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Dual Convergences of Iteration Processes for Nonexpansive Mappings in Banach Spaces

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Dual Convergences of Iteration Processes for Nonexpansive Mappings in Banach Spaces Czechoslovak Mathematical Journal Jong Soo Jung; Daya Ram Sahu Dual convergences of iteration processes for nonexpansive mappings in Banach spaces Czechoslovak Mathematical Journal, Vol. 53 (2003), No. 2, 397–404 Persistent URL: http://dml.cz/dmlcz/127808 Terms of use: © Institute of Mathematics AS CR, 2003 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 53 (128) (2003), 397–404 DUAL CONVERGENCES OF ITERATION PROCESSES FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES ¢¡ £ ¤¦¥§¡¨¡¦ © £ ¤ ¥ © , Busan, and , Bhilai (Received May 30, 2000) Abstract. In this paper we establish a dual weak convergence theorem for the Ishikawa iteration process for nonexpansive mappings in a reflexive and strictly convex Banach space with a uniformly Gˆateaux differentiable norm, and then apply this result to study the problem of the weak convergence of the iteration process. Keywords: Banach limit, dual convergence theorem, duality mapping, Ishikawa iteration process, nonexpansive mapping MSC 2000 : 47H09, 47H10 1. Introduction Let D be a nonempty subset of a Banach space E. A mapping T : D E is said → to be nonexpansive if T x T y x y k − k 6 k − k for all x, y in D. In [4], Ishikawa introduced a new iteration process as xn+1 = (1 tn)xn + tnT yn (1) − ( yn = (1 sn)xn + snT xn, n = 1, 2, . , − where t and s are sequences in [0, 1] satisfying certain restrictions. If s = 0, { n} { n} n the Ishikawa iteration process reduces to the Mann iteration process [7] x = (1 t )x + t T x , n = 1, 2, . n+1 − n n n n This paper was supported by Korea Research Foundation Grant (KRF-99-041-D00030). 397 The convergence of the sequence x defined by (1) for a nonexpansive self- { n} mapping T in a uniformly convex Banach space with a Fréchet differentiable norm has been studied by Deng [2], Reich [8], and Tan and Xu [10]. In this paper we first establish a dual weak convergence theorem for the Ishikawa iteration process (1) for nonexpansive mappings in a reflexive and strictly convex Banach space with a uniformly Gˆateaux differentiable norm. Then we apply this result to study the weak convergence of the iteration process (1) in the same Banach space. Further, we obtain the dual weak convergence for the iteration process (1) under different restrictions on t and s in the same Banach space. We should { n} { n} point out that the Banach space E in the main results is not assumed to be uniformly convex and that our results also apply to all Lp spaces (1 < p < ). ∞ 2. Preliminaries and Lemmas Let E be a real Banach space and let I denote the identity operator. Recall that a Banach space E is said to be smooth provided the limit x + ty x lim k k − k k t 0 t → exists for each x and y in U = x E : x = 1 . In this case, the norm of E is { ∈ k k } said to be Gˆateaux differentiable. It is said to be uniformly Gˆateaux differentiable if for each y U, this limit is attained uniformly for x U. The norm is said to be ∈ ∈ Fréchet differentiable if for each x U, this limit is attained uniformly for y U . ∈ ∈ Finally, the norm is said to be uniformly Fréchet differentiable if the limit is attained uniformly for (x, y) U U. In this case, E is said to be uniformly smooth. Since ∈ × the dual E∗ of E is uniformly convex if and only if the norm of E is uniformly Fréchet differentiable, every Banach space with a uniformly convex dual is reflexive and has a uniformly Gˆateaux differentiable norm. The reverse is false (cf. [9]). The duality mapping from E into the family of nonempty subsets of its dual E∗ is defined by 2 2 J(x) = x∗ E∗ : (x, x∗) = x = x∗ . { ∈ k k k k } It is single valued if and only if E is smooth. If E is smooth, the duality mapping J is said to be weakly