# Convergence theorems of solutions of a generalized variational inequality

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## Abstract

The convex feasibility problem (CFP) of finding a point in the nonempty intersection Open image in new window is considered, where *r* ≥ 1 is an integer and each *C*_{ m } is assumed to be the solution set of a generalized variational inequality. Let *C* be a nonempty closed and convex subset of a real Hilbert space *H*. Let *A*_{ m }, *B*_{ m } : *C* → *H* be relaxed cocoercive mappings for each 1 ≤ *m* ≤ *r*. It is proved that the sequence {*x*_{ n }} generated in the following algorithm:

where *u* ∈ *C* is a fixed point, {*α*_{ n }}, {*β*_{ n }}, {*γ*_{ n }}, {*δ*_{(1,n)}}, ..., and {*δ*_{(r,n)}} are sequences in (0, 1) and Open image in new window , Open image in new window are positive sequences, converges strongly to a solution of CFP provided that the control sequences satisfies certain restrictions.

**2000 AMS Subject Classification**: 47H05; 47H09; 47H10.

## Keywords

nonexpansive mapping fixed point relaxed cocoercive mapping variational inequality## Abbreviation

- CFP
convex feasibility problem.

## 1. Introduction and Preliminaries

*H*. That is,

where *r* ≥ 1 is an integer and each *C*_{ m } is a nonempty closed and convex subset of *H*. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [1, 2], computer tomography [3] and radiation therapy treatment planning [4].

*H*is a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ||·||. Let

*C*be a nonempty closed and convex subset of

*H*and

*A*:

*C*→

*H*a nonlinear mapping. Recall the following definitions:

- (a)
*A*is said to be monotone if - (b)
*A*is said to be*ρ*-strongly monotone if there exists a positive real number*ρ >*0 such that - (c)
*A*is said to be*η*-cocoercive if there exists a positive real number*η >*0 such that - (d)
*A*is said to be relaxed*η*-cocoercive if there exists a positive real number*η >*0 such that - (e)
*A*is said to be relaxed (*η*,*ρ*)-cocoercive if there exist positive real numbers*η*,*ρ >*0 such that

*A*:

*C*→

*H*and

*B*:

*C*→

*H*, find a

*u*∈

*C*such that

where *λ* and *τ* are two positive constants. In this paper, we use *GV I*(*C*, *B*, *A*) to denote the set of solutions of the generalized variational inequality (1.2).

*u*∈

*C*is a solution to the variational inequality (1.2) if and only if

*u*∈

*C*is a fixed point of the mapping

*P*

_{ C }(

*τB*-

*λA*), where

*P*

_{ C }denotes the metric projection from

*H*onto

*C*. Indeed, we have the following relations:

*B*=

*I*, the identity mapping and

*τ*= 1, then the generalized variational inequality (1.1) is reduced to the following. Find

*u*∈

*C*such that

The variational inequality (1.4) emerging as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences was introduced by Stam-pacchia [5]. In this paper, we use *V I*(*C*, *A*) to denote the set of solutions of the variational inequality (1.4).

*S*:

*C*→

*C*be a mapping. We use

*F*(

*S*) to denote the set of fixed points of the mapping

*S*. Recall that

*S*is said to be nonexpansive if

It is well known that if *C* is nonempty bounded closed and convex subset of *H*, then the fixed point set of the nonexpansive mapping *S* is nonempty, see [6] more details. Recently, fixed point problems of nonexpansive mappings have been considered by many authors; see, for example, [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

Recall that *S* is said to be demi-closed at the origin if for each sequence {*x*_{ n }} in *C*, *x*_{ n } ⇀ *x*_{0} and *Sx*_{ n } → 0 imply *Sx*_{0} = 0, where ⇀ and → stand for weak convergence and strong convergence.

Recently, many authors considered the variational inequality (1.4) based on iterative methods; see [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. For finding solutions to a variational inequality for a cocoercive mapping, Iiduka et al. [22] proved the following theorem.

