# Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces

## Abstract

We use strongly pseudocontractions to regularize a class of accretive variational inequalities in Banach spaces, where the accretive operators are complements of pseudocontractions and the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities.

Our results improve and generalize some recent results of Yao et al. (Fixed Point Theory Appl, doi:10.1155/2011/180534, 2011) and Lu et al. (Nonlinear Anal, 71(3-4), 1032-1041, 2009).

**2000 Mathematics subject classification** 47H05; 47H09; 65J15

## Keywords

hybrid method accretive operator variational inequality pseudocontraction## 1. Introduction

*E*is a real Banach space, 〈· , ·〉 is the dual pair between

*E*and

*E**, and 2

^{ E }denotes the family of all the nonempty subsets of

*E*. Let

*C*be a nonempty closed convex subset of

*E*and

*T*:

*C*→

*E*be a nonlinear mapping. Denote by

*Fix*(

*T*) the set of fixed points of

*T*, that is,

*Fix*(

*T*) = {

*x*∈

*C*:

*Tx*=

*x*}. The generalized duality mapping

*J*:

*E*→ 2

^{E*}is defined by

In the sequel, we shall denote the single-valued duality mapping by *j*. When {*x*_{ n } } is a sequence in *E*, *x*_{ n } → *x* (*x*_{ n } ⇀ *x*, *x*_{ n } ⇁ *x*) will denote strong (respectively, weak and weak*) convergence of the sequence {*x*_{ n } } to *x*.

*T*with domain

*D*(

*T*) and range

*R*(

*T*) in

*E*is called pseudocontractive if the inequality

holds for each *x*, *y* ∈ *D*(*T*) and for all *t >* 0. As a result of [1], it follows from (1.1) that *T* is pseudocontractive if and only if there exists *j* (*x - y*) ∈ *J*(*x - y*) such that 〈*Tx - Ty*, *j*(*x - y*)〉 *≤* ||*x - y*||^{2} for any *x*, *y* ∈ *D*(*T*). *T* is called strongly pseudocontractive if there exist *j*(*x - y*) ∈ *J* (*x - y*) and *β* ∈ (0, 1) such that 〈*Tx - Ty*, *j*(*x - y*)〉 *≤ β* ||*x - y*||^{2} for any *x*, *y* ∈ *D*(*T*). *T* is called Lipschitzian if there exists *L* ≥ 0 such that ||*Tx - Ty*|| *≤ L*||*x - y*||, ∀ *x*, *y* ∈ *D*(*T*). If *L* = 1, then *T* is called nonexpansive, and it is called contraction if *L* ∈ [0, 1).

*E*=

*H*be a Hilbert space with inner product 〈· , ·〉. Recall that

*T*:

*C*→

*H*is called monotone if 〈

*Tx - Ty*,

*x - y*〉

*≥*0, ∀

*x*,

*y*∈

*C*. A variational inequality problem, denoted by VI(

*T*,

*C*), is to find a point

*x** with the property

If the mapping *T* is a monotone operator, then we say that VI(*T*, *C*) is monotone.

where *T*, *S* : *C* → *C* are nonexpansive mappings and *Fix*(*T*) ≠ ∅. Let *W* denote the set of solutions of the VI(1.2).

Very recently, Yao et al. [3] considered VI(1.2) in Hilbert spaces when *T*, *S* : *C* → *C* are pseudocontractions.

where *T*, *S* : *C* → *C* are pseudocontractions. Let Ω denote the set of solutions of the VI(1.3) and assume that Ω is nonempty. Since *I - S* is accretive, then we say VI(1.3) is an accretive variational inequality.

*T*,

*C*), hybrid methods were studied by Yamada [4] where he assumed that

*T*is Lipschitzian and strongly monotone. However, his methods do not apply to the VI(1.2) since the mapping

*I - S*fails, in general, to be strongly monotone, though it is Lipschitzian. In fact the VI(1.2) is, in general, ill-posed, and thus regularization is needed. Let

*T*,

*S*:

*C*→

*C*be nonexpansive and

*f*:

*C*→

*C*be contractive. In 2006, Moudafi and Mainge [5] studied the VI(1.2) by regularizing the mapping

*tS*+ (1

*- t*)

*T*and defined {

*x*

_{s,t}} as follows:

*t*, the convergence of the scheme (1.4) is more complicated. So Lu et al. [2] defined {

*x*

_{s,t}} as follows by regularizing the mapping

*S*:

Note that Lu et al.'s regularization does no longer depend on *t*. And their result for the regularization (1.5) is under dramatically less restrictive conditions than Moudafi and Mainge's [5].

