Afr. Mat.DOI 10.1007/s13370-012-0120-8

Common fixed point of single valued and multivaluedmappings

Wajdi Chaker · Aref Jeribi

Received: 13 July 2012 / Accepted: 23 October 2012© African Mathematical Union and Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, we prove the existence of common fixed point of commuting andweakly commuting pair of single valued and multivalued mapping in some classes of Banachspaces. Our results extend previously known ones, we cites Theorem 2, Theorem 3 and Theo-rem 4 by Itoh and Takahashi (J Math Anal Appl 59(3):514–521, 1977), Theorem 4.2 by Kaew-charoen (Nonlinear Anal Hybrid Syst 4(3):389–394, 2010), Theorem 4.2 by Dhompongsaet al. (Nonlinear Anal 64(5):958–970, 2006) and Theorem 3.1 by Nanan and Dhompongsa(Fixed Point Theory Appl 54:10, 2011).

Keywords Common fixed point · Commute mappings · Weakly commute mappings

Mathematics Subject Classification (2010) Primary: 47H10, 55M20, 54H25; Secondary:47H08, 47H09

1 Introduction

Let K be a nonempty subset of a Banach space X . We denote by Co(K ), CoV (K ), C B(K )

and CV (K ) the sets of nonempty compact subsets of K , nonempty compact convex subsetsof K , nonempty closed bounded subsets of K and nonempty closed convex subsets of K ,respectively. Let f : K −→ K and T : K −→ C B(X) be single valued and multivaluedmappings, respectively. We denote by Fix( f ) := {x ∈ K , x = f (x)} and Fix(T ) := {x ∈K , x ∈ T (x)} the sets of fixed points of f and T , respectively. One of the most importantproblems is to prove the existence of common fixed point of f and T under certain conditionson X , K , f and T . Itoh and Takahashi have studied this problem and they have proved thefollowing theorems:

W. Chaker · A. Jeribi (B)Département de Mathématiques, Université de Sfax, Faculté des Sciences de Sfax,Route de Soukra Km 3.5, B.P. 1171, 3000 Sfax, Tunisiee-mail: aref.Jeribi@fss.rnu.tn

W. Chakere-mail: wajdi_chaker@yahoo.fr

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Theorem 1.1 [1] Let X be a Banach space which satisfies Opial’s condition. Let K be anonempty weakly compact convex subset of X, f be a nonexpansive asymptotically regularmapping of K into K , and T be a nonexpansive mapping of K into 2K such that for anyx ∈ K , T (x) is nonempty compact. If f and T commute, then there exists an element z ∈ Ksuch that f (z) = z ∈ T (z).

Theorem 1.2 [1] Let X be a strictly convex Banach space and K be a nonempty boundedclosed convex subset of X. Let f be a nonexpansive condensing mapping of K into K andT be an upper semicontinuous mapping of K into 2K such that for any x ∈ K , T (x) isnonempty closed convex. If f and T commute, then there exists an element z ∈ K such thatf (z) = z ∈ T (z).

Theses results have been improved, generalized under various conditions by the sameauthors in [5]. Recently, several authors are interested with this problem, for examples, wecite [2–4,6–15].

To pursue this analysis, we prove Theorem 3.1 which extends Theorem 1.1 to uniformlyconvex Banach spaces without Opial’s condition. More over, it generalizes Theorem 3 in [1],Theorem 4.2 in [2], Theorem 4.2 in [3] and Theorem 3.1 in [4]. Finally we extend Theorem1.2 for a larger variety of single valued mappings which commute weakly with an uppersemicontinuous closed convex valued multivalued mapping.

This paper is organized in the following way. In Sect. 2, we recall some definitions andresults needed in the rest of the paper, in particular the notion of Edelstein asymptotic center.The next section is devoted to state and prove our main results.

2 Preliminary

We denote by ‖.‖ the usual norm of the Banach space X and by H the Hausdorff distanceon C B(X) × C B(X) defined by:

H(E, F) = max{supt∈E

d(t, F), supt∈F

d(t, E)},

where d(t, E) = inf{‖t − y‖ : y ∈ E} is the distance from t ∈ X to the subset E of X .The Banach space X satisfies Opial’s condition if for every sequence {xn}, weakly

convergent to x0 and x �= x0, we have:

lim infn→∞ ‖xn − x‖ > lim inf

n→∞ ‖xn − x0‖ .

