Research Article A Generalized Mazur-Ulam Theorem for ...downloads.hindawi.com/journals/aaa/2014/624920.pdf · e theory of isometric mappings on classical normed spaces has its roots

Research ArticleA Generalized Mazur-Ulam Theorem for Fuzzy Normed Spaces

J J Font J Galindo S Macario and M Sanchis

Departament de Matematiques Institut Universitari de Matematiques i Aplicacions de Castello (IMAC) Universitat Jaume ICampus del Riu Sec sn 12071 Castello Spain

Correspondence should be addressed to S Macario macariomatujies

Received 3 April 2014 Accepted 19 May 2014 Published 2 June 2014

Academic Editor Salvador Romaguera

Copyright copy 2014 J J Font et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We introduce fuzzy norm-preserving maps which generalize the concept of fuzzy isometry Based on the ideas from Vogt 1973and Vaisala 2003 we provide the following generalized version of the Mazur-Ulam theorem in the fuzzy context let119883 119884 be fuzzynormed spaces and let 119891 119883 rarr 119884 be a fuzzy norm-preserving surjection satisfying 119891(0) = 0 Then 119891 is additive

1 Introduction

Studies on fuzzy normed spaces are relatively recent in thefield of fuzzy functional analysis It was Katsaras who in 1984[1] while studying topological vector spaces was the first tointroduce the idea of fuzzy normon a linear space Eight yearslater Felbin [2] offered an alternative definition With thisdefinition the induced fuzzy metric is of Kaleva and Seikkalatype [3] In 1994 Cheng and Mordeson [4] defined anothertype of fuzzy norm on a linear space whose associated fuzzymetric is of Kramosil and Michalek type [5] Finally in [6](see also [7]) Bag and Samanta redefined the concept of fuzzynorm given in [4] as follows

Definition 1 Let119883 be a real linear space A function119873 119883 times

R rarr [0 1] is said to be a fuzzy norm on119883 if for all 119909 119910 isin 119883

and all 119905 119904 119896 isin R it satisfies the following

(N1) 119873(119909 119905) = 0 for 119905 le 0

(N2) 119909 = 0 if and only if119873(119909 119905) = 1 for all 119905 gt 0

(N3) 119873(119896119909 119905) = 119873(119909 119905|119896|) if 119896 = 0

(N4) 119873(119909 + 119910 119905 + 119904) ge min119873(119909 119905)119873(119910 119904)

(N5) lim119905rarrinfin

119873(119909 119905) = 1

The pair (119883119873) is called a fuzzy normed space

We point out that classical normed spaces are strictlyincluded in the class of fuzzy normed spaces (see [6]) and that(N2) and (N4) imply that for a fixed 119909 isin 119883 the function119873(119909 sdot) is nondecreasing It is a well-known fact that everyfuzzy norm on a real linear space 119883 induces a topology on119883 defined as follows a subbase for the neighborhood systemat a point 119909 isin 119883 consists of the sets

119861119873(119909 120576 119905) = 119910 isin 119883 119873 (119909 minus 119910 119905) gt 1 minus 120576 (1)

for all 0 lt 120576 lt 1 and 119905 gt 0 It is straightforward to verifythat the filter of neighborhoods of the origin generated by thefamily 119861

119873(0 120576 119905) 0 lt 120576 lt 1 119905 gt 0 satisfies the properties

which make (119883119873) a Hausdorff topological vector spaceThe theory of isometric mappings on classical normed

spaces has its roots in the seminal paper by Mazur andUlam ([8] see also [9]) who proved that every bijectiveisometry between two real normed spaces is affine It isknown that the surjective assumption is essential in this resultand that it is not true for complex normed spacesTheMazur-Ulam theorem has been extended in many directions Forexample Baker [10] proved that the result remains true ifwe consider an (not necessarily onto) isometry between areal normed space and a strictly convex real normed spaceAnother direction was provided by Vogt [11] who replacedisometries by the more general notion of equality of distancepreserving maps (see also [12])

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 624920 4 pageshttpdxdoiorg1011552014624920

2 Abstract and Applied Analysis

In this paper following the ideas of Vogt we introduce ageneralization of the concept of fuzzy isometry as follows