**Theorem ITT**.

*Let C be a nonempty closed convex subset of a real Hilbert space H and let A be an α-cocoercive operator of H into H with V I*(

*C*,

*A*) ≠ ∅.

*Let*{

*x*

_{ n }}

*be a sequence defined as follows*.

*x*

_{1}=

*x*∈

*C and*

*for every n* = 1, 2, ..., *where C is the metric projection from H onto C*, {*α*_{ n }} *is a sequence in* [-1, 1], *and* {*λ*_{ n }} *is a sequence in* [0, 2*α*]. *If* {*α*_{ n }} *and* {*λ*_{ n }} *are chosen so that* {*α*_{ n }} ∈ [*a*, *b*] *for some a*, *b with* -1 *< a < b <* 1 *and* {*λ*_{ n }} ∈ [*c*, *d*] *for some c*, *d with* 0 *< c < d <* 2(1 + *a*)*α, then* {*x*_{ n }} *converges weakly to some element of V I*(*C*, *A*).

Subsequently, Iiduka and Takahashi [23] further studied the problem of finding solutions of the classical variational inequality (1.4) for cocoercive mappings (inverse-strongly monotone mappings) and nonexpansive mappings. They obtained a strong convergence theorem. More precisely, they proved the following theorem.

**Theorem IT**.

*Let C be a closed convex subset of a real Hilbert space H. Let S*:

*C*→

*C be a nonexpanisve mapping and A an α-cocoercive mapping of C into H such that F*(

*S*) ∩

*V I*(

*C*,

*A*) ≠ ∅.

*Suppose x*

_{1}=

*u*∈

*C and*{

*x*

_{ n }}

*is given by*

*for every n* = 1, 2, ..., *where* {*α*_{ n }} *is a sequence in* [0, 1) *and* {*λ*_{ n }} *is a sequence in* [*a*, *b*].

*If*{

*α*

_{ n }}

*and*{

*λ*

_{ n }}

*are chosen so that*{

*λ*

_{ n }} ∈ [

*a*,

*b*]

*for some a*,

*b with*0

*< a < b <*2

*α*,

*then* {*x*_{ n }} *converges strongly to P*_{F(S)∩V I(C,A)}*x*.

In this paper, motivated by research work going on in this direction, we study the CFP in the case that each *C*_{ m } is a solution set of generalized variational inequality (1.2). Strong convergence theorems of solutions are established in the framework of real Hilbert spaces.

In order to prove our main results, we need the following lemmas.

**Lemma 1.1**[33].

*Let*{

*x*

_{ n }}

*and*{

*y*

_{ n }}

*be bounded sequences in a Hilbert space H and*{

*β*

_{ n }}

*a sequence in*(0, 1)

*with*

*Suppose that x*

_{n+1}= (1 -

*β*

_{ n })

*y*

_{ n }+

*β*

_{ n }

*x*

_{ n }

*for all integers n*≥ 0

*and*

*Then* lim_{n→∞}||*y*_{ n } - *x*_{ n }|| = 0.

**Lemma 1.2**[34].

*Let C be a nonempty closed and convex subset of a real Hilbert space H. Let S*

_{1}:

*C*→

*C and S*

_{2}:

*C*→

*C be nonexpansive mappings on C. Suppose that F*(

*S*

_{1}) ∩

*F*(

*S*

_{2})

*is nonempty. Define a mapping S*:

*C*→

*C by*

*where a is a constant in* (0, 1). *Then S is nonexpansive with F*(*S*) = *F*(*S*_{1}) ∩ *F* (*S*_{2}).

**Lemma 1.3** [35]. *Let C be a nonempty closed and convex subset of a real Hilbert space H and S* : *C* → *C a nonexpansive mapping. Then I* - *S is demi-closed at zero*.

*where*{γ

_{ n }}

*is a sequence in*(0, 1)

*and*{

*δ*

_{ n }}

*is a sequence such that*

- (a)
- (b)
lim sup

_{n→∞}*δ*_{ n }/*γ*_{ n }≤ 0*or*Open image in new window .