It is well known that Hilbert spaces and �p(1 ≤ p < ∞) spaces satisfy the Opial’scondition. For further results and extensions of Opial’s condition we refer the reader to[16,17].

We recall some definitions needed in the sequel.

Definition 2.1 f is said to be nonexpansive with respect to a nonempty subset M of K iffor any x ∈ K , y ∈ M , we have ‖ f (x) − f (y)‖ ≤ ‖x − y‖. If M = K , f is said to benonexpansive and if M = Fix( f ), f is said to be quasi-nonexpansive.

Definition 2.2 [18] Let λ ∈ (0, 1), f is said to satisfy condition (Cλ) if for every x, y ∈ K ,we have

λ ‖x − f (x)‖ ≤ ‖x − y‖ �⇒ ‖ f (x) − f (y)‖ ≤ ‖x − y‖ .

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Definition 2.3 T is said to be nonexpansive if for every x, y ∈ K , H(T x, T y) ≤ ‖x − y‖.

Definition 2.4 f is said to be asymptotically regular if for any x ∈ K , ‖ f n(x)− f n+1(x)‖ →0 as n → ∞.

Definition 2.5 f and T commute (resp. commute weakly) if for each x ∈ K , f (T (x)) ⊂T ( f (x))(resp. f (∂T (x)) ⊂ T ( f (x))), where ∂T (x) denote the boundary of T (x) in K .

Definition 2.6 [19] Let D be a bounded subset of X . We define γ (D), the Kuratowskimeasure of noncompactness of D, to be inf{d > 0 such that D can be covered by a finitenumber of sets of diameter less than or equal to d}.The following proposition gives some properties of the Kuratowski measure of noncompact-ness which are frequently used.

Proposition 2.7 Let D and D′ two bounded subsets of X, then we have the following prop-erties.

(a) γ (D) = 0 if and only if D is relatively compact.(b) If D ⊆ D′, then γ (D) ≤ γ (D′).(c) γ (D + D′) ≤ γ (D) + γ (D′).(d) For every α ∈ C, γ (αD) = |α|γ (D).

Definition 2.8 [20] Let f : D( f ) ⊆ X −→ X be a continuous operator, γ (.) is the Kura-towski measure of noncompactness in X . Let k ≥ 0, f is said to be k-set-contraction if, forany bounded subset B of D( f ), f (B) is a bounded subset of X and γ ( f (B)) ≤ kγ (B). fis said to be condensing if, for any bounded subset B of D( f ) such that γ (B) > 0, f (B) isa bounded subset of X and γ ( f (B)) < γ (B).

Remark 2.9 It is well known that:

(a) Every k-set-contraction operator such that k < 1 is condensing.(b) Every condensing operator is 1-set-contraction.

Let D be a nonempty bounded subset of a Banach space X . The asymptotic radius of {xn}with respect to D is the scalar

r(D, {xn}) := inf{lim sup ‖x − xn‖ : x ∈ D}.z ∈ D is said asymptotic center of {xn} with respect to D if

lim sup ‖z − xn‖ = r(D, {xn}).The set of asymptotic center of {xn} with respect to D is denoted by Z(D, {xn}).

The notion of asymptotic center is introduced by Edelstein [21], it is well known thatif D is convex then Z(D, {xn}) is convex, and if D is weakly compact then Z(D, {xn}) isnonempty .

Definition 2.10 [16] A bounded sequence {xn} in X is said to be regular with respect to Dif for every subsequence {xni } of {xn}, we have

r(D, {xn}) = r(D, {xni }).A regular sequence {xn} is said to be asymptotically uniform with respect to D if for everysubsequence {xni } of {xn}, we have

Z(D, {xn}) = Z(D, {xni }).

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Goebel [22] and Lim [23] have proved the following lemma.

Lemma 2.11 Let {xn} be a bounded sequence in X and D be a nonempty subset of X.

(i) If D is closed convex, then there exists a subsequence of {xn} which is regular relativeto D.

(ii) Furthermore, if D is separable, then {xn} have a subsequence which is asymptoticallyuniform relative to D.

3 Main result

We start this section with a common fixed point theorem for commuting pair of single valuedand multivalued mappings in uniformly convex Banach space without Opial’s condition.

Theorem 3.1 Let K be a nonempty closed bounded convex subset of a uniformly convexBanach space X, f : K −→ K be a continuous asymptotically regular mapping, andT : K −→ Co(K ) be a nonexpansive mapping. If f and T commute, then there existsz ∈ K such that f (z) = z ∈ T (z).