Definition 2 Let (119883119873) and (1198841198731015840) be two fuzzy normed

spaces One says that 119891 119883 rarr 119884 is a fuzzy norm-preservingmapping if given 119909 119910 1199091015840 1199101015840 isin 119883 then for all 119905 gt 0

119873(119909 minus 119910 119905) = 119873 (1199091015840minus 1199101015840 119905) 997904rArr 119873

1015840(119891 (119909) minus 119891 (119910) 119905)

= 1198731015840(119891 (119909

1015840) minus 119891 (119910

1015840) 119905)

(2)

Let us recall here the definition of a fuzzy isometry


spaces It is said that 119891 119883 rarr 119884 is a fuzzy isometry if1198731015840(119891(119909) minus 119891(119910) 119905) = 119873(119909 minus 119910 119905) for all 119909 119910 isin 119883 and 119905 gt 0We provide the following generalized version of the

Mazur-Ulam theorem in the fuzzy context let 119883 119884 be fuzzynormed spaces and let 119891 119883 rarr 119884 be a fuzzy norm-preserving surjection satisfying 119891(0) = 0 Then 119891 is additiveAs a corollary we deduce that if such an119891 is a fuzzy isometrythen 119891 is affine

2 The Results

Let (119883119873) be a fuzzy normed space Fix 119888 isin 119883 and define afunction 120601

119888 119883 rarr 119883 as 120601

119888(119909) = 2119888 minus 119909 It is apparent that

120601119888is bijective indeed 120601

119888∘ 120601119888= 119868119889 where 119868119889 stands for the

identity map on119883 Furthermore 120601119888is a fuzzy isometry since

119873(120601119888(119909) minus 120601

119888(119910) 119905) = 119873(2119888 minus 119909 minus 2119888 + 119910 119905) = 119873(119909 minus 119910 119905) for

all 119909 119910 isin 119883 and 119905 gt 0Let 119871(119888119873) = 119909 isin 119883 119873(119909 119905) = 119873(2119888 minus 119909 119905) ge

119873(119888 1199052) for all 119905 gt 0 It is clear that 119888 isin 119871(119888119873) = 0 andthat 120601

119888(119871(119888119873)) sube 119871(119888119873)

Lemma 4 Every fuzzy isometry from 119871(119888119873) onto 119871(119888119873)

fixes 119888

Proof Let 119868 = 119892 119871(119888119873) rarr 119871(119888119873)

119892 is an onto fuzzy isometry This is a nonempty set sincethe identity map belongs to it Fix 119905 gt 0 and let us definefor each 119894 = 1 2 3

120582119905

119894= inf 119873(119892 (119888) minus 119888

119905

119894) 119892 isin 119868 (3)

If 119892 isin 119868 then we define 1198921015840 = 119892minus1∘ 120601119888∘ 119892 which is in 119868 Then

119873(1198921015840(119888) minus 119888 119905) = 119873 ((119892

minus1∘ 120601119888∘ 119892) (119888) minus 119888 119905)

= 119873 (120601119888(119892 (119888)) minus 119892 (119888) 119905)

= 119873 (2 (119892 (119888) minus 119888) 119905)

= 119873(119892 (119888) minus 119888119905

2)

(4)

Consequently119873(119892(119888) minus 119888 1199052) ge 120582119905

1

Moreover for all 119892 isin 119868 we have

119873(119892 (119888) minus 119888 119905) ge 119873(119892 (119888) minus 119888119905

2) ge 120582

119905

1 (5)

which yields 1205821199051ge 120582119905

2ge 120582119905

1 That is 120582119905

1= 120582119905

2 which leads us

to the following equalities

120582119905

1= 120582119905

2= sdot sdot sdot = 120582

119905

119899= sdot sdot sdot (6)

On the other hand given 119892 isin 119868 we have

119873(119888119905

2) le 120582

119905

119899119905

le 119873(1198920(119888) minus 119888

119905

119899119905

) le 119873 (1198920(119888) minus 119888 119905

0)

= 119896 lt 1

(7)

since 119892(119888) isin 119871(119888119873)Let us suppose that there exists 119892

