Convergence Theorems for a Finite Family of Relatively Quasi-nonexpansive Mappings and System of Equilibrium Problems

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we employ an iteration scheme introduced in [13], to prove strong convergence to a common element of the set of fixed points of a family {T i } N i=1 of L i −Lipschitzian mappings and the set of common solutions to a system of equilibrium problems, in a uniformly convex Banach space which is also uniformly smooth. We do not require that the family {T i } N i=1 be uniformly continuous, as in [13]. Introduction Let E be a real Banach space and let C be a nonempty subset of E.


Introduction
Let E be a real Banach space and let C be a nonempty subset of E. A mapping T : C → C is called Lipschitzian, if there exists L > 0 such that for all x ∈ C, p ∈ F (T ), then T is called Quasi-Nonexpansive.Observe that every nonepansive mapping with a nonempty fixed point set is a quasinonexpansive mapping.
Let E E * , be the dual space of E, defined by If E is smooth, then J is single-valued.Throughout this paper, we denote by φ, the functional on E × E defined by φ(x, y) := ||x|| 2 − 2 x, j(y) + ||y|| 2 , for all x, y ∈ C. A point y ∈ C is said to be an asymptotic fixed point of T, if C contains a sequence {x n } which converges weakly to y and lim ||x n − T x n || = 0.The set of asymptotic fixed points of T is denoted by F .We say that a mapping T is relatively nonexpansive (see for example [3][4][5] and the references therein) if the following conditions are satisfied: then T is said to be relatively quasi-nonexpansive.It is easily seen that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive maps.Several authors (see for example [9], [10] and the references therein) have studied the approximations of fixed points of relatively quasi-nonexpansive mappings.It is easy to see that in a Hilbert space,H, the classes of relatively quasi-nonexpansive and relatively nonexpansive mappings coincide.This is because φ(x, y) = ||x−y|| 2 , ∀x, y ∈ H and this implies φ(p, T x) Examples of relatively quasi-nonexpansive mappings are given in [10].Let F be a bi-function of C × C → .The equilibrium problem is to find for all y ∈ C. The set of solutions to (1.1) is denoted by EP (F ) := {x * ∈ C : F (x * , y) ≥ 0 ∀y ∈ C}.Many problems in Engineering, Economics and optimization are reduced to finding a solution of (1.1).Several methods have been proposed for solving (1.1) (see for example [2], [6] and the references therein).Takahashi and Zembayashi [12] introduced a hybrid iterative scheme for the approximation of fixed points of relatively nonexpansive mappings which are also solutions to equilibrium problems, in a uniformly smooth real Banach space which is also uniformly convex.Recently, Yekini [13] proved strong convergence theorems for a finite family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a real uniformly convex Banach space which is also uniformly smooth.More precisely, he proved the following theorem: Theorem 1 [13]: Let E be a real uniformly convex Banach space which is also uniformly smooth.Let C be a nonempty closed convex subset of E. For each k = 1, 2, ..., m let F k be a function from C × C → , satisfying (A 1 ) − (A 4 ) and let {T i } N i=1 be a finite family of closed relatively quasi-nonexpansive mappings of C into itself such that Assume that T i is uniformly continuous for each i = 1, 2, 3, ..., N. Let {x n } be iteratively generated by where J is the duality mapping on E and T n := T n(modN) .Suppose {α n } is a sequence in (0, 1) such that lim inf α n (1 − α n ) > 0 and {r k,n }, (k = 1, 2, ..., m) satisfying lim inf r k,n > 0, (k = 1, 2, ..., m).Then {x n } converges strongly to Π F x 0 .Motivated by the above theorem, we prove strong convergence theorems for a finite family {T i } N i=1 of L i -Lipschitzian relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex Banach space which is also uniformly smooth.Our finite family {T i } N i=1 is different from that in [13], as ours is a family of L i -Lipschitzian mappings as opposed to the uniform continuity assumed for the family {T i } N i=1 in [13].

Preliminaries
Let E be a real Banach space.The modulus of smoothness of E is the function

E is uniformly smooth if and only if lim
E is said to be uniformly convex if for any ∈ (0, 2], there exists a δ = δ( Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Then the generalized projection Π C from satisfying (A 1 ) − (A 4 ).Let r > 0 and x ∈ E. Then there exists z ∈ C such that F (z, y) + 1 r y − z, Jz − Jx ≥ 0, for all y ∈ C. Takahashi and Zembayashi [11], proved the following lemma: Lemma 7 [11]: Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space, E. Assume that F : C × C → satisfies (A 1 ) − (A 4 ).For r > 0 and x ∈ E, define a mapping T F r : E → C as follows: for all z ∈ E.Then, the following hold: 1. T F r is single -valued 2. T F r is a firmly nonexpansive-type mapping, i.e. for all x, y ∈ E,

Main Results
Theorem 2: Let E be a real uniformly convex Banach space which is also uniformly smooth.Let C be a nonempty closed convex subset of E. For each k = 1, 2, ..., m let F k be a function from C × C → , satisfying (A 1 ) − (A 4 ) and let {T i } N i=1 be a finite family of closed relatively quasi-nonexpansive mappings of C into itself such that where J is the duality mapping on E and T n := T n(modN) .Suppose {α n } is a sequence in (0, 1) such that lim inf α n (1 − α n ) > 0 and {r k,n }, (k = 1, 2, ..., m) satisfying lim inf r k,n > 0, (k = 1, 2, ..., m).Then {x n } converges strongly to Π F x 0 .Proof: The facts that C n is closed and convex, {x n } is a Cauchy sequence in C, lim n→∞ ||x n+l − x n || = 0 ∀ l = 1, 2, ..., N; lim ||x n − T n x n || = 0 all follow from the proof of theorem 3.1 of [13].The rest of the proof now follows as from (3.7) of [13].
Remark: The classes of lipschitzian and uniformly continuous maps are independent of each other.Hence our results complement the results in [13].