*Then* lim_{n→∞}*α*_{ n } = 0.

## 2. Main results

**Theorem 2.1**.

*Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A*

_{ m }:

*C*→

*H be a relaxed*(

*η*

_{ m },

*ρ*

_{ m })

*-cocoercive and μ*

_{ m }

*-Lipschitz continuous mapping and B*

_{ m }:

*C*→

*H a relaxed*Open image in new window

*-cocoercive and*Open image in new window

*-Lipschitz continuous mapping for each*1 ≤

*m*≤

*r. Assume that*Open image in new window .

*Let*{

*x*

_{ n }}

*be a sequence generated in the following manner:*

*where u*∈

*C is a fixed point*, {

*α*

_{ n }}, {

*β*

_{ n }}, {

*γ*

_{ n }}, {

*δ*

_{(1,n)}}, ...,

*and*{

*δ*

_{(r,n)}}

*are sequences in*(0, 1)

*satisfying the following restrictions:*

- (a)
- (b)
0

*<*lim inf_{n→∞}*β*_{ n }≤ lim sup_{n→∞}*β*_{ n }*<*1; - (c)
lim

_{n→∞}*α*_{ n }= 0*and*Open image in new window ; - (d)
lim

_{n→∞}*δ*_{(m,n)}=*δ*_{ m }∈ (0, 1), ∀1 ≤*m*≤*r*,

*Then the sequence*{

*x*

_{ n }}

*generated in the iterative process*(ϒ)

*converges strongly to a common element*Open image in new window ,

*which uniquely solves the following variational inequality*.

**Proof**. First, we prove that the mapping

*P*

_{ C }(

*τ*

_{ m }

*B*

_{ m }-

*λ*

_{ m }

*A*

_{ m }) is nonexpansive for each 1 ≤

*m*≤

*r*. For each

*x*,

*y*∈

*C*, we have

*A*

_{ m }is relaxed (

*η*

_{ m },

*ρ*

_{ m })-cocoercive and

*μ*

_{ m }-Lipschitz continuous that

*P*

_{ C }(

*τ*

_{ m }

*B*

_{ m }-

*λ*

_{ m }

*A*

_{ m }) is nonexpansive for each 1 ≤

*m*≤

*r*. Put

*P*

_{ C }(

*τ*

_{ m }

*B*

_{ m }-

*λ*

_{ m }

*A*

_{ m }) is nonexpansive for each 1 ≤

*m*≤

*r*, we see that

*M*is an appropriate constant such that

_{n→∞}||

*l*

_{ n }-

*x*

_{ n }|| = 0. In view of (2.5), we see that

*x*

_{ n }+1

*x*

_{ n }= (1 -

*β*

_{ n })(

*l*

_{ n }-

*x*

_{ n }). It follows that

*x*

_{ n }+1 -

*x*

_{ n }=

*α*

_{ n }(

*u*-

*x*

_{ n }) + γ

_{ n }(

*y*

_{ n }-

*x*

_{ n }). It follows from (2.6) and the conditions (b), (c) that

*x*

_{ n }} such that

*q*. Without loss of generality, we may assume that Open image in new window . Next, we show that Open image in new window . Define a mapping

*R*:

*C*→

*C*by

*δ*

_{ m }= lim

_{n→∞}

*δ*

_{(m,n)}. From Lemma 1.2, we see that

*R*is nonexpansive with

_{n→∞}||

*Rx*

_{ n }-

*x*

_{ n }|| = 0. From Lemma 1.3, we see that

This completes the proof.

If *B*_{ m } ≡ *I*, the identity mapping and *τ*_{ m } ≡ 1, then Theorem 2.1 is reduced to the following result on the classical variational inequality (1.4).

**Corollary 2.2**.

*Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A*

_{ m }:

*C*→

*H be a relaxed*(

*η*

_{ m },

*ρ*

_{ m })

*-cocoercive and μ*

_{ m }

*-Lipschitz continuous mapping for each*1 ≤

*m*≤

*r. Assume that*Open image in new window .

