119887 (119909)119863 (119909 119865119909) (32)

(iii) the function 119901 119883 rarr [0 +infin) defined by 119901119909 =

119863(119909 119865119909) for all 119909 isin 119883 is lower semicontinuous(iv) there exists 120578 gt 0 such that

inf 119902 (119909) minus 119887 (119909) 119902 (119909) minus 119886 (119909) 119909 isin 119883

119863 (119909 119865119909) le inf119911isin119883

119863 (119911 119865119911) + 120578 gt 0

(33)

Then 119865 has a selection 119891 that is Caristirsquos mapping on a closedsubset of 119883

Remark 21 If in Theorems 18 and 20 and Corollary 19we assume that the multivalued mapping 119865 is ℎ-uppersemicontinuous then (iii) holds true In this case we canreformulate the statements of these results requiring that 119865

satisfies only conditions (i) (ii) and (iv)

4 Generalization of Caristirsquos Theorem

We denote by Φ the set of all functions 120577 [0 +infin) rarr

[0 +infin) such that there exist 120576 gt 0 and 119888 isin (0 1) satisfying120575120576

= sup 120577minus1

([0 120576]) lt +infin 120577(119905) ge 119888119905 for all 119905 isin [0 120575120576] and

120577(119905) gt 120576 for all 119905 gt 120575120576

Remark 22 Given a nondecreasing function 120577 [0 +infin) rarr

[0 +infin) continuous at 119905 = 0 with 120577(0) = 0 consider the rightlower Dini derivative of 120577 at 119905 isin [0 +infin) that is

[119863+120577] (119905) = lim inf

119904rarr 119905+

120577 (119904) minus 120577 (119905)

119904 minus 119905 (34)

Then 120577 isin Φ provided that [119863+120577](0) gt 0 see [8] Also each

function 120577 [0 +infin) rarr [0 +infin) that is nondecreasingsubadditive and continuous at 119905 = 0 with 120577(0) = 0 belongs toΦ

Inspired by Khamsi [13] and Jachymski [8] we givetwo fixed point theorems In particular our first theoremfurnishes an alternative proof to Theorem 3 of [13] and therelated Kirkrsquos problem without using order relations (seeSection 3 in [13] for more details)

Theorem 23 Let (119883 119889) be a complete metric space Let 119891

119883 rarr 119883 be a mapping Suppose that there exist a lowersemicontinuous function 120601 119883 rarr [0 +infin) and a function120577 isin Φ such that

120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) forall119909 isin 119883 (35)

Then 119891 has a fixed point in 119883

Proof Let 120576 gt 0 119888 isin (0 1) and let 120575120576be as stated above Let

119884 = 119909 isin 119883 120601 (119909) le inf119911isin119883

120601 (119911) + 120576 (36)

The set119884 is closed since 120601 is lower semicontinuous and hencecomplete Now from (35) we get that 119891119909 isin 119884 whenever 119909 isin

119884 Also120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) le 120601 (119909) minus inf

119911isin119883

120601 (119911) le 120576

(37)

for all 119909 isin 119884 we obtain that 119889(119909 119891119909) isin [0 120575120576] whenever 119909 isin

119884 Hence

119888119889 (119909 119891119909) le 120577 (119889 (119909 119891119909)) le 120601 (119909) minus 120601 (119891119909) (38)

that is

119889 (119909 119891119909) le1

119888120601 (119909) minus

1

119888120601 (119891119909) forall119909 isin 119884 (39)

Since the function (1119888)120601 is lower semicontinuous byTheorem 3 the mapping 119891 119884 rarr 119884 has a fixed point in119884 and so in 119883

Example 24 Let119883 = minus3 minus1cup[0 +infin) be endowed with theusual metric 119889(119909 119910) = |119909 minus 119910| for all 119909 119910 isin 119883 so that (119883 119889) isa complete metric space Also let 119891 119883 rarr 119883 be defined by

119891119909 = 0 if 119909 notin [1 3]

1 if 119909 isin [1 3] (40)

It follows that

119889 (119909 119891119909) = |119909| if 119909 notin [1 3]

119909 minus 1 if 119909 isin [1 3] (41)

Notice that 120601 119883 rarr [0 +infin) defined by 120601(119909) = |119909| forall 119909 isin 119883 is a lower semicontinuous function such that120577(119889(119909 119891119909)) le 120601(119909) minus 120601(119891119909) where 120577 [0 +infin) rarr [0 +infin) isgiven by 120577(119905) = 119888119905 for all 119905 ge 0 where 119888 isin (0 1) Thus we canapplyTheorem 23 to conclude that119891 has a fixed point clearly0 and 1 are fixed points of 119891