0isin 119868 such that 119892

0(119888) = 119888

Then there exists 1199050gt 0 such that

119873(1198920(119888) minus 119888 119905

0) = 119896 lt 1 (8)

In addition for each 119905 gt 0 there exists 119899119905such that119873(119892

0(119888) minus

119888 119905119899119905) le 119873(119892

0(119888) minus 119888 119905

0) Hence by (6) and (7) we have that

119873(119888119905

2) le 120582

119905

119899119905

le 119873(1198920(119888) minus 119888

119905

119899119905

) le 119873 (1198920(119888) minus 119888 119905

0)

= 119896 lt 1

(9)

but

1 = lim119905rarrinfin

119873(119888119905

2) le 119896 lt 1 (10)

a contradiction which completes the proof

Remark 5 It can be checked that a fuzzy norm-preservingmapping is associated for each 119905 gt 0 with a function 120588

119905

119863119905sube [0 1] rarr [0 1] such that

1198731015840(119891 (119909) minus 119891 (119910) 119905) = 120588

119905(119873 (119909 minus 119910 119905)) (11)

It is then apparent that fuzzy isometries are fuzzy norm-preserving mappings taking 120588

119905= 119868119889 119905 gt 0

Theorem 6 Let (119883119873) and (1198841198731015840) be two fuzzy normed

spaces and let 119891 119883 rarr 119884 be a fuzzy norm-preservingsurjection satisfying 119891(0) = 0 Then 119891 is additive

Proof Fix 1199090isin 119883 and let

119871 (119891 (1199090) 1198731015840)

= 119910 isin 119884 1198731015840(119910 119905) = 119873

1015840(2119891 (119909

0) minus 119910 119905)

ge 1198731015840(119891 (119909

0)

119905

2) 119905 gt 0

(12)

We know that 119891(1199090) isin 119871(119891(119909

0)1198731015840) = 0

We now define map ℎ 119871(119891(1199090)1198731015840) rarr 119871(119891(119909

0)1198731015840) as

ℎ(119910) = 119891(1199091minus 1199091015840) for a fixed 119909

1isin 119891minus1(2119891(119909

0)) and 119909

1015840isin

119891minus1(119910)

Abstract and Applied Analysis 3

Claim 1 ℎ is a fuzzy isometry from 119871(119891(1199090)1198731015840) to

119871(119891(1199090)1198731015840) which does not depend on the choice of 1199091015840 isin

119891minus1(119910)Let us first check that ℎ does not depend on the choice

of 1199091015840 isin 119891minus1(119910) To this end suppose that 1199091015840 11990910158401015840 sube 119891

minus1(119910)

Then

1198731015840(119891 (119909

1minus 1199091015840) minus 119891 (119909

1minus 11990910158401015840) 119905)

= 120588119905(119873 (119909

10158401015840minus 1199091015840 119905)) = 119873

1015840(119891 (119909

10158401015840) minus 119891 (119909

1015840) 119905)

= 1198731015840(0 119905) = 1

(13)

which is to say that 119891(1199091minus 1199091015840) = 119891(119909

1minus 11990910158401015840)

Let us next prove that ℎ is a fuzzy isometry Suppose that119891(1199091015840) = 1199101015840 and 119891(11990910158401015840) = 119910

10158401015840 with 1199091015840 11990910158401015840 isin 119883 and 1199101015840 11991010158401015840 isin 119884Then

1198731015840(ℎ (1199101015840) minus ℎ (119910

10158401015840) 119905) = 119873

1015840(119891 (119909

1minus 1199091015840) minus 119891 (119909

1minus 11990910158401015840) 119905)

= 120588119905(119873 (119909

10158401015840minus 1199091015840 119905))

= 1198731015840(119891 (119909

10158401015840) minus 119891 (119909

1015840) 119905)

= 1198731015840(11991010158401015840minus 1199101015840 119905)

(14)

Finally let us check that ℎ maps 119871(119891(1199090)1198731015840) onto

119871(119891(1199090)1198731015840) Fix 1199101015840 isin 119871(119891(119909

0)1198731015840) and let 1199091015840 isin 119883 such that

119891(1199091015840) = 1199101015840 From the definition of119871(119891(119909

0)1198731015840) we know that

1198731015840(1199101015840 119905) = 119873

1015840(2119891 (119909

0) minus 1199101015840 119905) ge 119873

1015840(119891 (119909

0)

