Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications

and Applied Analysis 3 Remark 1.1. If E is a reflexive strictly convex and smooth Banach space, then for x, y ∈ E, φ x, y 0 if and only if x y. It is sufficient to show that if φ x, y 0 then x y. From 1.6 , we have ‖x‖ ‖y‖. This implies 〈x, Jy〉 ‖x‖2 ‖Jy‖2. From the definitions of j, we have Jx Jy, that is, x y, see 8, 9 for more details. Let C be a nonempty closed convex subset of a smooth Banach space E and T : C → C a multivalued mapping such that Tx is nonempty for all x ∈ C. A point p is called a fixed point of T , if p ∈ Tp. The set of fixed points of T is represented by F T . A point p ∈ C is called an asymptotic fixed point of T , if there exists a sequence {xn} in Cwhich converges weakly to p and limn→∞d xn, Txn 0, where d xn, Txn infu∈Txn‖xn − u‖. The set of asymptotic fixed points of T is represented by F̂ T . Moreover, a multivalued mapping T : C → C is called relatively nonexpansive multivalued mapping, if the following conditions are satisfied: 1 F T / ∅, 2 φ p, z ≤ φ p, x , for all p ∈ F T , for all x ∈ C, for all z ∈ Tx, 3 F̂ T F T . In 4 , authors also give an example which is a relatively nonexpansive multivalued mapping but not a nonexpansive multivalued mapping. The purpose of this paper is to present the notion of weak relatively nonexpansive multivalued mapping and to prove the strong convergence theorems for the weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivaluedmapping but not a relatively nonexpansivemultivaluedmapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized metric projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. We first give the definition of weak relatively nonexpansive multivalued mapping as follows. Let C be a nonempty closed convex subset of a smooth Banach space E and T : C → C a multivalued mapping such that Tx is nonempty for all x ∈ C. A point p ∈ C is called an strong asymptotic fixed point of T , if there exists a sequence {xn} in C which converges strongly to p and limn→∞d xn, Txn 0, where d xn, Txn infu∈Txn‖xn − u‖. The set of the strong asymptotic fixed points of T is represented by F̃ T . Moreover, a multivaluedmapping T : C → C is called weak relatively nonexpansive multivalued mapping, if the following conditions are satisfied: I F T / ∅, II φ p, z ≤ φ p, x , for all p ∈ F T , for all x ∈ C, for all z ∈ Tx, III F̃ T F T . We need the following Lemmas for the proof of our main results. Lemma 1.2 see Kamimura and Takahashi 7 . Let E be a uniformly convex and smooth Banach space and let {xn},{yn} be two sequences of E. If φ xn, yn → 0 and either {xn} or {yn} is bounded, then xn − yn → 0. 4 Abstract and Applied Analysis Lemma 1.3 see Alber 5 . Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E. Then, x0 ΠCx if and only if 〈 x0 − y, Jx − Jx0 〉 ≥ 0, for y ∈ C. 1.7 Lemma 1.4 see Alber 5 . Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E, and let x ∈ E. Then φ ( y,Πcx ) φ Πcx, x ≤ φ ( y, x ) , ∀y ∈ C. 1.8 Lemma 1.5. Let E be a uniformly convex and smooth Banach space, let C be a closed convex subset of E, and let T : C → C be a weak relatively nonexpansive multivalued mapping. Then F T is closed and convex. Proof. First, we show F T is closed. Let {xn} be a sequence in F T such that xn → q. Since T is weak relatively nonexpansive multivalued mapping, we have φ xn, z ≤ φ ( xn, q ) , ∀z ∈ Tq, n 1, 2, 3, . . . . 1.9