Very recently, Yao et al. [3] extended Lu et al.'s result to a general case, i.e., in the scheme (1.5), *S*, *T* are extended to Lipschitz pseudocontractive and *f* is extended to strongly pseudocontractive. But in [3], after careful discussion, we observe that a continuity condition on *f* is necessary. So, in this paper, we modify it.

Motivated and inspired by the above work, in this paper, we use strongly pseudocontrations to regularize the ill-posed accretive VI(1.3), and analyze the convergence of the scheme (1.5). The results we obtained improve and extend the corresponding results in [2, 3].

## 2 Preliminaries

If Banach space *E* admits sequentially continuous duality mapping *J* from weak topology from weak* topology, then by [6, Lemma 1], we get that duality mapping *J* is single-valued. In this case, duality mapping *J* is also said to be weakly sequentially continuous, i.e., for each {*x*_{ n } } ⊂ *E* with *x*_{ n } ⇀ *x*, then *J*(*x*_{ n } ) ⇁ *Jx*[6, 7].

*E*is said to be satisfying

*Opial's condition*if for any sequence {

*x*

_{ n }} in

*E*,

*x*

_{ n }⇀

*x*(

*n*→ ∞) implies that

By [6, Lemma 1], we know that if *E* admits a weakly sequentially continuous duality mapping, then *E* satisfies *Opial's condition*.

**Lemma 2.1**([7]) Let

*C*be a nonempty closed convex subset of a reflexive Banach space

*E*, which satisfies

*Opial's condition*, and suppose

*T*:

*C*→

*E*is nonexpansive. Then, the mapping

*I - T*is demiclosed at zero, i.e.,

Recall that *S* : *C* → *C* is called accretive if *I* - *S* is pseudocontractive. We denote by *J*_{ r } the resolvent of *S*, i.e., *J*_{ r } = (*I* + *rS*)^{-1}. It is well known that *J*_{ r } is nonexpansive, single-valued and *Fix*(*J*_{ r }) = *S*^{-1}(0) = {*z* ∈ *D*(*S*) : 0 = *Sz*} for all *r* > 0. For more details, see[8, 9, 10].

Let *T* : *C* → *C* be a pseudocontractive mapping; then, *I - T* is accretive. We denote *A* = *J*_{1} = (2*I - T*)^{-1}. Then, *Fix*(*A*) = *Fix*(*T*) and *A* : *R*(2*I - T*) → *K* is nonexpansive and single-valued. The following lemma can be found in [11].

**Lemma 2.2**([11]) Let

*C*be a nonempty closed convex subset of a real Banach space

*E*and

*T*:

*C*→

*C*be a continuous pseudocontractive map. We denote

*A*=

*J*

_{1}= (2

*I - T*)

^{-1}. Then,

- (i)[12, Theorem 6] The map
*A*is a nonexpansive self-mapping on*C*, i.e., for all*x*,*y*∈*C*, there hold$\parallel Ax-Ay\parallel \le \parallel x-y\parallel \phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}Ax\in C;$ - (ii)
If lim

_{n→∞}||*x*_{ n }-*Tx*_{ n }|| = 0, then lim_{n→∞}||*x*_{ n }-*Ax*_{ n }|| = 0.

We also need the following lemma.

**Lemma 2.3**Let

*C*be a nonempty closed convex subset of a real Banach space

*E*. Assume that

*F*:

*C*→

*E*is accretive and weakly continuous along segments; that is

*F*(

*x*+

*ty*) ⇀

*F*(

*x*) as

*t*→ 0. Then, the variational inequality

**Proof**(2.1) ⇒ (2.2) Since

*F*is accretive, we have

*x*∈

*C*, put

*w*=

*tx*+ (1

*- t*)

*x**, ∀

*t*∈ (0, 1). Then,

*w*∈

*C*. Taking

*x*=

*w*in (2.2), we have

Letting *t* → 0 in the above inequality, since *F* is weakly continuous along segments, it follows that (2.1) holds.

