A New Iterative Scheme for Countable Families of Weak Relatively Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth using the properties of generalized -projection operator. Using this result, we discuss strong convergence theorem concerning general -monotone mappings and system of generalized mixed equilibrium problems in Banach spaces. Our results extend many known recent results in the literature. 1. Introduction Let be a real Banach space with dual , and let be nonempty, closed and convex subset of . A mapping is called nonexpansive if A point is called a fixed point of if . The set of fixed points of is denoted by . We denote by the normalized duality mapping from to defined by The following properties of are well known (the reader can consult [1–3] for more details). (1)If is uniformly smooth, then is norm-to-norm uniformly continuous on each bounded subset of . (2) , ？ . (3)If is reflexive, then is a mapping from onto .(4)If is smooth, then is single valued. Throughout this paper, we denote by , the functional on defined by From [4], in uniformly convex and uniformly smooth Banach spaces, we have Definition 1.1. Let be a nonempty subset of and let be a countable family of mappings from into . A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by . One says that is countable family of relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1) ;(R2) , for all , ？ , ？ ;(R3) . Definition 1.2. A point is said to be a strong asymptotic fixed point of if contains a sequence which converges strongly to and . The set of strong asymptotic fixed points of is denoted by . One says that a mapping is countable family of weak relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1) ;(R2) , for all , ？ , ？ ;(R3) . Definition 1.3. Let be a nonempty subset of and let be a mapping from into . A point is said to be an asymptotic fixed point of if contains a sequence which converges weakly to and . The set of asymptotic fixed points of is denoted by . We say that a mapping is relatively nonexpansive (see, e.g., [6–11]) if the following conditions are satisfied:(R1) ;(R2) , for all , ？ ;(R3) . Definition 1.4. A point is said to be an strong asymptotic fixed