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# Some results on a viscosity splitting algorithm in Hilbert spaces

- Yunpeng Zhang
^{4}Email author

**2015**:2

https://doi.org/10.1186/1029-242X-2015-2

© Zhang; licensee Springer. 2015

**Received:**25 August 2014**Accepted:**5 December 2014**Published:**6 January 2015

## Abstract

In this paper, a viscosity splitting for common solution problems is proposed.Strong convergence theorems are obtained in the framework of Hilbert spaces.Applications are also provided to support the main results.

## Keywords

- zero point
- fixed point
- variational inclusion
- nonexpansive mapping

## 1 Introduction; preliminaries

In this paper, we always assume that *H* is a real Hilbert space with theinner product
and the induced norm
for
. Recall that a set-valued mapping
is said to be *monotone* iff, for all
,
, and
imply
. In this paper, we use
to denote the zero point set of *M*. Amonotone mapping
is *maximal* iff the graph
of *M* is not properly contained in the graphof any other monotone mapping. It is well known that a monotone mapping *M*is maximal if and only if, for any
,
, for all
implies
. For a maximal monotone operator *M* on*H*, and
, we may define the single-valued resolvent
, where
denotes the domain of *M*. It is well knownthat
is firmly nonexpansive, and
.

The proximal point algorithm, which was proposed by Martinet [1, 2] and generalized by Rockafellar [3, 4] is one of the classical methods for solving zero points of maximalmonotone operators. In this paper, we investigate the problem of finding a zero ofthe sum of two monotone operators. The problem is very general in the sense that itincludes, as special cases, convexly constrained linear inverse problems, splitfeasibility problem, convexly constrained minimization problems, fixed pointproblems, variational inequalities, Nash equilibrium problem in noncooperative gamesand others. Because of their importance, splitting methods, which were proposed byLions and Mercier [5] and Passty [6], for zero problems have been studied extensively recently; see, forinstance, [7–17] and the references therein.

*C*be a nonempty closed and convex subset of

*H*. Let be a mapping. Recall that the classical variationalinequality problem is to find a point such that

*A*is said to be

*monotone*iff

For such a case, we also call *A* is *κ*-inverse-stronglymonotone. It is also not hard to see that every inverse-strongly monotone mapping ismonotone and continuous.

*S*.

*S*is said to be

*contractive*iff there exists a constant such that

*C*is nonempty closed convex of

*H*, then is not empty.

*S*is said to be

*firmlynonexpansive*iff

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

**Lemma 1.1**[18]

*Let**A**be a maximal monotone operator on**H*. *For*
,
, *and*
, *we have*
, *where*
*and*
.

**Lemma 1.2**[19]

*Then*
.

**Lemma 1.3**[20]

*Let*
*be a sequence of nonnegative numbers satisfying the condition*
,
, *where*
*is a number sequence in*
*such that*
*and*
,
*is a number sequence such that*
. *Then*
.

**Lemma 1.4**[21]

*Let**C**be a nonempty closed convex subset of**H*. *Let*
*be a mapping and let*
*be a maximal monotone operator*. *Then*
*for all*
.

**Lemma 1.5**[22]

*Let*
*be a real sequence that does not decreasing at infinity*, *in the sensethat there exists a subsequence*
*such that*
*for all*
. *For every*
, *define an integer sequence*
*as*
. *Then*
*and for all*
.

**Lemma 1.6**[23]

*Let**C**be a nonempty closed convex subset of**H*. *Let*
*be a nonexpansive mapping with a nonempty fixed point set*. *If*
*converges weakly to**x**and*
*converges to zero*. *Then*
.

## 2 Main results

Now, we are in a position to state our main results.

**Theorem 2.1**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a nonexpansive mapping with fixed points and let*

*be a*

*β*-

*contractive mapping*.

*Let*

*be an*

*α*-

*inverse*-

*strongly monotone mapping and let*

*B*

*be a maximal monotone operator on*

*H*.

*Assume that*

*and*

*is not empty*.

*Let*

*and*

*be real number sequences in*

*and let*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (a)
, ;

- (b)
;

- (c)
*and*,

*where*

*a*

*and*

*b*

*are two real numbers*.