The inspiration of our next theorem is Theorem 10 Inparticular our result does not use a monotonic conditionFor a comprehensive discussion we refer the reader to thefundamental paper of Jachymski [8]

6 Journal of Function Spaces

Theorem 25 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 119862119897(119883) be a multivalued mapping Suppose that 119865 is a120583-contraction with 120583 right upper semicontinuous such that thefunction 120577(119905) = (119905 minus 120583(119905))2 for all 119905 ge 0 belongs to Φ Then 119865

has a fixed point

Proof Let ] [0 +infin) rarr [0 +infin) be the function definedby ](119905) = (119905 + 120583(119905))2 for all 119905 ge 0 Clearly ] is right uppersemicontinuous and ](119905) lt 119905 for all 119905 gt 0 Therefore the set

119910 isin 119865119909 ] (119889 (119909 119910)) le 119863 (119909 119865119909) = 0 (42)

for all 119909 isin 119883 In fact if 119910119899 sub 119865119909 is a sequence such that

119889(119909 119910119899) darr 119863(119909 119865119909) the right upper semicontinuity of the

function ] ensures that

lim sup119899rarr+infin

] (119889 (119909 119910119899)) le ] (119863 (119909 119865119909)) (43)

and hence there exists 119910119899such that

] (119889 (119909 119910119899)) lt 119863 (119909 119865119909) (44)

The axiom of choice ensures that there is a mapping 119891 119883 rarr

119883 such that

119891119909 isin 119865119909 ] (119889 (119909 119891119909)) le 119863 (119909 119865119909) for 119909 isin 119883 (45)

Since 119863(119891119909 119865119891119909) le 119867(119865119909 119865119891119909) le 120583(119889(119909 119891119909)) we get

120577 (119889 (119909 119891119909)) = ] (119889 (119909 119891119909)) minus 120583 (119889 (119909 119891119909))

le 119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)

= 120601 (119909) minus 120601 (119891119909)

(46)

where the function 120601 119883 rarr [0 +infin) is defined by120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 Since the function 120601 is lowersemicontinuous then we get that 119891 has a fixed point whichis a fixed point for 119865

5 Conclusion

Under suitable hypotheses for multivalued mappings weestablished the existence of Caristi type selections Also weproved fixed point and quasi-fixed point theorems by usingweaker andmodified hypotheses on some classes of functionspresent in the literature Our results extend and complementmany theorems in the literature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

The first author is a Member of the Gruppo Nazionale perlrsquoAnalisi Matematica la Probabilita e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM) The second author is a Member of the GruppoNazionale per le Strutture Algebriche Geometriche e le loroApplicazioni (GNSAGA) of the Istituto Nazionale di AltaMatematica (INdAM)

References

[1] D Repovs and P V Semenov Continuous Selections of Multi-valued Mappings vol 455 of Mathematics and Its Applications1998

[2] E Michael ldquoContinuous selections Irdquo Annals of MathematicsSecond Series vol 63 pp 361ndash382 1956

[3] A Petrusel ldquoMultivalued operators and continuous selectionsrdquoPure Mathematics and Applications vol 9 no 1-2 pp 165ndash1701998

[4] A Petrusel andG Petrusel ldquoSelection theorems formultivaluedgeneralized contractionsrdquo Mathematica Moravica vol 9 pp43ndash52 2005

[5] A Sıntamarian ldquoSelections and common fixed points for somegeneralized multivalued contractionsrdquo Demonstratio Mathe-matica vol 39 no 3 pp 609ndash617 2006

[6] A Petrusel and A Sıntamarian ldquoSingle-valued and multi-valued Caristi type operatorsrdquo Publicationes MathematicaeDebrecen vol 60 no 1-2 pp 167ndash177 2002

[7] F E Browder ldquoThefixed point theory ofmulti-valuedmappingsin topological vector spacesrdquo Mathematische Annalen vol 177pp 283ndash301 1968

[8] J R Jachymski ldquoCaristirsquos fixed point theorem and selections ofset-valued contractionsrdquo Journal of Mathematical Analysis andApplications vol 227 no 1 pp 55ndash67 1998

[9] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[10] H Covitz and S B Nadler Jr ldquoMulti-valued contractionmappings in generalized metric spacesrdquo Israel Journal of Math-ematics vol 8 pp 5ndash11 1970

[11] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969

[12] H-K Xu ldquoSome recent results and problems for set-valuedmappingsrdquo in Advances in Mathematics Research vol 1 pp 31ndash49 2002

[13] M A Khamsi ldquoRemarks on Caristirsquos fixed point theoremrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 227ndash231 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