119905

2) (15)

Hence

1198731015840(ℎ (1199101015840) 119905) = 119873

1015840(119891 (119909

1minus 1199091015840) minus 119891 (0) 119905)

= 120588119905(119873 (119909

1minus 1199091015840minus 0 119905))

= 1198731015840(119891 (1199091) minus 119891 (119909

1015840) 119905)

= 1198731015840(2119891 (119909

0) minus 1199101015840 119905)

= 1198731015840(1199101015840 119905)

ge 1198731015840(119891 (119909

0)

119905

2)

(16)

Furthermore

1198731015840(2119891 (119909

0) minus ℎ (119910

1015840) 119905) = 119873

1015840(2119891 (119909

0) minus 119891 (119909

1minus 1199091015840) 119905)

= 1198731015840(119891 (1199091) minus 119891 (119909

1minus 1199091015840) 119905)

= 120588119905(119873 (119909

1minus 1199091+ 1199091015840minus 0 119905))

= 1198731015840(119891 (119909

1015840) minus 119891 (0) 119905)

= 1198731015840(1199101015840 119905)

= 1198731015840(ℎ (1199101015840) 119905)

(17)

As a consequence we deduce that ℎ(119871(119891(1199090)1198731015840)) sube

119871(119891(1199090)1198731015840) Since it is a routinematter to verify that ℎminus1 = ℎ

we infer that ℎ(119871(119891(1199090)1198731015840)) = 119871(119891(119909

0)1198731015840) and the claim

is provedThanks to Claim 1 we can apply Lemma 4 and conclude

that ℎ fixes 119891(1199090) Hence

119891 (1199090) = ℎ (119891 (119909

0)) = 119891 (119909

1minus 1199090) (18)

and then

1 = 1198731015840(119891 (1199090) minus 119891 (119909

1minus 1199090) 119905)

= 120588119905(119873 (2119909

0minus 1199091 119905))

= 1198731015840(119891 (2119909

0) minus 119891 (119909

1) 119905)

= 1198731015840(119891 (2119909

0) minus 2119891 (119909

0) 119905)

(19)

for all 119905 gt 0 which is to say that

119891 (21199090) = 2119891 (119909

0) (20)

Next let us define for a fixed 119911 isin 119883 the following map

119891119911(119909) = 119891 (119909 + 119911) minus 119891 (119911) (21)

for all 119909 isin 119883It is clear that 119891

119911(0) = 0 and 119891

119911is surjective since 119891 is

assumed to be also surjective Furthermore for all 119909 119910 isin 119883

and 119905 gt 0

1198731015840(119891119911(119909) minus 119891

119911(119910) 119905)

= 1198731015840(119891 (119909 + 119911) minus 119891 (119910 + 119911) 119905) = 120588

119905(119873 (119909 minus 119910 119905))

(22)

which is to say that 119891119911is also a fuzzy norm-preserving map

Hence by (20) we infer that 119891119911(2119909) = 2119891

119911(119909) for all 119909 isin 119883

Then for all 119909 119910 isin 119883 we have

119891 ((119909 minus 119910) + 119910) minus 119891 (119910)

= 119891119910(119909 minus 119910) = 2119891

119910(119909 minus 119910

2)

= 2 (119891(119909 minus 119910

2+ 119910) minus 119891 (119910))

(23)


which yields

119891 (119909) + 119891 (119910) = 2119891(119909 + 119910

2) = 119891 (119909 + 119910) (24)

that is 119891 is additive

Let us recall here that additivity yieldsQ-linearity whichin presence of continuity implies linearity Hence as astraightforward corollary of Theorem 6 we obtain a fuzzyversion of the Mazur-Ulam theorem

Corollary 7 Let (119883119873) and (1198841198731015840) be two fuzzy normed

spaces and let 119891 119883 rarr 119884 be a surjective fuzzy isometryThen119891 is affine