sequentially continuous at 0 if J(x ) converges to 0 in the { n } sense of the weak-star topology of E∗, whenever x converges weakly to 0 in E. { n} Let µ be a mean on positive integers , i.e., a continuous linear functional on `∞ satisfying µ = 1 = µ(1). Then we know that µ is a mean on if and only if k k inf a : n µ(a) sup a : n { n ∈ } 6 6 { n ∈ } 398 for every a = (a , a , . .) `∞. According to time and circumstances, we use µ (a ) 1 2 ∈ n n instead of µ(a). A mean µ on is called a Banach limit if µn(an) = µn(an+1) for every a = (a , a , . .) `∞. Using the Hahn-Banach theorem, we can prove the 1 2 ∈ existence of a Banach limit. We know that if µ is a Banach limit, then lim inf an 6 µn(an) 6 lim sup an n n →∞ →∞ for every a = (a , a , . .) `∞. Let x be a bounded sequence in E. Then we can 1 2 ∈ { n} define a real valued continuous convex function ϕ on E by ϕ(z) = µ x z 2 nk n − k for each z E. The following lemma was proved in [5] (cf. [9]). ∈ Lemma 1. Let E be a Banach space with a uniformly Gˆateaux differentiable norm and let x be a bounded sequence in E. Let µ be a Banach limit and u E. { n} ∈ Then 2 2 µn xn u = inf µn xn z z E k − k ∈ k − k if and only if µ (z, J(x u)) = 0 n n − for all z E, where J is the duality mapping of E into E∗. ∈ In the sequel, we need the following lemmas for the proof of our main results. Lemma 2 ([2]). Let D be a subset of a normed space E and let T : D E be a → nonexpansive mapping. Let a sequence x in E be defined by { n} x = (1 t )x + t T y , n+1 − n n n n ( yn = (1 sn)xn + snT xn, n = 1, 2, . , − where two real sequences t and s satisfy the following conditions { n} { n} ∞ (i) 0 6 tn 6 t < 1 and tn = , n=1 ∞ ∞ P (ii) 0 6 sn 6 1 and sn < . n=1 ∞ If xn is bounded, thenP xn T xn 0 as n . { } k − k → → ∞ 399 Lemma 3 ([10]). Let D be a subset of a normed space E, and T : D X a → nonexpansive mapping. Let a sequence x in D be defined by { n} x = (1 t )x + t T y , n+1 − n n n n ( yn = (1 sn)xn + snT xn, n = 1, 2, . , − where 0 6 tn, sn 6 1 for all n > 1. Then x p x p k n+1 − k 6 k n − k for all n 1 and all p F (T ), where F (T ) denotes the set of fixed points of T . > ∈ Finally, we recall the following definition: a sequence a `∞ is said to { n} ∈ be almost convergent if all Banach limits agree (Lorentz’s characterization is n that lim ai+k /n exists uniformly in k 0 [6]). We also say that a se- n > →∞ i=1 quence xnPin a Banac h space E is weakly almost convergent to z E if the weak n{ } ∈ lim xi+k /n = z uniformly in k 0. n > →∞ i=1 P 3. Main results Now, we establish a dual weak convergence theorem for the Ishikawa iteration process in a reflexive and strictly convex Banach space. Theorem 1. Let E be a reflexive and strictly convex Banach space with a uni- formly Gˆateaux differentiable norm and let T : E E be a nonexpansive mapping. → Let a sequence x in E be defined by { n} x = (1 t )x + t T y , n+1 − n n n n ( yn = (1 sn)xn + snT xn, n = 1, 2, . , − where two real sequences t and s satisfy the following conditions: { n} { n} ∞ (i) 0 6 tn 6 t < 1 and tn = , n=1 ∞ ∞ P (ii) 0 6 sn 6 1 and sn < . n=1 ∞ If the fixed point setPF (T ) of T is nonempty, then there exists a point v in F (T ) such that J(x v) weakly almost converges to 0. { n − } . Let p F (T ) and let x be an initial point of x . Then x p ∈ 0 { n} k n − k 6 x p for all n and hence x is bounded by Lemma 3. Let µ be a Banach limit k 0 − k { n} and define a real valued function ϕ on E by ϕ(z) = µ x z 2 nk n − k 400 for each z E. Then ϕ is a continuous convex function and ϕ(z) as z . ∈ → ∞ k k → ∞ Since X is reflexive, ϕ attains its infimum over E (cf. [3, p. 12]). Let K = u E : ϕ(u) = inf ϕ(z) . { ∈ z E } ∈ Then it is easy to verify that K is nonempty closed convex and bounded. Further- more, K is invariant under T . In fact, by Lemma 2, lim xn T xn = 0, and so n k − k we have, for each u K, →∞ ∈ ϕ(T u) = µ x T u 2 = µ T x T u 2 µ x u = ϕ(u).
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