Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings

and Applied Analysis 3 Very recently, Qin et al. 16 proposed the composite Halpern type iterative scheme in a uniformly smooth Banach space as follows: x0 x, u ∈ C, zn γnxn ( 1 − γn ) Txn, yn βnxn ( 1 − βn ) Tzn, xn 1 αnu 1 − αn yn, n ≥ 0, 1.6 and showed strong convergence of the sequence {xn} generated by 1.6 under the following control conditions: i ∑∞ n 0αn ∞; ii limn→∞αn 0, limn→∞βn 0 and 0 < a ≤ γn for some a ∈ 0, 1 ; iii ∑∞ n 0|αn 1 − αn| < ∞, ∑∞ n 0|βn 1 − βn| < ∞, ∑∞ n 0|γn 1 − γn| < ∞. In this paper, under the framework of a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that everyweakly compact convex subset ofE has the fixed point property for nonexpansive mappings, we consider a new composite iterative scheme for a nonexpansive mapping T as the viscosity approximation method: for f ∈ ΣC and the initial guess x0 x ∈ C, zn γnxn ( 1 − γn ) Txn, yn βnxn ( 1 − βn ) Tzn, xn 1 αnf xn 1 − αn yn, n ≥ 0, IS where {αn}, {βn} and {γn} are sequences in 0, 1 . First, we prove under certain control conditions on the sequences {αn}, {βn} and {γn} different from those of Qin et al. 16 that the sequence {xn} generated by IS converges strongly to a fixed point of T , which is a solution of a certain variational inequality. Next we study the composite iterative scheme IS with the weakly contractive mapping instead of the contractions. The main results develop and complement the corresponding results of 2, 3, 8, 9, 11, 12, 15, 16 . In particular, if βn 0 for all n ≥ 0 in IS , then IS reduces a new viscosity iterative scheme for finding a fixed point of T : zn γnxn ( 1 − γn ) Txn, xn 1 αnf xn 1 − αn Tzn, n ≥ 0. 1.7 2. Preliminaries and Lemmas Let E be a real Banach space with norm ‖ · ‖, and let E∗ be its dual. The value of f ∈ E∗ at x ∈ E will be denoted by 〈x, f〉. When {xn} is a sequence in E, then xn → x resp., xn ⇀ x will denote strong resp., weak convergence of the sequence {xn} to x. 4 Abstract and Applied Analysis The normalized duality mapping J from E into the family of nonempty by HahnBanach theorem weak-star compact subsets of its dual E∗ is defined by J x { f ∈ E∗ : x, f ‖x‖ ∥f∥2 } , 2.1 for each x ∈ E 17 . The norm of E is said to be Gâteaux differentiable and E is said to be smooth if lim t→ 0 ∥x ty ∥∥ − ‖x‖ t 2.2 exists for each x, y in its unit sphere U {x ∈ E : ‖x‖ 1}. The norm is said to be uniformly Gâteaux differentiable if for y ∈ U, the limit is attained uniformly for x ∈ U. The space E is said to have a uniformly Fréchet differentiable norm and E is said to be uniformly smooth if the limit in 2.2 is attained uniformly for x, y ∈ U×U. It is known that E is smooth if and only if each duality mapping J is single-valued. It is also well-known that if E has a uniformly Gâteaux differentiable norm, J is uniformly norm-to-weak∗ continuous on each bounded subsets of E 17 . Let C be a nonempty closed convex subset of E. C is said to have the fixed point property for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex subset D of C has a fixed point in D. Let D be a subset of C. Then a mapping Q : C → D is said to be a retraction from C ontoD ifQx x for all x ∈ D. A retractionQ : C → D is said to be sunny if Q Qx t x −Qx Qx for all x ∈ C and t ≥ 0 with Qx t x −Qx ∈ C. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C ontoD. In a smooth Banach space E, it is well-known 18, page 48 that Q is a sunny nonexpansive retraction from C onto D if and only if the following condition holds 〈x −Qx, J z −Qx 〉 ≤ 0, x ∈ C, z ∈ D. 2.3 We need the following lemmas for the proof of our main results. Lemma 2.1 was also given in Jung and Morales 19 , Lemma 2.2 is Lemma 2 of Suzuki 20 and Lemma 2.3 is essentially Lemma 2 of Liu 21 also see 10 . Lemma 2.1. Let E be a real Banach space and let J be the duality mapping. Then, for any given x, y ∈ E, one has ∥x y ∥∥2 ≤ ‖x‖ 2y, jx y, 2.4 for all j x y ∈ J x y . Lemma 2.2. Let {xn} and {wn} be bounded sequences in a Banach space E and let {δn} be a sequence in 0, 1 which satisfies the following condition: 0 < lim inf n→∞ δn ≤ lim sup n→∞ δn < 1. 2.5 Abstract and Applied Analysis 5and Applied Analysis 5