Proof It suffices to apply Theorem 6 to 119892(119909) = 119891(119909) minus 119891(0)119909 isin 119883

Conflict of Interests

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

Acknowledgments

This research is supported by the Spanish Ministry of Educa-tion and Science (Grant noMTM2011-23118) and byBancaixa(Project P1-1B2011-30)

References

[1] A K Katsaras ldquoFuzzy topological vector spaces IIrdquo Fuzzy Setsand Systems vol 12 no 2 pp 143ndash154 1984

[2] C Felbin ldquoFinite-dimensional fuzzy normed linear spacerdquoFuzzy Sets and Systems vol 48 no 2 pp 239ndash248 1992

[3] O Kaleva and S Seikkala ldquoOn fuzzy metric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[4] S C Cheng and J N Mordeson ldquoFuzzy linear operators andfuzzy normed linear spacesrdquo Bulletin of the Calcutta Mathemat-ical Society vol 86 no 5 pp 429ndash436 1994

[5] I Kramosil and J Michalek ldquoFuzzy metrics and statisticalmetric spacesrdquo Kybernetika vol 11 no 5 pp 326ndash334 1975

[6] T Bag and S K Samanta ldquoFinite dimensional fuzzy normedlinear spacesrdquo Journal of Fuzzy Mathematics vol 11 no 3 pp687ndash705 2003

[7] T Bag and S K Samanta ldquoFuzzy bounded linear operatorsrdquoFuzzy Sets and Systems vol 151 no 3 pp 513ndash547 2005

[8] S Mazur and S Ulam ldquoSur les transformations isometriquesdrsquoespaces vectoriels normesrdquo Comptes Rendus de lrsquoAcademie desSciences vol 194 pp 946ndash948 1932

[9] J Vaisala ldquoA proof of the Mazur-Ulam theoremrdquoTheAmericanMathematical Monthly vol 110 no 7 pp 633ndash635 2003

[10] J A Baker ldquoIsometries in normed spacesrdquo The AmericanMathematical Monthly vol 78 pp 655ndash658 1971

[11] A Vogt ldquoMaps which preserve equality of distancerdquo StudiaMathematica vol 45 pp 43ndash48 1973

[12] R Hu ldquoOn the maps preserving the equality of distancerdquoJournal of Mathematical Analysis and Applications vol 343 no2 pp 1161ndash1165 2008

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Stochastic AnalysisInternational Journal of


In this paper following the ideas of Vogt we introduce ageneralization of the concept of fuzzy isometry as follows


spaces One says that 119891 119883 rarr 119884 is a fuzzy norm-preservingmapping if given 119909 119910 1199091015840 1199101015840 isin 119883 then for all 119905 gt 0

119873(119909 minus 119910 119905) = 119873 (1199091015840minus 1199101015840 119905) 997904rArr 119873

1015840(119891 (119909) minus 119891 (119910) 119905)

= 1198731015840(119891 (119909

1015840) minus 119891 (119910

1015840) 119905)

(2)

Let us recall here the definition of a fuzzy isometry


spaces It is said that 119891 119883 rarr 119884 is a fuzzy isometry if1198731015840(119891(119909) minus 119891(119910) 119905) = 119873(119909 minus 119910 119905) for all 119909 119910 isin 119883 and 119905 gt 0We provide the following generalized version of the

Mazur-Ulam theorem in the fuzzy context let 119883 119884 be fuzzynormed spaces and let 119891 119883 rarr 119884 be a fuzzy norm-preserving surjection satisfying 119891(0) = 0 Then 119891 is additiveAs a corollary we deduce that if such an119891 is a fuzzy isometrythen 119891 is affine

2 The Results

Let (119883119873) be a fuzzy normed space Fix 119888 isin 119883 and define afunction 120601

119888 119883 rarr 119883 as 120601

119888(119909) = 2119888 minus 119909 It is apparent that

120601119888is bijective indeed 120601

119888∘ 120601119888= 119868119889 where 119868119889 stands for the

identity map on119883 Furthermore 120601119888is a fuzzy isometry since

119873(120601119888(119909) minus 120601

119888(119910) 119905) = 119873(2119888 minus 119909 minus 2119888 + 119910 119905) = 119873(119909 minus 119910 119905) for

all 119909 119910 isin 119883 and 119905 gt 0Let 119871(119888119873) = 119909 isin 119883 119873(119909 119905) = 119873(2119888 minus 119909 119905) ge

