Modified Iterative Scheme for Multivalued Nonexpansive Mappings, Equilibrium Problems and Fixed Point Problems in Banach
In this research, we modified iterative scheme for finding common element of the set of fixed point of total quasi-\(\phi \) -asymptotically nonexpansive multivalued mappings, the set of solution of an equilibrium problem and the set of fixed point of rel
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Abstract In this research, we modified iterative scheme for finding common element of the set of fixed point of total quasi-φ-asymptotically nonexpansive multivalued mappings, the set of solution of an equilibrium problem and the set of fixed point of relatively nonexpansive mappings in Banach spaces. In addition, the strong convergence for approximating common solution of our mentioned problems is proved under some mild conditions. Our results extend and improve some recent results announced by some authors. We divide our research details into three main sections including Introduction, Preliminaries, Main Results. First, we introduce the backgrounds and motivations of this research and follow with the second section, Preliminaries, which mention about the tools that will be needed to prove our main results. In the last section, Main Results, we propose the theorem and corollary which is the most important part in our research. Keywords Banach space · Equilibrium problem · Fixed point problem · Hybrid projection method · Multivalued nonexpansive mapping · Strong convergence
1 Introduction Let C be a nonempty closed convex subset of a real Banach space E. A mapping t : C → C is said to be nonexpansive if t x − t y ≤ x − y for all x, y ∈ C. U. Witthayarat · K. Wattanawitoon · P. Kumam (B) King Mongkut’s University of Technology Thonburi, 126 Pracha-Uthit Rd., Bang Mod, Thung Khru, Bangkok, Thailand e-mail: [email protected] K. Wattanawitoon e-mail: [email protected] U. Witthayarat e-mail: [email protected] G.-C. Yang et al. (eds.), Transactions on Engineering Technologies, Lecture Notes in Electrical Engineering 275, DOI: 10.1007/978-94-007-7684-5_20, © Springer Science+Business Media Dordrecht 2014
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We denote by F(t) the set of fixed points of t, that is F(t) = {x ∈ C : x = t x}. A mapping t is said to be an asymptotic fixed point of t (see [1]) if C contains a sequence {xn } which converges weakly to p such that limn→∞ xn − t xn = 0. The set of asymptotic fixed points of t will be denoted by F(t). A mapping t from C into = F(t) and φ( p, t x) ≤ itself is said to be relatively nonexpansive [2–4] if F(t) φ( p, x) for all x ∈ C and p ∈ F(t). The asymptotic behavior of a relatively nonexpansive mapping was studied in [5, 6]. Let N (C) and C B(C) denote the family of nonempty subsets and nonempty closed bounded subsets of C, respectively. Let H : C B(C) × C B(C) → R+ be the Hausdorff distance on C B(C), that is H (A, B) = max{sup dist (a, A), sup dist (b, B)}, a∈A
b∈B
for every A, B ∈ C B(C), where dist (a, B) = inf{a − b : b ∈ B} is the distance from the point a to the subset B of C. A multi-valued mapping T : E → C B(C) is said to be nonexpansive if H (T x, T y) ≤ x − y, for all x, y ∈ C. An element p ∈ C is called a fixed point of T : C → C B(C), if p ∈ T p. The set of fixed point T is denoted by F(T ). A point p ∈ C is said to be an asymptotic fixed point of T : C → C B(C), if there exists a sequence {xn } ⊂ C such that xn x ∈ E and d(xn , T x
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