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Hybrid method for a class of accretive variational inequalities involving nonexpansive mappings
Journal of Inequalities and Applications volume 2014, Article number: 217 (2014)
Abstract
In this paper we use contractions to regularize a class of accretive variational inequalities and prove the strongly convergence in Banach spaces. We extend the result of Lu et al. (Nonlinear Anal. 71:1032-1041, 2009) to the framework of Banach spaces.
MSC:47H05, 47H09, 65J15.
1 Introduction
Let H be a Hilbert space, C be a nonempty closed convex subset of H, and a nonlinear mapping. The set of fixed points of F is denoted by , i.e., . A monotone variational inequality problem is to find a point with the property
where F is a monotone operator.
Recently, Lu et al. [1] were concerned with a special class of variational inequalities in which the mapping F is the complement of a nonexpansive mapping and the constraint set is the set of fixed points of another nonexpansive mapping. Namely, they considered the following type of monotone variational inequality (VI) problem:
where are nonexpansive mappings and .
Hybrid methods for solving VI (1) were studied by Yamada [2], where F is Lipschitzian and strongly monotone. However, his methods do not apply to the variational inequality (2) since the mapping fails, in general, to be strongly monotone, though it is Lipschitzian. Therefore, other hybrid methods have to be sought. Recently, Moudafi and Mainge [3] studied VI (2) by regularizing the mapping and defined as the unique fixed point of the equation
Since Moudafi and Mainge’s regularization depends on t, the convergence of the scheme (3) is more complicated. Very recently, Lu et al. [1] studied VI (2) by regularizing the mapping S and defined as the unique fixed point of the equation
Note that Lu et al.’s regularization (4) no longer depends on t.
Motivated and inspired by the result of Lu et al. [1], we put forward a question: Can this implicit hybrid method [1] in Hilbert spaces be extended to the framework of Banach spaces? In this paper, we give a positive answer.
Throughout this paper, we always assume that E is a real Banach space. Let C be a nonempty closed convex subset of E. Let be a nonlinear mapping.
In this paper, we consider the following type of accretive variational inequality problem:
where are two nonexpansive mappings with the set of fixed point . Let Ω denote the set of solutions of VI (5) and assume that Ω is nonempty.
2 Preliminaries
Let E be a real Banach space and J be the normalized duality mapping from E into given by
for all , where denotes the dual space of E and the generalized duality pairing between E and .
Let C be a nonempty closed convex subset of a real Banach space E. Recall the following concepts of mappings.
-
(i)
A mapping is a ρ-contraction if and the following property is satisfied:
-
(ii)
A mapping is nonexpansive provided
-
(iii)
A mapping is
-
(a)
accretive if for any there exists such that
-
(b)
strictly accretive if F is accretive and the equality in (a) holds if and only if ;
-
(c)
β-strongly accretive if for any there exists such that
-
(a)
for some real constant .
Let be a continuous strictly increasing function such that and as . This function φ is called a gauge function. The duality mapping associated with a gauge function φ is defined by
In the case that , , where J is the normalized duality mapping. Clearly, the relation , holds (see [4]).
Following Browder [4], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping is single valued and weak-to-weak∗ sequentially continuous (i.e., if is a sequence in E weakly convergent to a point x, then the sequence converges weakly∗ to ). It is well known that has a weakly continuous duality mapping with a gauge function for all . Set
then
where ∂ denotes the sub-differential in the sense of convex analysis.
Remark 2.1 If is weak-to-weak∗ sequentially continuous, then J is strong-to-weak∗ sequentially continuous.
Indeed, if strongly, then weakly, converges weakly∗ to and strongly. Since , , for any , we have
Letting , we have
i.e., J is strong-to-weak∗ sequentially continuous.
Lemma 2.1 ([[5], Lemma 2.1])
Assume that a Banach space E has a weakly continuous duality mapping with a gauge φ. For all , the following inequality holds:
In particular, for all ,
Lemma 2.2 (see [6])
Let C be a nonempty closed convex subset of a real Banach space E. Assume that is accretive and weakly continuous along segments; that is as . Then the variational inequality
is equivalent to the dual variational inequality
3 Main results
In this section, we introduce an implicit algorithm and prove this algorithm converges strongly to which solves VI (5). Let C be a nonempty closed convex subset of a real Banach space E. Let be a contraction and be two nonexpansive mappings. For , we define the following mapping:
It is obvious that is a contraction. So the contraction has a unique fixed point which is denoted . Namely,
Theorem 3.1 Let C be a nonempty closed convex subset of a reflexive Banach space E which has a weakly continuous duality map with the gauge φ. Let be a contraction with constant and be two nonexpansive mappings with . Suppose that the solution set Ω of VI (5) is nonempty. Let, for each , be defined implicitly by (6). Then, for each fixed , the net converges in norm, as , to a point . Moreover, as , the net converges in norm to the unique solution of the following variational inequality:
Hence, for each null sequence in , there exists another null sequence in , such that the sequence in norm as .
Proof Step 1. For each fixed , the net is bounded.
For any , we have
Combining the above inequality and Lemma 2.1, we obtain
which implies that
Taking , then and , from (8) we have
which implies that
So for each fixed , is bounded, furthermore , and are all bounded.
Step 2. as .
From (6) and the boundedness of the sequences , and , for each fixed we have
Assume that is such that (). From (8), for any , we have
Since is bounded, without loss of generality, we may assume that converges weakly to a point as . This together with (10) implies that . Taking in (11), we have
Since is weakly continuous, it follows from (12) that as , which implies that strongly. This has proved the relative norm compactness of the net as .
Taking in (9), we have
Since is weakly continuous, then by Remark 2.1, J is strong-to-weak∗ sequentially continuous. Let in the above inequality, we have
Hence we obtain
This together with Lemma 2.2, we have
Next, we prove that the entire net converges strongly to as . We assume that where . Similar to the above proof, we have and
Taking and in (13) and (14), respectively, we have
Adding up the above two inequalities yields
Since
we obtain
i.e., . So the entire net converges in norm to as .
Step 3. The net is bounded.
For any , taking in (13), we have
which together with the fact of implies that
Since is strongly accretive and is accretive, we obtain
It follows from (15)-(17) that
Hence we have
Step 4. The net which solves VI (7).
First, the uniqueness of the solution of VI (7) is obvious. We denote the unique solution by .
Next we prove that , i.e., if is a null sequence in such that weakly as , then . Indeed, since , then . Since is accretive, for any we have
It follows from (13) that
By virtue of (19) and (20), we have
furthermore, we get
Letting () in the above inequality, since is bounded and φ is a continuous strictly increasing function, we have
This implies that
hence from the above inequality and Lemma 2.2, we have
i.e., .
Next we show that is the solution of VI (7). Taking and in (18), we obtain
which implies that
Since weakly and is weakly continuous, let in (21), we get
which together with the property of φ implies that in norm. It follows from (15) and (17) that
Since J is strong-to-weak∗ sequentially continuous and f is a contraction, we have
Letting () in (22) and combining (23) we have
So is the solution of VI (7). By uniqueness, we have . Therefore, in norm as . The proof is complete. □
References
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Acknowledgements
The first author was supported by Zhejiang Provincial Natural Science Foundation of China under Grant (no. LQ13A010007, no. LY14A010006) and the China Postdoctoral Science Foundation Funded Project (no. 2012M511928).
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Wang, Y., Xu, Hk. Hybrid method for a class of accretive variational inequalities involving nonexpansive mappings. J Inequal Appl 2014, 217 (2014). https://doi.org/10.1186/1029-242X-2014-217
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DOI: https://doi.org/10.1186/1029-242X-2014-217