1. 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 powerful tools of nonlinear analysis. Further, by the development of accurate and efficient techniques for computing fixed points, the effectiveness of the concept for applications has been increased enormously. In recent years, the theory of fixed points has grown rapidly into a flourishing and dynamic field of study both in pure and applied mathematics. It has become one of the most essential tools in the study of nonlinear phenomena. The iterative methods for approximating 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].
2. Preliminaries
Throughout this paper, we assume that E is a real Banach space with a dual , is the set of real numbers, is the generalized duality pairing between E and , I is the identity mapping on E, and is the set of nonnegative integers. We denote by and the strong and weak convergence of the sequence , respectively. And denote the set of weak limit points of the sequence . The set of fixed points of is denoted by . The (normalized) duality mapping of E is denoted by J, that is,
for all . If E is a Hilbert space, then , where I is the identity mapping.
A Banach space E is said to be smooth if the limit
exists for all on the unit sphere .
Assume defined is a continuous strictly increasing function such that and . This function is called a gauge function. The duality mapping defined by
In the case that , where J is the normalized duality mapping. Clearly, the relation holds (see, e.g., [13] ). Browder [14] initiated the study . Set, for ,
As we know that is the subdifferential of the convex function at x. Following Browder [14], we say that Banach space E has a weakly continuous duality map if there exists a gauge such that the duality map is single-valued and continuous from E with the weak topology to with weak* topology. Every space has a weakly continuous duality map with the gauge (see, e.g., [15] [16] ). A space with a weakly continuous duality map is easily seen to satisfy Opial’s condition (see [14] ). Conversely, if Banach space satisfies Opial’s condition and has a uniformly Gâteaux differentiable norm, then it has a weakly continuous duality mapping.
Remark 2.1. It is well known that is single-valued if and only if is smooth (see, e.g., [17] ).
Let E be a real Banach space, C a nonempty closed convex subset of a mapping on C and .
Definition 2.1. A mapping is said to be:
1) Contractive if there exists a constant such that
2) Nonexpansive if for all .
3) Quasinonexpansive if .
Definition 2.2. A mapping is said to satisfy condition (C) on C if for all with , one has that .
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-generalized nonexpansive mappings or Garcia-Falset mappings).
Definition 2.3. Let C be a nonempty subset of a Banach space E and . A mapping which satisfies the inequality
is said to be endowed with ( )-property. Moreover,we say that T satisfies condition (E)on C,whenever T satisfies condition ( ),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 be a mapping satisfying condition (E) on C. Then T is demiclosed at zero, i.e., for any sequence , if and , then .
Recall that, if C and D are nonempty subsets of a Banach space E such that C is closed convex and , then a mapping is sunny [4] provided
for all and , whenever . A mapping is called a retraction if for all . 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;
2) ;
3) .
Lemma 2.6. [20] Let E be a uniformly convex Banach space, a positive number, and a closed ball of E. Then, for any given subset and for any positive numbers with , there exists a continuous, strictly increasing, and convex function with such that, for any with ,
Lemma 2.7. [21] Let be a sequence of nonnegative real numbers, let be a sequence of [0, 1] with , let be a sequence of nonnegative real numbers with , and let be a sequence of real numbers with . Suppose that
Then, .
Lemma 2.8. [22] Let be a sequence of real numbers such that there exists a subsequence of which satisfies for all . Also consider the sequence of positive integers defined by
for all (for some large enough). Then is a nondecreasing sequence such that and it holds that
3. 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 be a contractive mapping with constant , be a mapping satisfying condition (E) with . For arbitrarily given , let be the sequence generated iteratively by:
(3.1)
where , , , and are real number sequences in [0, 1] satisfying:
1) and ,
2) ,
3) , and .
Then the sequence 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 ,that is, .
Proof. We divide the proof into two steps.
Step 1. Firstly, we prove that the sequence is bounded. Taking arbitrarily, it follows by Definition 2.3 that T is quasi-nonexpansive. Hence, we get from (3.1) that
(3.2)
In the same way, we get that
(3.3)
It follows from (3.1), (3.2) and (3.3) that
This implies that the sequence is bounded, so are and .
Step 2. We show that . Here again , which is also the unique solution of the hierarchical variational inequality
We analyze this step by considering the following two cases.
Case A: Put for all and assume that for all (for large enough). In this case, it is easily seen that the exists. Now we prove that
To see this, we apply Lemma 2.6 to (3.1) to get
(3.4)
which is reduced to the inequality
Then, by using the conditions (1)-(3) and the assumption , we derive that
(3.5)
It follows from the property of g that
(3.6)
Repeat the argument for (3.4) to obtain
(3.7)
which implies that
(3.8)
Hence, by using the conditions (1)-(3), the assumption and the property of g, we derive that
(3.9)
Write
and apply the condition (1) and (3.9) to get
(3.10)
Note that
(3.11)
Apply the condition (1), (3.6) and (3.10) to get
(3.12)
Since is bounded, there exists a subsequence of such that converges weakly to a point q and moreover
(3.13)
Apply (3.6), (3.12) and Lemma 2.4 to infer that converges weakly to a point q and . This together with the property of the sunny nonexpansive retraction implies
Finally, we prove that . Using (3.1) and the assumption , we have that, for all ,
By virtue of (3.13) and Lemma 2.7 and noticing (3.14), we get
Case B. Assume that is nondecreasing. From Lemma 2.8, there exists a nondecreasing sequence such that
(3.14)
Following an argument similar to that in Case A and noticing (3.14), we derive that
(3.15)
and
(3.16)
Repeat the argument for (3.13) to obtain
(3.17)
Finally, we show that . It follows (3.1) and (3.17) that
(3.18)
After simplifying, we have
This together with the (3.17) implies that
Further, Lemma 2.8 implies
that is, as . This completes the proof.
Remark 3.2. The main results in this paper extend and generalize corresponding results in [7] [8] in the following senses:
1) The subset C of Banach space E does not have to be compact in our Theorem 3.1. However,this assumption is very necessary in Theorem 3.4 of Usurelu et al. [7] and Corollary 2of Houmani and Turcanu [8].
2)Our result is new and the proofs are simple and different from those in [7] [8].
4. An Extension of Our Main Results
From Theorem 3.1, we deduce immediately the following results
Corollary 4.1. 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, be a contractive mapping with constant , be a nonexpansive mapping with . For arbitrarily given , let be the sequence generated iteratively by:
(4.1)
where , , , and are real number sequences in [0, 1] satisfying:
1) and ,
2) ,
3) , and .
Then the sequence 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 , that is, .
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 be a nonexpansive mapping with . For arbitrarily given , let be the sequence generated iteratively by:
(4.2)
where , , , and are real number sequences in [0, 1] satisfying:
1) and ,
2) ,
3) , and .
Then the sequence 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 ,that is, .
Corollary 4.3. Let C be a nonempty closed convex subset of a Hilbert space H, be a contractive mapping with constant , be a nonexpansive mapping with . For arbitrarily given , let be the sequence generated iteratively by:
(4.3)
where , , , and are real number sequences in [0, 1] satisfying:
1) and ,
2) ,
3) , and .
Then the sequence 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 ,that is, .
5. 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.
Acknowledgements
This research was supported by the Key Scientific Research Projects of Higher Education Institutions in Henan Province (20A110038).
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