Some results on a viscosity splitting algorithm in Hilbert spaces

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.

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 Mis maximal if and only if, for any , , for all implies . For a maximal monotone operator M onH, 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.

Let 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

(1.1)

Such a point is called a solution of variational inequality (1.1).In this paper, we use to denote the solution set of variational inequality(1.1). Recall that A is said to be monotone iff

Recall that A is said to be inverse-strongly monotone iff thereexists a constant such that

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.

Let be a mapping. In this paper, we use to denote the fixed point set of S.S is said to be contractive iff there exists a constant such that

We also call S is β-contractive. S is said to benonexpansive iff

It is well known if 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]

LetAbe a maximal monotone operator onH. For, , and, we have, whereand.

Lemma 1.2[19]

Letandbe bounded sequences inH. Letbe a sequence inwith. Suppose that, and

Then.

Lemma 1.3[20]

Letbe a sequence of nonnegative numbers satisfying the condition, , whereis a number sequence insuch thatand, is a number sequence such that. Then.

Lemma 1.4[21]

LetCbe a nonempty closed convex subset ofH. Letbe a mapping and letbe a maximal monotone operator. Thenfor all.

Lemma 1.5[22]

Letbe a real sequence that does not decreasing at infinity, in the sensethat there exists a subsequencesuch thatfor all. For every, define an integer sequenceas. Thenand for all.

Lemma 1.6[23]

LetCbe a nonempty closed convex subset ofH. Letbe a nonexpansive mapping with a nonempty fixed point set. Ifconverges weakly toxandconverges to zero. Then.

2 Main results

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

Theorem 2.1LetCbe a nonempty closed convex subset ofH. Letbe a nonexpansive mapping with fixed points and letbe aβ-contractive mapping. Letbe anα-inverse-strongly monotone mapping and letBbe a maximal monotone operator onH. Assume thatandis not empty. Letandbe real number sequences inand letbe a positive real number sequence in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (a) , ;

• (b) ;

• (c) and,

whereaandbare two real numbers. Thenconverges strongly to a point, which is also a unique solution to the followingvariational inequality:

Proof Note that the mapping is nonexpansive. Indeed, we have

In light of restriction (c), one finds that is nonexpansive. It is obvious that. Fix . It follows that

Putting , we see that

By mathematical induction, we find that the sequence is bounded. Note that

Putting , we find from Lemma 1.1 that

This yields

It follows from restrictions (a) and (c) that

Using Lemma 1.2, we have . It follows that

(2.1)

Since , we find that

(2.2)

Since is convex, we find that

(2.3)

It follows that

Hence, we have

In view of restrictions (a), (b), and (c), we find from (2.1) that

(2.4)

Since is firmly nonexpansive, we have

This implies from (2.3) that

On the other hand, we have

This implies that

In view of restrictions (a) and (b), we find from (2.1) and (2.4) that

(2.5)

Next, we show that , where . To show it, we can choose a subsequence of such that

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.

Now, we are in a position to show that . Set . It follows that . Since B is monotone, we get, for any,

Replacing n by and letting , we obtain from (2.5) that

This gives , that is, . This proves that .

Now, we are in a position to prove that . Notice that

This implies that . This implies from (2.5) that . Since is demiclosed at zero, we find that. This complete the proof that . It follows that

Finally, we show that . Notice that

This implies that

It follows 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 2.2LetCbe a nonempty closed convex subset ofH. Letbe a nonexpansive mapping with fixed points and letbe aβ-contractive mapping. Letandbe real number sequences in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (a) , ;

• (b) .

Thenconverges strongly to a point, which is also a unique solution to the followingvariational inequality:

Corollary 2.3LetCbe a nonempty closed convex subset ofH. Letbe aβ-contractive mapping. Letbe anα-inverse-strongly monotone mapping and letBbe a maximal monotone operator onH. Assume thatandis not empty. Letandbe real number sequences inandbe a positive real number sequence in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (a) , ;

• (b) ;

• (c) and,

whereaandbare two real numbers. Thenconverges 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 andB based on a different method.

Theorem 2.4LetCbe a nonempty closed convex subset ofH. Letbe aβ-contractive mapping. Letbe anα-inverse-strongly monotone mapping and letBbe a maximal monotone operator onH. Assume thatandis not empty. Letandbe real number sequences inandbe a positive real number sequence in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (a) , ;

• (b) ;

• (c) ,

where, a, andbare real numbers. Thenconverges 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

It follows that

(2.6)

Since is firmly nonexpansive and A isinverse-strongly monotone, we find that

(2.7)

Substituting (2.6) into (2.7), we find that

It follows that

(2.8)

Next, we consider the following possible two cases.

