Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces

https://doi.org/10.1016/j.na.2006.03.023Get rights and content

Abstract

Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T:CK(E) a nonexpansive mapping. For uC and t(0,1), let xt be a fixed point of a contraction Gt:CK(E), defined by GtxtTx+(1t)u,xC. It is proved that if C is a nonexpansive retract of E, {xt} is bounded and Tz={z} for any fixed point z of T, then the strong limt1xt exists and belongs to the fixed point set of T. Furthermore, we study the strong convergence of {xt} with the weak inwardness condition on T in a reflexive Banach space with a uniformly Gâteaux differentiable norm.

Introduction

Let E be a Banach space and C a nonempty closed subset of E. We shall denote by F(E) the family of nonempty closed subsets of E, by CB(E) the family of nonempty closed bounded subsets of E, by K(E) the family of nonempty compact subsets of E, and by KC(E) the family of nonempty compact convex subsets of E. Let H(,) be the Hausdorff distance on CB(E), that is, H(A,B)=max{supaAd(a,B),supbBd(b,A)} for all A,BCB(E), where d(a,B)=inf{ab:bB} is the distance from the point a to the subset B. A multivalued mapping T:CF(E) is said to be a contraction if there exists a constant k[0,1) such that H(Tx,Ty)kxy for all x,yC. If (1) is valid when k=1, the T is called nonexpansive. A point x is a fixed point for a multivalued mapping T if xTx. Banach’s Contraction Principle was extended to a multivalued contraction by Nadler [13] in 1969. Given a uC and a t(0,1), we can define a contraction Gt:CK(C) by GtxtTx+(1t)u,xC. Then Gt is multivalued and hence it has a (non-unique, in general) fixed point xtC (see [13]): that is xttTxt+(1t)u. If T is a single valued, we have xt=tTxt+(1t)u. (Such a sequence {xt} is said to be an approximating fixed point of T since it possesses the property that if {xt} is bounded, then limt1Txtxt=0.) The strong convergence of {xt} as t1 for a single-valued nonexpansive self- or nonself-mapping T was studied in Hilbert space or certain Banach spaces by many authors (see for instance, Browder [2], Halpern [7], Jung and Kim [8], Jung and Kim [9], Kim and Takahashi [10], Reich [17], Singh and Watson [20], Takahashi and Kim [23], Xu [25], and Xu and Yin [28]).

Let ytTxt be such that xt=tyt+(1t)u. Now a natural question arises of whether Browder’s theorem can be extended to the multivalued case. A simple example given by Pietramala [14] shows that the answer is negative even if E is Euclidean.

Example 1

Let C=[0,1]×[0,1] be the square in the real plane and T:CK(C) be defined by T(a,b)=the triangle with vertices(0,0),(a,0),(0,b),(a,b)C. Then it is easy to see that for any (ai,bi)C,i=1,2, H(T(a1,b1),T(a2,b2))=max{|a1a2|,|b1b2|}(a1,b1)(a2,b2), showing that T is nonexpansive. It is also easy to see that the fixed point set of T is F(T)={(a,0):0a1}{(0,b):0b1}. Let u=(1,0). Then the mapping Gt defined by (2) has the fixed point set F(Gt)={(a,0):1ta1}. Let xt={(1n,0),ift=11n(1,0)otherwise. Then {xt} satisfies (3) but does not converge. This example also shows that the sequence {F(Gt)} of fixed point sets of Gt’s does not converge as t1 to the fixed point set F(T) of T under the Hausdorff metric. However, López Acedo and Xu [12] gave the strong convergence of {xt} under the restriction F(T)={z} in Hilbert space. Kim and Jung [11] extended the result of López Acedo and Xu [12] to a Banach space with a sequentially continuous duality mapping. Recently, Sahu [19] also studied the multivalued case in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm.

In this paper, we establish the strong convergence of {xt} defined by (3) for the multivalued nonexpansive nonself-mapping T in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We also study the strong convergence of {xt} for the multivalued nonexpansive nonself-mapping T satisfying the inwardness condition in a reflexive Banach space with a uniformly Gâteaux differentiable norm. Our results improve and extend the results in [8], [9], [25], [28] to the multivalued case. We also point out that the condition F(T)={z} should be included in the main results of Sahu [19].

Section snippets

Preliminaries

Let E be a real Banach space with norm and let E be its dual. The value of xE at xE will be denoted by x,x.

A Banach space E is called uniformly convex if δ(ε)>0 for every ε>0, where the modulus δ(ε) of convexity of E is defined by δ(ε)=inf{1x+y2:x1,y1,xyε} for every ε with 0ε2. It is well known that if E is uniformly convex, then E is reflexive and strictly convex (cf. [5]).

The norm of E is said to be Gâteaux differentiable (and E is said to be smooth) if limt0x+ty

Main results

In this section, we first prove a strong convergence theorem for multivalued nonexpansive nonself-mappings in a Banach space with a uniformly Gâteaux differentiable norm.

Theorem 1

Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E , and T:CK(E) a nonexpansive nonself-mapping. Suppose that C is a nonexpansive retract of E . Suppose that T(y)={y} for any fixed point y of T and that for each uC and t(0,1) , the contraction Gt

Acknowledgement

The author thanks the anonymous referee for his/her careful reading and helpful comments and suggestions, which improved the presentation of this manuscript.

References (28)

  • V. Barbu et al.

    Convexity and Optimization in Banach spaces

    (1978)
  • F.E. Browder

    Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces

    Arch. Ration. Mech. Anal.

    (1967)
  • M.M. Day

    Normed Linear Spaces

    (1973)
  • K. Deimling

    Multivalued Differential Equations

    (1992)
  • View full text