Proof By Lim’s fixed point Theorem [23], there exists a fixed point x of T . By the factthat f and T commute, we have { f n(x)}n>0 is a bounded sequence of Fix(T ). Since X isuniformly convex, { f n(x)} has a subsequence that all of its subsequences have the same andunique asymptotic center z with respect to K . For the simplicity, we denote this subsequenceby { f n(x)}. Since T (z) is compact, there exists wn ∈ T (z) such that

‖ f n(x) − wn‖ ≤ H(T f n(x), T (z))

≤ ‖ f n(x) − z‖.Since {wn} has a convergent subsequence {wni } which converge to v ∈ T (z), we obtain that

‖ f ni (x) − v‖ ≤ ‖ f ni (x) − wni ‖ + ‖wni − v‖≤ ‖ f ni (x) − z‖ + ‖wni − v‖.

Hence

lim sup ‖ f ni (x) − v‖ ≤ lim sup ‖ f ni (x) − z‖.We conclude that z = v ∈ T (z). It remains to prove that z = f (z). Let p ∈ N

∗, we have

‖ f n+p(x) − f n(x)‖ ≤ ‖ f n+p(x) − f n+p−1(x)‖ + · · · + ‖ f n+1(x) − f n(x)‖.Since f is asymptotically regular, { f n(x)} is a Cauchy sequence and so convergent to v ∈ K .

lim sup ‖ f n(x) − z‖ ≤ lim sup ‖ f n(x) − v‖ = 0

then z = v. By the continuity of f , we have z = f (z). ��As a direct consequence of Theorem 3.1, we obtain the following corollary which extends

Theorem 2 in [1] to the case of uniformly convex Banach spaces without Opial’s condition.

Corollary 3.2 Let K be a nonempty closed bounded convex subset of a uniformly convexBanach space X. Let f : K −→ K be a nonexpansive asymptotically regular mapping, andT : K −→ Co(K ) be a nonexpansive mapping. If f and T commute, then there exists z ∈ Ksuch that f (z) = z ∈ T (z).

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Remark 3.3 Since f is continuous and T may be non convex valued, Theorem 3.1 generalizesTheorem 4.2 in [2], Theorem 3.1 in [4], Theorem 5.3 in [10], Theorem 3.1 in [15], Theorem5.2 in [11].

Remark 3.4 Since K is a closed bounded convex and T may be non convex valued, Theorem3.1 generalizes Theorem 3 in [1], Theorem 4.2 in [3] and Theorem 3.3 in [6].

In the following, we generalize Theorem 1.2 to a weakly commuting pair of single andmultivalued mappings.

Theorem 3.5 Let K be a nonempty closed bounded convex subset of a strictly convex Banachspace X. Let f : K −→ K be a quasi-nonexpansive condensing mapping and T : K −→CV (K ) be an upper semicontinuous mapping. If f and T commute weakly, then there existsz ∈ K such that f (z) = z ∈ T (z).

The following Lemma is needed for establish Theorem 3.5, his proof is similar to Theorem5.2.27 in [16].

Lemma 3.6 Let K be a convex subset of a strictly convex Banach space X and f be aquasi-nonexpansive mapping of K into X. If Fi x( f ) is nonempty then it is a convex subsetof K .

Proof of Theorem 3.5. By Furi and Vignoli’s fixed point theorem [24], Fix( f ) is a nonemptyclosed subset of K . Since f is condensing and by Lemma 3.6, one can deduce that Fix( f ) isa nonempty compact convex subset of K . Let x ∈ Fix( f ), since f and T commute weaklyand f is quasi-nonexpansive, the unique nearest point of T (x) to x is a fixed point of f . Itfollows that Fix( f ) ∩ T (x) is nonempty. Define a multivalued mapping S of Fix( f ) intoCoV (Fix( f )) by S(x) = Fix( f )∩T (x). Since S is upper semi-continuous, by Bohnenblustand Karlin’s fixed point Theorem, there exists z ∈ S(z). Thus f (z) = z ∈ T (z). ��Corollary 3.7 Let K be a nonempty closed bounded convex subset of a strictly convex Banachspace X. Let f : K −→ K be a condensing mapping which satisfies condition (Cλ) andT : K −→ CV (K ) be an upper semicontinuous mapping. If f and T commute weakly, thenthere exists z ∈ K such that f (z) = z ∈ T (z).

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