119873(119888 1199052) for all 119905 gt 0 It is clear that 119888 isin 119871(119888119873) = 0 andthat 120601

119888(119871(119888119873)) sube 119871(119888119873)

Lemma 4 Every fuzzy isometry from 119871(119888119873) onto 119871(119888119873)

fixes 119888

Proof Let 119868 = 119892 119871(119888119873) rarr 119871(119888119873)

119892 is an onto fuzzy isometry This is a nonempty set sincethe identity map belongs to it Fix 119905 gt 0 and let us definefor each 119894 = 1 2 3

120582119905

119894= inf 119873(119892 (119888) minus 119888

119905

119894) 119892 isin 119868 (3)

If 119892 isin 119868 then we define 1198921015840 = 119892minus1∘ 120601119888∘ 119892 which is in 119868 Then

119873(1198921015840(119888) minus 119888 119905) = 119873 ((119892

minus1∘ 120601119888∘ 119892) (119888) minus 119888 119905)

= 119873 (120601119888(119892 (119888)) minus 119892 (119888) 119905)

= 119873 (2 (119892 (119888) minus 119888) 119905)

= 119873(119892 (119888) minus 119888119905

2)

(4)

Consequently119873(119892(119888) minus 119888 1199052) ge 120582119905

1

Moreover for all 119892 isin 119868 we have

119873(119892 (119888) minus 119888 119905) ge 119873(119892 (119888) minus 119888119905

2) ge 120582

119905

1 (5)

which yields 1205821199051ge 120582119905

2ge 120582119905

1 That is 120582119905

1= 120582119905

2 which leads us

to the following equalities

120582119905

1= 120582119905

2= sdot sdot sdot = 120582

119905

119899= sdot sdot sdot (6)

On the other hand given 119892 isin 119868 we have

119873(119888119905

2) le 120582

119905

119899119905

le 119873(1198920(119888) minus 119888

119905

119899119905

) le 119873 (1198920(119888) minus 119888 119905

0)

= 119896 lt 1

(7)

since 119892(119888) isin 119871(119888119873)Let us suppose that there exists 119892

0isin 119868 such that 119892

0(119888) = 119888

Then there exists 1199050gt 0 such that

119873(1198920(119888) minus 119888 119905

0) = 119896 lt 1 (8)

In addition for each 119905 gt 0 there exists 119899119905such that119873(119892

0(119888) minus

119888 119905119899119905) le 119873(119892

0(119888) minus 119888 119905

0) Hence by (6) and (7) we have that

119873(119888119905

2) le 120582

119905

119899119905

le 119873(1198920(119888) minus 119888

119905

119899119905

) le 119873 (1198920(119888) minus 119888 119905

0)

= 119896 lt 1

(9)

but

1 = lim119905rarrinfin

119873(119888119905

2) le 119896 lt 1 (10)

a contradiction which completes the proof

Remark 5 It can be checked that a fuzzy norm-preservingmapping is associated for each 119905 gt 0 with a function 120588

119905

119863119905sube [0 1] rarr [0 1] such that

1198731015840(119891 (119909) minus 119891 (119910) 119905) = 120588

119905(119873 (119909 minus 119910 119905)) (11)

It is then apparent that fuzzy isometries are fuzzy norm-preserving mappings taking 120588

119905= 119868119889 119905 gt 0

Theorem 6 Let (119883119873) and (1198841198731015840) be two fuzzy normed

spaces and let 119891 119883 rarr 119884 be a fuzzy norm-preservingsurjection satisfying 119891(0) = 0 Then 119891 is additive

Proof Fix 1199090isin 119883 and let

119871 (119891 (1199090) 1198731015840)

= 119910 isin 119884 1198731015840(119910 119905) = 119873

1015840(2119891 (119909

0) minus 119910 119905)

ge 1198731015840(119891 (119909

0)

119905

2) 119905 gt 0

(12)

