A Modified Thakur Three-Step Iterative Algorithm to Garcia-Falset Mappings and Variational Inequalities

In this paper, we suggest and analyze a modified Thakur three-step iterative algorithm to approximate a common element of the set of common fixed points of Garcia-Falset mappings and the set of solutions of some variational inequalities in Banach spaces. We also establish strong convergence theorems for a common solution of the above-said problems by the proposed iterative algorithm without the compactness assumption. The methods in this paper are novel and different from those given in many other papers. And the results are the extension and improvement of the recent results announced by many others.


Introduction
In the early 1960s, Stampacchia [1] introduced the variational inequalities theory, which has emerged as an interesting branch of applied mathematics with a wide range of applications in industry, physics, optimization, social and science. The variational inequalities are closely related with many general problems of Nonlinear Analysis, such as fixed point, complementarity and optimization problems.
It has been extended and generalized in several directions using novel and new techniques.
On the other hand, the theory of fixed points has become one of the very po-fixed points are of great importance for modern numerical mathematics (see, e.g., [2] [3] [4] [5]).
The study for variational inequalities, fixed points and approximation algorithms became a topic of intensive research efforts in recent years. Nowadays, this is still one of the most active fields in mathematics. Meanwhile, the nature of many practical problems arouses an iterative approach to the solution. Recently, Garcia-Falset et al. [6] introduced a new class of mappings satisfying the so-called condition (E) (in the sequel, the class of mappings satisfying condition (E) will be referred to as Garcia-Falset mappings). The class of Garcia-Falset mappings covers the class of Suzuki mappings and nonexpansive mappings.
However it is still included by quasi-nonexpansiveness. The study for Garcia-Falset mappings with iterative processes using a Banach space as underlying setting is only at the beginning (more precisely; there are just two research papers on uniformly convex Banach spaces that connect mappings endowed with property (E) (see [7] [8]). In order to establish strong convergence results for approximation of fixed points of Garcia-Falset mappings in Banach space, Gabriela et al. [7] and Houmani and Turcanu [8] adopt different iteration schemes, respectively. However, the compactness assumption imposed on C is indispensable in both two papers.
Motivated and inspired by the work in the literature, we suggest and analyze a modified Thakur three-step iterative algorithm to approximate a common element of the set of common fixed points of Garcia-Falset mappings and the set of solutions of some variational inequalities in Banach spaces. We also establish strong convergence theorems for a common solution of the above-said problems by the proposed iterative algorithm without the compactness assumption. The methods in this paper are novel and different from those given in many other papers. And the results are the extension and improvement of the recent results in the literature; see [9] [10] [11] [12].

Preliminaries
Throughout this paper, we assume that E is a real Banach space with a dual * E ,  is the set of real numbers, , ⋅ ⋅ is the generalized duality pairing between E for all x E ∈ . If E is a Hilbert space, then J I = , where I is the identity mapping.
A Banach space E is said to be smooth if the limit x y on the unit sphere ( ) { } Assume ϕ defined 2 In the case that ( ) , t t J J ϕ ϕ = = , where J is the normalized duality mapping.
Remark 2.1. It is well known that J ϕ is single-valued if and only if ( ) , E ⋅ is smooth (see, e.g., [17] is said to be endowed with ( E µ )-property. Moreover, we say that T satisfies condition (E) on C, whenever T satisfies condition ( E µ ), for some Clearly, condition (E) is weaker than condition (C).
Lemma 2.4. [6] Let E be a real Banach space with the Opial condition. Let C be a nonempty closed convex subset of E and : T C E → be a mapping satisfying condition (E) on C. Then T is demiclosed at zero, i.e., for any sequence Recall that, if C and D are nonempty subsets of a Banach space E such that C is closed convex and D C ⊂ , then a mapping : Furthermore, P is a sunny nonexpansive retraction from C onto D if P is retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma collects some properties of the sunny nonexpansive retraction. Lemma 2.5. [18] [19] Let C be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C. Let : be a retraction and let j be the normalized duality mapping and generalized duality mapping on E, respectively. Then the following are equivalent: 1) P is sunny and nonexpansive; Lemma 2.6. [20] Let E be a uniformly convex Banach space,

Main Results
Theorem 3.1. Let C be a nonempty closed convex subset of a uniformly convex and smooth Banach space E which admits a weakly continuous duality mapping J. Let Then the sequence { } n x converges strongly to a point ( ) , which is also the unique solution of the hierarchical variational inequality In other words, p is the unique fixed point of the mapping Proof. We divide the proof into two steps.
Step 1. Firstly, we prove that the sequence { } In the same way, we get that   Step 2. We show that lim 0 which is reduced to the inequality which implies that  Following an argument similar to that in Case A and noticing (3.14), we derive  Then the sequence { } n x converges strongly to a point ( ) , which is also the unique solution of the hierarchical variational inequality In other words, p is the unique fixed point of the mapping Corollary 4.2. Let C be a nonempty closed convex subset of a uniformly convex and smooth Banach space E which admits a weakly continuous duality mapping J. Let : T C C → be a nonexpansive mapping with

( )
Fix T ≠ ∅ . For arbitrarily given 0 , be the sequence generated iteratively by: Then the sequence { } n x converges strongly to a point ( ) , which is also the unique solution of the hierarchical variational inequality In other words, p is the unique fixed point of the mapping

Conclusion
The present work has been aimed to theoretically establish a new iterative scheme for finding a common element of the set of common fixed points of generalized nonexpansive mappings enriched with property (E) and the set of solutions of some variational inequalities in Banach spaces without the compactness assumption. Our results can be viewed as improvement, supplementation, development and extension of the corresponding results in some references to a great extent.