*Then*

*converges strongly to a point*,

*which is also a unique solution to the followingvariational inequality*:

Since
is bounded, we can choose a subsequence
of
which converges weakly some point *x*. We mayassume, without loss of generality, that
converges weakly to *x*.

*B*is monotone, we get, for any ,

This gives , that is, . This proves that .

In view of restrictions (a) and (b), we find from Lemma 1.3 that . This completes the proof. □

From Theorem 2.1, we have the following results immediately.

**Corollary 2.2**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a nonexpansive mapping with fixed points and let*

*be a*

*β*-

*contractive mapping*.

*Let*

*and*

*be real number sequences in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (a)
, ;

- (b)
.

*Then*

*converges strongly to a point*,

*which is also a unique solution to the followingvariational inequality*:

**Corollary 2.3**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a*

*β*-

*contractive mapping*.

*Let*

*be an*

*α*-

*inverse*-

*strongly monotone mapping and let*

*B*

*be a maximal monotone operator on*

*H*.

*Assume that*

*and*

*is not empty*.

*Let*

*and*

*be real number sequences in*

*and*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (a)
, ;

- (b)
;

- (c)
*and*,

*where*

*a*

*and*

*b*

*are two real numbers*.

*Then*

*converges strongly to a point*,

*which is also a unique solution to the followingvariational inequality*:

Next, we give a result on the zeros of the sum of the operators *A* and*B* based on a different method.

**Theorem 2.4**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a*

*β*-

*contractive mapping*.

*Let*

*be an*

*α*-

*inverse*-

*strongly monotone mapping and let*

*B*

*be a maximal monotone operator on*

*H*.

*Assume that*

*and*

*is not empty*.

*Let*

*and*

*be real number sequences in*

*and*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (a)
, ;

- (b)
;

- (c)
,

*where*,

*a*,

*and*

*b*

*are real numbers*.

*Then*

*converges strongly to a point*,

*which is also a unique solution to the followingvariational inequality*:

*Proof*From the proof of Theorem 2.1, we find that is bounded. Since is contractive, it has a unique fixed point. Next, weuse to denote the unique fixed point. Note that

Next, we consider the following possible two cases.

*m*such that thesequence is eventually decreasing. Then exists. By (2.8), we find that

Since
is bounded, we can choose a subsequence
of
which converges weakly to some point *x*. Wemay assume, without loss of generality, that
converges weakly to *x*. Since the mapping
is nonexpansive, we find that
. It follows that
. In view of (2.8), we find from Lemma 1.3 that
.

Next, we consider another case.

By use of (2.10), we find that . Since , we find that . This completes the proof. □

## 3 Applications

In this section, we investigate solutions of equilibrium problems, variationalinequalities and convex minimization problems, respectively.

*F*be a bifunction of into ℝ, where ℝ denotes the set of realnumbers. Recall the following equilibrium problem:

In this paper, we use to denote the solution set of the equilibriumproblem.

To study equilibrium problems (3.1), we may assume that *F* satisfies thefollowing conditions:

(A1) for all ;

(A2) *F* is monotone, *i.e.*,
for all
;

(A4) for each , is convex and weakly lower semi-continuous.

**Lemma 3.1**[24]

*Let*

*C*

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

*H*.

*Let*

*F*

*be a bifunction from*

*to*ℝ

*which satisfies*(A1)-(A4)

*and let*

*B*

*be a multivalued mapping of*

*H*

*into itself defined by*

**Theorem 3.2**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a nonexpansive mapping with fixed points and let*

*be a*

*β*-

*contractive mapping*.

*Let*

*be an*

*α*-

*inverse*-

*strongly monotone mapping and let*

*F*

*be a bifunction from*

*to*ℝ

*which satisfies*(A1)-(A4).

*Assume that*

*is not empty*.

*Let*

*and*

*be real number sequences in*

*and let*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*where*

*B*

*is a mapping defined as in*(3.2).

*Assume that the control sequencessatisfy the following restrictions*:

- (a)
, ;

- (b)
;

- (c)
*and*,

*where*

*a*

*and*

*b*

*are two real numbers*.