We know that 119891(1199090) isin 119871(119891(119909

0)1198731015840) = 0

We now define map ℎ 119871(119891(1199090)1198731015840) rarr 119871(119891(119909

0)1198731015840) as

ℎ(119910) = 119891(1199091minus 1199091015840) for a fixed 119909

1isin 119891minus1(2119891(119909

0)) and 119909

1015840isin

119891minus1(119910)






minus1(119910)

Then

1198731015840(119891 (119909

1minus 1199091015840) minus 119891 (119909

1minus 11990910158401015840) 119905)

= 120588119905(119873 (119909

10158401015840minus 1199091015840 119905)) = 119873

1015840(119891 (119909

10158401015840) minus 119891 (119909

1015840) 119905)

= 1198731015840(0 119905) = 1

(13)


1minus 11990910158401015840)



1198731015840(ℎ (1199101015840) minus ℎ (119910

10158401015840) 119905) = 119873

1015840(119891 (119909

1minus 1199091015840) minus 119891 (119909

1minus 11990910158401015840) 119905)

= 120588119905(119873 (119909

10158401015840minus 1199091015840 119905))

= 1198731015840(119891 (119909

10158401015840) minus 119891 (119909

1015840) 119905)

= 1198731015840(11991010158401015840minus 1199101015840 119905)

(14)


119871(119891(1199090)1198731015840) Fix 1199101015840 isin 119871(119891(119909



0)1198731015840) we know that

1198731015840(1199101015840 119905) = 119873

1015840(2119891 (119909

0) minus 1199101015840 119905) ge 119873

1015840(119891 (119909

0)

119905

2) (15)

Hence

1198731015840(ℎ (1199101015840) 119905) = 119873

1015840(119891 (119909

1minus 1199091015840) minus 119891 (0) 119905)

= 120588119905(119873 (119909

1minus 1199091015840minus 0 119905))

= 1198731015840(119891 (1199091) minus 119891 (119909

1015840) 119905)

= 1198731015840(2119891 (119909

0) minus 1199101015840 119905)

= 1198731015840(1199101015840 119905)

ge 1198731015840(119891 (119909

0)

119905

2)

(16)

Furthermore

1198731015840(2119891 (119909

0) minus ℎ (119910

1015840) 119905) = 119873

1015840(2119891 (119909

0) minus 119891 (119909

1minus 1199091015840) 119905)

= 1198731015840(119891 (1199091) minus 119891 (119909

1minus 1199091015840) 119905)

= 120588119905(119873 (119909

1minus 1199091+ 1199091015840minus 0 119905))

= 1198731015840(119891 (119909

1015840) minus 119891 (0) 119905)

= 1198731015840(1199101015840 119905)

= 1198731015840(ℎ (1199101015840) 119905)

(17)



we infer that ℎ(119871(119891(1199090)1198731015840)) = 119871(119891(119909

0)1198731015840) and the claim



119891 (1199090) = ℎ (119891 (119909

0)) = 119891 (119909

1minus 1199090) (18)

and then

1 = 1198731015840(119891 (1199090) minus 119891 (119909

1minus 1199090) 119905)

= 120588119905(119873 (2119909

0minus 1199091 119905))

= 1198731015840(119891 (2119909

0) minus 119891 (119909

1) 119905)

= 1198731015840(119891 (2119909

0) minus 2119891 (119909

0) 119905)

(19)


119891 (21199090) = 2119891 (119909

0) (20)


119891119911(119909) = 119891 (119909 + 119911) minus 119891 (119911) (21)


119911(0) = 0 and 119891



and 119905 gt 0

1198731015840(119891119911(119909) minus 119891

119911(119910) 119905)

= 1198731015840(119891 (119909 + 119911) minus 119891 (119910 + 119911) 119905) = 120588

119905(119873 (119909 minus 119910 119905))

(22)



119911(119909) for all 119909 isin 119883


119891 ((119909 minus 119910) + 119910) minus 119891 (119910)

= 119891119910(119909 minus 119910) = 2119891

119910(119909 minus 119910

2)

= 2 (119891(119909 minus 119910

2+ 119910) minus 119891 (119910))