## 3. Main Results

*C*be a nonempty closed convex subset of a real Banach space

*E*. Let

*f*:

*C*→

*C*be a Lipschitz strongly pseudocontraction and

*T*,

*S*:

*C*→

*C*be two continuous pseudocontractions. For

*s*,

*t*∈ (0, 1), we define the following mapping

*W*

_{s,t}:

*C*→

*C*is a continuous strongly pseudocontractive mapping. So, by [13],

*W*

_{s,t}has a unique fixed point which is denoted

*x*

_{s,t}∈

*C*; that is

**Theorem 3.1**Let

*E*be a reflexive Banach space that admits a weakly sequentially continuous duality mapping from

*E*to

*E**. Let

*C*be a nonempty closed convex subset of

*E*. Let

*f*:

*C*→

*C*be a Lipschitz strongly pseudocontraction,

*S*:

*C*→

*C*be a Lipschitz pseudocontraction, and

*T*:

*C*→

*C*be a continuous pseudocontraction with

*Fix*(

*T*) ≠ ∅. Suppose that the solution set Ω of the VI(1.3) is nonempty. Let for each (

*s*,

*t*) ∈ (0, 1)

^{2}, {

*x*

_{s,t}} be defined by (3.1). Then, for each fixed

*t*∈ (0, 1), the net {

*x*

_{s,t}} converges in norm, as

*s*→ 0, to a point

*x*

_{ t }∈

*Fix*(

*T*). Moreover, as

*t*→ 0, the net {

*x*

_{ t }} converges in norm to the unique solution

*x** of the following inequality variational(denoted by VI(3.2)):

Hence, for each null sequence {*t*_{ n } } in (0,1), there exists another null sequence {*s*_{ n } } in (0,1), such that the sequence ${{x}_{{s}_{n},}}_{{t}_{n}}\to {x}^{*}$ in norm as *n* → ∞.

**Proof** We divide our proofs into several steps as follows.

**Step 1** For each fixed *t* ∈ (0, 1), the net {*x*_{s,t}} is bounded.

*z*∈

*Fix*(

*T*), for all

*s*,

*t*∈ (0, 1), by (3.1), we have

*t*∈ (0, 1), {

*x*

_{s,t}} is bounded. Furthermore, by the Lipschitz continuity of

*f*and

*S*, we obtain {

*f*(

*x*

_{s,t})} and {

*Sx*

_{s,t}}, which are both bounded for each

*t*∈ (0, 1). From (3.1), we have

So {*Tx*_{s,t}} is also bounded as *s* → 0 for each *t* ∈ (0, 1).

**Step 2** *x*_{s,t}→ *x*_{ t }∈ *Fix*(*T*) as *s* → 0.

*t*∈ (0, 1), we get

*s*

_{ n }} ⊂ (0, 1) is such that

*s*

_{ n }→ 0(

*n*→ ∞), by the above inequality we have

*s*

_{ n }→ 0, ${x}_{{s}_{n},t}\rightharpoonup {x}_{t}$. Combining (3.3), Lemma 2.1 and 2.2, we obtain

*x*

_{ t }∈

*Fix*(

*A*) =

*Fix*(

*T*). Taking

*z*=

*x*

_{ t }in (3.4), we get

Since ${x}_{{s}_{n},t}\rightharpoonup {x}_{t}$ and *J* is weakly sequentially continuous, by (3.5) as *s*_{ n } → 0, we obtain ${x}_{{s}_{n},t}\to {x}_{t}$. This has proved that the relative norm compactness of the net {*x*_{s,t}} as *s* → 0.

