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Some results on a viscosity splitting algorithm in Hilbert spaces

Journal of Inequalities and Applications20152015:2

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

  • Received: 25 August 2014
  • Accepted: 5 December 2014
  • Published:

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 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, [717] 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 , where and .

Lemma 1.2[19]

Let and be bounded sequences inH. Let be a sequence in with . Suppose that , and

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]

LetCbe a nonempty closed convex subset ofH. 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]

LetCbe a nonempty closed convex subset ofH. Let be a nonexpansive mapping with a nonempty fixed point set. If converges weakly toxand converges 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. Let be a nonexpansive mapping with fixed points and let be aβ-contractive mapping. Let be anα-inverse-strongly monotone mapping and letBbe a maximal monotone operator onH. 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:
  1. (a)

    , ;

     
  2. (b)

    ;

     
  3. (c)

    and ,

     
whereaandbare two real numbers. Then converges 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. Let be a nonexpansive mapping with fixed points and let be aβ-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:
  1. (a)

    , ;

     
  2. (b)

    .

     
Then converges strongly to a point , which is also a unique solution to the followingvariational inequality:
Corollary 2.3LetCbe a nonempty closed convex subset ofH. Let be aβ-contractive mapping. Let be anα-inverse-strongly monotone mapping and letBbe a maximal monotone operator onH. 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:
  1. (a)

    , ;

     
  2. (b)

    ;

     
  3. (c)

    and ,

     
whereaandbare 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 andB based on a different method.

Theorem 2.4LetCbe a nonempty closed convex subset ofH. Let be aβ-contractive mapping. Let be anα-inverse-strongly monotone mapping and letBbe a maximal monotone operator onH. 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:
  1. (a)

    , ;

     
  2. (b)

    ;

     
  3. (c)

    ,

     
where , a, andbare 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
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:
  1. (i)

    Our proofs are different from theirs.

     
  2. (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 , where 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 from to which satisfies (A1)-(A4) and letBbe a multivalued mapping ofHinto itself defined by
(3.2)
ThenBis a maximal monotone operator with the domain , , and , , , where is defined as
Theorem 3.2LetCbe a nonempty closed convex subset ofH. Let be a nonexpansive mapping with fixed points and let be aβ-contractive mapping. Let be anα-inverse-strongly monotone mapping and letFbe a bifunction from to 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
whereBis a mapping defined as in (3.2). Assume that the control sequencessatisfy the following restrictions:
  1. (a)

    , ;

     
  2. (b)

    ;

     
  3. (c)

    and ,

     
whereaandbare two real numbers. Then converges strongly to a point , which is a unique solution to the followingvariational inequality:
Theorem 3.3LetCbe a nonempty closed convex subset ofH. Let be aβ-contractive mapping. Let be anα-inverse-strongly monotone mapping and LetFbe a bifunction from to 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:
  1. (a)

    , ;

     
  2. (b)

    ;

     
  3. (c)

    ,

     
where , a, andbare real numbers. Then converges 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 let be the metric projection fromHontoC. Let be the subdifferential of , where is as defined in (3.3). Then , , ,

Theorem 3.5LetCbe a nonempty closed convex subset ofH. Let be a nonexpansive mapping with fixed points and let be aβ-contractive mapping. Let be anα-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. (1)

    , ;

     
  2. (2)

    ;

     
  3. (3)

    and ,

     
whereaandbare 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 3.6LetCbe a nonempty closed convex subset ofH. Let be aβ-contractive mapping and let be anα-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. (a)

    , ;

     
  2. (b)

    ;

     
  3. (c)

    ,

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

Theorem 3.7Let be a convex and differentiable function such that Wis -Lipschitz continuous and let be a convex and lower semi-continuous function such that is not empty. Letfbe aβ-contractive mapping onH. 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. (a)

    , ;

     
  2. (b)

    ;

     
  3. (c)

    ,

     
where , a, andbare 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

(1)
College of Electric Power, North China University of Water Resources andElectric Power, Zhengzhou, 450011, China

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Copyright

© Zhang; licensee Springer. 2015

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly credited.

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