(23)


which yields

119891 (119909) + 119891 (119910) = 2119891(119909 + 119910

2) = 119891 (119909 + 119910) (24)








Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














minus1(119910)

Then

1198731015840(119891 (119909

1minus 1199091015840) minus 119891 (119909

1minus 11990910158401015840) 119905)

= 120588119905(119873 (119909

10158401015840minus 1199091015840 119905)) = 119873

1015840(119891 (119909

10158401015840) minus 119891 (119909

1015840) 119905)

= 1198731015840(0 119905) = 1

(13)


1minus 11990910158401015840)



1198731015840(ℎ (1199101015840) minus ℎ (119910

10158401015840) 119905) = 119873

1015840(119891 (119909

1minus 1199091015840) minus 119891 (119909

1minus 11990910158401015840) 119905)

= 120588119905(119873 (119909

10158401015840minus 1199091015840 119905))

= 1198731015840(119891 (119909

10158401015840) minus 119891 (119909

1015840) 119905)

= 1198731015840(11991010158401015840minus 1199101015840 119905)

(14)


119871(119891(1199090)1198731015840) Fix 1199101015840 isin 119871(119891(119909



0)1198731015840) we know that

1198731015840(1199101015840 119905) = 119873

1015840(2119891 (119909

0) minus 1199101015840 119905) ge 119873

1015840(119891 (119909

0)

119905

2) (15)

Hence

1198731015840(ℎ (1199101015840) 119905) = 119873

1015840(119891 (119909

1minus 1199091015840) minus 119891 (0) 119905)

= 120588119905(119873 (119909

1minus 1199091015840minus 0 119905))

= 1198731015840(119891 (1199091) minus 119891 (119909

1015840) 119905)

= 1198731015840(2119891 (119909

0) minus 1199101015840 119905)

= 1198731015840(1199101015840 119905)

ge 1198731015840(119891 (119909

0)

119905

2)

(16)

Furthermore

1198731015840(2119891 (119909

0) minus ℎ (119910

1015840) 119905) = 119873

1015840(2119891 (119909

0) minus 119891 (119909

1minus 1199091015840) 119905)

= 1198731015840(119891 (1199091) minus 119891 (119909

1minus 1199091015840) 119905)

= 120588119905(119873 (119909

1minus 1199091+ 1199091015840minus 0 119905))

= 1198731015840(119891 (119909

1015840) minus 119891 (0) 119905)

= 1198731015840(1199101015840 119905)

= 1198731015840(ℎ (1199101015840) 119905)

(17)



we infer that ℎ(119871(119891(1199090)1198731015840)) = 119871(119891(119909

0)1198731015840) and the claim



119891 (1199090) = ℎ (119891 (119909

0)) = 119891 (119909

1minus 1199090) (18)

and then

1 = 1198731015840(119891 (1199090) minus 119891 (119909

1minus 1199090) 119905)

= 120588119905(119873 (2119909

0minus 1199091 119905))

= 1198731015840(119891 (2119909

0) minus 119891 (119909

1) 119905)

= 1198731015840(119891 (2119909

0) minus 2119891 (119909

0) 119905)

(19)


119891 (21199090) = 2119891 (119909

0) (20)


119891119911(119909) = 119891 (119909 + 119911) minus 119891 (119911) (21)


119911(0) = 0 and 119891



and 119905 gt 0

1198731015840(119891119911(119909) minus 119891

119911(119910) 119905)

= 1198731015840(119891 (119909 + 119911) minus 119891 (119910 + 119911) 119905) = 120588

119905(119873 (119909 minus 119910 119905))

(22)



119911(119909) for all 119909 isin 119883


119891 ((119909 minus 119910) + 119910) minus 119891 (119910)

= 119891119910(119909 minus 119910) = 2119891

119910(119909 minus 119910

2)

= 2 (119891(119909 minus 119910

2+ 119910) minus 119891 (119910))

(23)


which yields

119891 (119909) + 119891 (119910) = 2119891(119909 + 119910

2) = 119891 (119909 + 119910) (24)








Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










which yields

119891 (119909) + 119891 (119910) = 2119891(119909 + 119910

2) = 119891 (119909 + 119910) (24)








Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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