*Then*

*converges strongly to a point*,

*which is a unique solution to the followingvariational inequality*:

**Theorem 3.3**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a*

*β*-

*contractive mapping*.

*Let*

*be an*

*α*-

*inverse*-

*strongly monotone mapping and Let*

*F*

*be a bifunction from*

*to*ℝ

*which satisfies*(A1)-(A4).

*Assume that*

*is not empty*.

*Let*

*and*

*be real number sequences in*

*and let*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (a)
, ;

- (b)
;

- (c)
,

*where*,

*a*,

*and*

*b*

*are real numbers*.

*Then*

*converges strongly to a point*,

*which is also a unique solution to the followingvariational inequality*:

*∂g*of

*g*is defined as follows:

*∂g*is maximal monotone. It is easy to verifythat if and only if . Let be the indicator function of

*C*,

*i.e.*,

Since
is a proper lower semi-continuous convex function on*H*, we see that the subdifferential
of
is a maximal monotone operator.

**Lemma 3.4**[24]

*Let**C**be a nonempty closed convex subset of**H**and let*
*be the metric projection from**H**onto**C*. *Let*
*be the subdifferential of*
, *where*
*is as defined in* (3.3). *Then*
,
,
,

**Theorem 3.5**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a nonexpansive mapping with fixed points and let*

*be a*

*β*-

*contractive mapping*.

*Let*

*be an*

*α*-

*inverse*-

*strongly monotone mapping*.

*Assumethat*

*is not empty*.

*Let*

*and*

*be real number sequences in*

*and let*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (1)
, ;

- (2)
;

- (3)
*and*,

*where*

*a*

*and*

*b*

*are two real numbers*.

*Then*

*converges strongly to a point*,

*which is a unique solution to the followingvariational inequality*:

*Proof* Putting
, we find from Theorem 2.1 and Lemma 3.4 thedesired conclusion immediately. □

**Theorem 3.6**

*Let*

*C*

*be a nonempty closed convex subset of*

*H*.

*Let*

*be a*

*β*-

*contractive mapping and let*

*be an*

*α*-

*inverse*-

*strongly monotone mapping*.

*Assumethat*

*is not empty*.

*Let*

*and*

*be real number sequences in*

*and let*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (a)
, ;

- (b)
;

- (c)
,

*where*,

*a*,

*and*

*b*

*are real numbers*.

*Then*

*converges strongly to a point*,

*which is also a unique solution to the followingvariational inequality*:

*Proof* Putting
, we find from Theorem 2.4 and Lemma 3.4 thedesired conclusion immediately. □

Let
be a convex and differentiable function and
is a convex function. Consider the convexminimization problem
. From [25], we know if ∇*W* is
-Lipschitz continuous, then it is*L*-inverse-strongly monotone. Hence, we have the following results.

**Theorem 3.7**

*Let*

*be a convex and differentiable function such that*∇

*W*

*is*-

*Lipschitz continuous and let*

*be a convex and lower semi*-

*continuous function such that*

*is not empty*.

*Let*

*f*

*be a*

*β*-

*contractive mapping on*

*H*.

*Let*

*and*

*be real number sequences in*

*and let*

*be a positive real number sequence in*.

*Let*

*be a sequence generated in the following process*:

*and*

*Assume that the control sequences satisfy the following restrictions*:

- (a)
, ;

- (b)
;

- (c)
,

*where*,

*a*,

*and*

*b*

*are real numbers*.

*Then*

*converges strongly to a point*,

*which is also a unique solution to the followingvariational inequality*:

*Proof* Putting
and
, we find from Theorem 2.4 the desired conclusionimmediately. □

## 4 Conclusions

In this paper, we study a convex feasibility problem via two monotone mappings and anonexpansive mapping. The common solution is also a unique solution of anothervariational inequality. The restrictions imposed on the sequence are mild. The results presented in this paper mainlyimprove the corresponding results in [7] and [11].

## Declarations

### Acknowledgements

The author is very grateful to the editor and anonymous reviewers’suggestions which improved the contents of the article.

## Authors’ Affiliations

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