Case 1. Suppose that there exists some nonnegative integer m such that thesequence is eventually decreasing. Then exists. By (2.8), we find that

By use of restrictions (b) and (c), we have . It also follows from (2.8) that

From restrictions (b) and (c), we obtain

Hence, we have . From Lemma 1.4, we find that. This implies that . Next, we show that . To show it, we can choose a subsequence of such 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.

Case 2. Suppose that the sequence is not eventually decreasing. There exists asubsequence such that for all . We define an integer sequence as in Lemma 1.5. By use of (2.8), we have

(2.9)

It follows that

(2.10)

Hence, we have . In view of (2.9), we find that

Note that

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

Remark 2.5 Comparing Theorem 2.4 with the recent results announced in [7, 11] and [24], we have the following:

• (i) Our proofs are different from theirs.

• (ii) We remove the additional restriction .

3 Applications

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

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

(3.1)

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 ;

(A3) for each ,

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

Lemma 3.1[24]

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetFbe a bifunction fromto ℝ which satisfies (A1)-(A4) and letBbe a multivalued mapping ofHinto itself defined by

(3.2)

ThenBis a maximal monotone operator with the domain, , and, , , whereis defined as

Theorem 3.2LetCbe a nonempty closed convex subset ofH. Letbe a nonexpansive mapping with fixed points and letbe aβ-contractive mapping. Letbe anα-inverse-strongly monotone mapping and letFbe a bifunction fromto ℝ which satisfies (A1)-(A4). Assume thatis not empty. Letandbe real number sequences inand letbe a positive real number sequence in. Letbe a sequence generated in the following process: and

whereBis a mapping defined as in (3.2). Assume that the control sequencessatisfy the following restrictions:

• (a) , ;

• (b) ;

• (c) and,

whereaandbare two real numbers. Thenconverges strongly to a point, which is a unique solution to the followingvariational inequality:

Theorem 3.3LetCbe a nonempty closed convex subset ofH. Letbe aβ-contractive mapping. Letbe anα-inverse-strongly monotone mapping and LetFbe a bifunction fromto ℝ which satisfies (A1)-(A4). Assume thatis not empty. Letandbe real number sequences inand letbe a positive real number sequence in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (a) , ;

• (b) ;

• (c) ,

where, a, andbare real numbers. Thenconverges strongly to a point, which is also a unique solution to the followingvariational inequality:

Let be a proper convex lower semi-continuous function.Then the subdifferential ∂g of g is defined as follows:

From Rockafellar [4], we know that ∂g is maximal monotone. It is easy to verifythat if and only if . Let be the indicator function of C,i.e.,

(3.3)

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

Lemma 3.4[24]

LetCbe a nonempty closed convex subset ofHand letbe the metric projection fromHontoC. Letbe the subdifferential of, whereis as defined in (3.3). Then, , ,

Theorem 3.5LetCbe a nonempty closed convex subset ofH. Letbe a nonexpansive mapping with fixed points and letbe aβ-contractive mapping. Letbe anα-inverse-strongly monotone mapping. Assumethatis not empty. Letandbe real number sequences inand letbe a positive real number sequence in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (1) , ;

• (2) ;

• (3) and,

whereaandbare two real numbers. Thenconverges 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 3.6LetCbe a nonempty closed convex subset ofH. Letbe aβ-contractive mapping and letbe anα-inverse-strongly monotone mapping. Assumethatis not empty. Letandbe real number sequences inand letbe a positive real number sequence in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (a) , ;

• (b) ;

• (c) ,

where, a, andbare real numbers. Thenconverges 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 isL-inverse-strongly monotone. Hence, we have the following results.

Theorem 3.7Letbe a convex and differentiable function such that ∇Wis-Lipschitz continuous and letbe a convex and lower semi-continuous function such thatis not empty. Letfbe aβ-contractive mapping onH. Letandbe real number sequences inand letbe a positive real number sequence in. Letbe a sequence generated in the following process: and

Assume that the control sequences satisfy the following restrictions:

• (a) , ;

• (b) ;

• (c) ,

where, a, andbare real numbers. Thenconverges 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].