Theorem 25 Let (119883 119889) be a complete metric space Let 119865

119883 rarr 119862119897(119883) be a multivalued mapping Suppose that 119865 is a120583-contraction with 120583 right upper semicontinuous such that thefunction 120577(119905) = (119905 minus 120583(119905))2 for all 119905 ge 0 belongs to Φ Then 119865

has a fixed point

Proof Let ] [0 +infin) rarr [0 +infin) be the function definedby ](119905) = (119905 + 120583(119905))2 for all 119905 ge 0 Clearly ] is right uppersemicontinuous and ](119905) lt 119905 for all 119905 gt 0 Therefore the set

119910 isin 119865119909 ] (119889 (119909 119910)) le 119863 (119909 119865119909) = 0 (42)

for all 119909 isin 119883 In fact if 119910119899 sub 119865119909 is a sequence such that

119889(119909 119910119899) darr 119863(119909 119865119909) the right upper semicontinuity of the

function ] ensures that

lim sup119899rarr+infin

] (119889 (119909 119910119899)) le ] (119863 (119909 119865119909)) (43)

and hence there exists 119910119899such that

] (119889 (119909 119910119899)) lt 119863 (119909 119865119909) (44)

The axiom of choice ensures that there is a mapping 119891 119883 rarr

119883 such that

119891119909 isin 119865119909 ] (119889 (119909 119891119909)) le 119863 (119909 119865119909) for 119909 isin 119883 (45)

Since 119863(119891119909 119865119891119909) le 119867(119865119909 119865119891119909) le 120583(119889(119909 119891119909)) we get

120577 (119889 (119909 119891119909)) = ] (119889 (119909 119891119909)) minus 120583 (119889 (119909 119891119909))

le 119863 (119909 119865119909) minus 119863 (119891119909 119865119891119909)

= 120601 (119909) minus 120601 (119891119909)

(46)

where the function 120601 119883 rarr [0 +infin) is defined by120601(119909) = 119863(119909 119865119909) for all 119909 isin 119883 Since the function 120601 is lowersemicontinuous then we get that 119891 has a fixed point whichis a fixed point for 119865

5 Conclusion

Under suitable hypotheses for multivalued mappings weestablished the existence of Caristi type selections Also weproved fixed point and quasi-fixed point theorems by usingweaker andmodified hypotheses on some classes of functionspresent in the literature Our results extend and complementmany theorems in the literature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

All authors contributed equally and significantly in writingthis paper All authors read and approved the final paper

Acknowledgments

The first author is a Member of the Gruppo Nazionale perlrsquoAnalisi Matematica la Probabilita e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM) The second author is a Member of the GruppoNazionale per le Strutture Algebriche Geometriche e le loroApplicazioni (GNSAGA) of the Istituto Nazionale di AltaMatematica (INdAM)

References

[1] D Repovs and P V Semenov Continuous Selections of Multi-valued Mappings vol 455 of Mathematics and Its Applications1998

[2] E Michael ldquoContinuous selections Irdquo Annals of MathematicsSecond Series vol 63 pp 361ndash382 1956

[3] A Petrusel ldquoMultivalued operators and continuous selectionsrdquoPure Mathematics and Applications vol 9 no 1-2 pp 165ndash1701998

[4] A Petrusel andG Petrusel ldquoSelection theorems formultivaluedgeneralized contractionsrdquo Mathematica Moravica vol 9 pp43ndash52 2005

[5] A Sıntamarian ldquoSelections and common fixed points for somegeneralized multivalued contractionsrdquo Demonstratio Mathe-matica vol 39 no 3 pp 609ndash617 2006

[6] A Petrusel and A Sıntamarian ldquoSingle-valued and multi-valued Caristi type operatorsrdquo Publicationes MathematicaeDebrecen vol 60 no 1-2 pp 167ndash177 2002

[7] F E Browder ldquoThefixed point theory ofmulti-valuedmappingsin topological vector spacesrdquo Mathematische Annalen vol 177pp 283ndash301 1968

[8] J R Jachymski ldquoCaristirsquos fixed point theorem and selections ofset-valued contractionsrdquo Journal of Mathematical Analysis andApplications vol 227 no 1 pp 55ndash67 1998

[9] J Caristi ldquoFixed point theorems for mappings satisfyinginwardness conditionsrdquo Transactions of the American Mathe-matical Society vol 215 pp 241ndash251 1976

[10] H Covitz and S B Nadler Jr ldquoMulti-valued contractionmappings in generalized metric spacesrdquo Israel Journal of Math-ematics vol 8 pp 5ndash11 1970

[11] S B Nadler Jr ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 pp 475ndash488 1969

[12] H-K Xu ldquoSome recent results and problems for set-valuedmappingsrdquo in Advances in Mathematics Research vol 1 pp 31ndash49 2002

[13] M A Khamsi ldquoRemarks on Caristirsquos fixed point theoremrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no1-2 pp 227ndash231 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of