*n*→ ∞ in (3.4), we obtain

*x*

_{ t }is a solution of the following variational inequality:

*C*=

*Fix*(

*T*),

*F*=

*t*(

*I - f*) + (1

*- t*)(

*I - S*), by Lemma 2.3, we have the equivalent dual variational inequality:

*t*∈ (0, 1), as

*s*→ 0, {

*x*

_{s,t}} converges in norm to

*x*

_{ t }∈

*Fix*(

*T*). Assume ${x}_{{{s}^{\prime}}_{n},t}\to {x}_{t}^{\prime}$ as ${s}_{n}^{\prime}\to 0$. Similar to the above proof, we have ${x}_{t}^{\prime}\in Fix\left(T\right)$, which solves the following variational inequality:

*z*=

*x*

_{ t }in (3.8), we have

*f*is strongly pseudocontractive and

*S*is pseudocontractive, we have

which implies that ${x}_{t}^{\prime}={x}_{t}$. Hence, the net {*x*_{s,t}} converges in norm to *x*_{ t } ∈ *Fix*(*T*) as *s* → 0.

**Step 3** {*x*_{ t } } is bounded.

*Fix*(

*T*), for any

*y*∈ Ω, taking

*z*=

*y*in (3.7) we obtain

*I - S*is accretive, for any

*y*∈ Ω, we have

So {*x*_{ t } } is bounded.

**Step 4** *x*_{ t } → *x** ∈ Ω which is a solution of variational inequality (3.2).

Since *f* is strongly pseudocontractive, it is easy to see that the solution of the variational inequality (3.2) is unique.

_{ w }(

*x*

_{ t }) ⊂ Ω; namely, if (

*t*

_{ n }) is a null sequence in (0,1) such that ${x}_{{t}_{n}}\rightharpoonup {x}^{\prime}$ as

*n*→ ∞, then

*x*' ∈ Ω. Indeed, it follows from (3.7) that

*I - S*is accretive, from the above inequality, we have

*t*=

*t*

_{ n }→ 0 in (3.15), we have

Since *Fix*(*T*) is closed convex, then *Fix*(*T*) is weakly closed. Thus, *x*' ∈ *Fix*(*T*) by virtue of *x*_{ t } ∈ *Fix*(*T*). So, *x*' ∈ Ω.

*x*' =

*x**, the unique solution of VI(3.2). In fact, taking

*t*=

*t*

_{ n }and

*y*=

*x*' in (3.14), we obtain

*t*

_{ n }→ 0. Let

*t*=

*t*

_{ n }→ 0 in (3.13), we have

It follows from (3.16) and *x*' ∈ Ω that *x*' is a solution of VI(3.2). By uniqueness, we have *x*' = *x**. Therefore, *x*_{ t } → *x** as *t* → 0.

By Theorem 3.1, we have the following corollary directly.

**Corollary 3.1**([2, Theorem 3.3]) Let

*C*be a nonempty closed convex subset of a real Hilbert space

*H*. Let

*f*:

*C*→

*C*be a contraction,

*S*,

*T*:

*C*→

*C*be nonexpansive with

*Fix*(

*T*) ≠ ∅. Suppose that the solution set

*W*of the VI(1.2) is nonempty. Let for each (

*s*,

*t*) ∈ (0, 1)

^{2}, {

*x*

_{s,t}} be defined by (3.1). Then, for each fixed

*t*∈ (0, 1), the net {

*x*

_{s,t}} converges in norm, as

*s*→ 0, to a point

*x*

_{ t }∈

*Fix*(

*T*). Moreover, as

*t*→ 0, the net {

*x*

_{ t }} converges in norm to the unique solution

*x** of the following inequality variational:

Hence, for each null sequence {*t*_{ n } } in (0,1), there exists another null sequence {*s*_{ n } } in (0,1), such that the sequence ${{x}_{{s}_{n},}}_{{t}_{n}}\to {x}^{*}$ in norm as *n* → ∞.

**Remark**Theorem 3.1 improves and generalizes Theorem 3.1 of Yao et al.[3] in the following aspects:

- (i)
Theorem 3.1 generalizes Theorem 3.1 in [3] from Hilbert spaces to more general Banach spaces;

- (ii)
The mappings

*T*in [3, Theorem 3.1] is weakened from Lipschitzian to continuous; - (iii)
We modify the condition of

*f*, i.e., we suppose that*f*is Lipschitz strongly pseudocontractive.

The authors declare that they have no competing interests.

## Notes

### Acknowledgements

The first author was supported by the Research Project of Shaoxing University(No. 09LG1002), and the second author was supported partly by NSFC Grants(No.11071279).

## Supplementary material

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