*Let*{

*x*

_{ n }}

*be a sequence generated by the following manner*:

*where u*∈

*C is a fixed point*, {

*α*

_{ n }}, {

*β*

_{ n }}, {

*γ*

_{ n }}, {

*δ*

_{(1,n)}}, ...,

*and*{

*δ*

_{(r,n)}}

*are sequences in*(0, 1)

*satisfying the following restrictions*.

- (a)
- (b)
0

*<*lim inf_{n→∞}*β*_{ n }≤ lim sup_{n→∞}*β*_{ n }< 1; - (c)
lim

_{n→∞}*α*_{ n }= 0*and*Open image in new window ; - (d)
lim

_{n→∞}*δ*_{(m,n)}=*δ*_{ m }∈ (0, 1), ∀1 ≤*m*≤*r*,*and*Open image in new window*is a positive sequence such that* - (e)
Open image in new window , ∀1 ≤

*m*≤*r*.

*Then the sequence*{

*x*

_{ n }}

*converges strongly to a common element*Open image in new window ,

*which uniquely solves the following variational inequality*

If *r* = 1, then Theorem 2.1 is reduced to the following.

**Corollary 2.3**.

*Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A*:

*C*→

*H be a relaxed*(

*η*,

*ρ*)

*-cocoercive and μ-Lipschitz continuous mapping and B*:

*C*→

*H a relaxed*Open image in new window

*-cocoercive and*Open image in new window

*-Lipschitz continuous mapping. Assume that GV I*(

*C*,

*B*,

*A*)

*is not empty. Let*{

*x*

_{ n }}

*be a sequence generated in the following manner*:

*where u*∈

*C is a fixed point*, {

*α*

_{ n }}, {

*β*

_{ n }}

*and*{

*γ*

_{ n }}

*are sequences in*(0, 1)

*satisfying the following restrictions*.

- (a)
*α*_{ n }+*β*_{ n }+ γ_{ n }= 1, ∀_{ n }≥ 1; - (b)
0

*<*lim inf_{n→∞}*β*_{ n }≤ lim sup_{n→∞}*β*_{ n }*<*1; - (c)
lim

_{n→∞}*α*_{ n }= 0*and*Open image in new window - (d)

*Then the sequence*{

*x*

_{ n }}

*converges strongly to a common element*Open image in new window ,

*which uniquely solves the following variational inequality*

For the variational inequality (1.4), we can obtain from Corollary 2.3 the following immediately.

**Corollary 2.4**.

*Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A*:

*C*→

*H be a relaxed*(

*η*,

*ρ*)

*-cocoercive and μ-Lipschitz continuous mapping. Assume that V I*(

*C*,

*A*)

*is not empty. Let*{

*x*

_{ n }}

*be a sequence generated in the following manner*:

*where u*∈

*C is a fixed point*, {

*α*

_{ n }}, {

*β*

_{ n }}

*and*{

*γ*

_{ n }}

*are sequences in*(0, 1)

*satisfying the following restrictions*.

- (a)
*α*_{ n }+*β*_{ n }+ γ_{ n }= 1, ∀*n*≥ 1; - (b)
0

*<*lim inf_{n→∞}*β*_{ n }≤ lim sup_{n→∞}*β*_{ n }< 1; - (c)
lim

_{n→∞}*α*_{ n }= 0*and*Open image in new window ; - (d)

*Then the sequence*{

*x*

_{ n }}

*converges strongly to a common element*Open image in new window ,

*which uniquely solves the following variational inequality*

**Remark 2.5**. In this paper, the generalized variational inequality (1.2), which includes the classical variational inequality (1.4) as a special case, is considered based on iterative methods. Strong convergence theorems are established under mild restrictions imposed on the parameters. It is of interest to extend the main results presented in this paper to the framework of Banach spaces.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant no. 70871081 and Important Science and Technology Research Project of Henan province, China (102102210022).

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