 Research
 Open Access
 Published:
Some results on a viscosity splitting algorithm in Hilbert spaces
Journal of Inequalities and Applications volume 2015, Article number: 2 (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.
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 setvalued 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 singlevalued 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
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 inversestrongly monotone iff thereexists a constant such that
For such a case, we also call A is κinversestronglymonotone. It is also not hard to see that every inversestrongly 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 βcontractive. 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 aβcontractive mapping. Letbe anαinversestrongly 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
Since , we find that
Since is convex, we find that
It follows that
Hence, we have
In view of restrictions (a), (b), and (c), we find from (2.1) that
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
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 aβcontractive 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 aβcontractive mapping. Letbe anαinversestrongly 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 aβcontractive mapping. Letbe anαinversestrongly 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
Since is firmly nonexpansive and A isinversestrongly monotone, we find that
Substituting (2.6) into (2.7), we find that
It follows that
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
It follows that
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:
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 semicontinuous.
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
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 aβcontractive mapping. Letbe anαinversestrongly 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 aβcontractive mapping. Letbe anαinversestrongly 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 semicontinuous 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.,
Since is a proper lower semicontinuous 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 aβcontractive mapping. Letbe anαinversestrongly 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 aβcontractive mapping and letbe anαinversestrongly 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 isLinversestrongly monotone. Hence, we have the following results.
Theorem 3.7Letbe a convex and differentiable function such that ∇WisLipschitz continuous and letbe a convex and lower semicontinuous function such thatis not empty. Letfbe aβcontractive 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].
References
Martinet B: Regularisation d’inéquations variationelles par approximationssuccessives.Rev. Fr. Inform. Rech. Oper. 1970, 4:154–158.
Martinet B: Determination approchée d’un point fixe d’une applicationpseudocontractante.C. R. Acad. Sci. Paris Sér. AB 1972, 274:163–165.
Rockafellar RT: Monotone operators and the proximal point algorithm.SIAM J. Control Optim. 1976, 14:877–898. 10.1137/0314056
Rockafellar RT: Augmented Lagrangians and applications of the proximal point algorithm inconvex programming.Math. Oper. Res. 1976, 1:97–116. 10.1287/moor.1.2.97
Lions PL, Mercier B: Splitting algorithms for the sum of two nonlinear operators.SIAM J. Numer. Anal. 1979, 16:964–979. 10.1137/0716071
Passty GB: Ergodic convergence to a zero of the sum of monotone operators in Hilbertspace.J. Math. Anal. Appl. 1979, 72:383–390. 10.1016/0022247X(79)902348
Cho SY, Qin X, Wang L: Strong convergence of a splitting algorithm for treating monotoneoperators.Fixed Point Theory Appl. 2014., 2014: Article ID 94
Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variationalinequality and fixed point problem.Adv. Fixed Point Theory 2012, 2:374–397.
Cho SY, Kang SM: On iterative solutions of a common element problem.J. Nonlinear Funct. Anal. 2014., 2014: Article ID 3
Zhang M: An algorithm for treating asymptotically strict pseudocontractions andmonotone operators.Fixed Point Theory Appl. 2014., 2014: Article ID 52
Qin X, Cho SY, Wang L: A regularization method for treating zero points of the sum of two monotoneoperators.Fixed Point Theory Appl. 2014., 2014: Article ID 75
Hecai Y: On weak convergence of an iterative algorithm for common solutions ofinclusion problems and fixed point problems in Hilbert spaces.Fixed Point Theory Appl. 2013., 2013: Article ID 155
Zhao J, Zhang Y, Yang Q: Modified projection methods for the split feasibility problem and themultiplesets split feasibility problem.Appl. Math. Comput. 2012, 219:1644–1653. 10.1016/j.amc.2012.08.005
Qin X, Cho SY, Wang L: Convergence of splitting algorithms for the sum of two accretive operatorswith applications.Fixed Point Theory Appl. 2014., 2014: Article ID 166
Yu L, Liang M: Convergence of iterative sequences for fixed point and variational inclusionproblems.Fixed Point Theory Appl. 2011., 2011: Article ID 368137
Yang S: Zero theorems of accretive operators in reflexive Banach spaces.J. Nonlinear Funct. Anal. 2013., 2013: Article ID 2
Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces.J. Math. Anal. Appl. 2007, 329:415–424. 10.1016/j.jmaa.2006.06.067
Barbu V: Nonlinear Semigroups and Differential Equations in Banach Space. Noordhoff, Groningen; 1976.
Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences foroneparameter nonexpansive semigroups without Bochner integrals.J. Math. Anal. Appl. 2005, 305:227–239. 10.1016/j.jmaa.2004.11.017
Liu LS: Ishikawa and Mann iterative process with errors for nonlinear stronglyaccretive mappings in Banach spaces.J. Math. Anal. Appl. 1995, 194:114–125. 10.1006/jmaa.1995.1289
Lopez G, Marquez VM, Wang F, Xu HK: Forwardbackward splitting methods for accretive operators in Banachspaces.Abstr. Appl. Anal. 2012., 2012: Article ID 109236
Mainge PE: Strong convergence of projected subgradient methods for nonsmooth andnonstrictly convex minimization.SetValued Anal. 2008, 16:899–912. 10.1007/s112280080102z
Browder FE: Nonexpansive nonlinear operators in a Banach space.Proc. Natl. Acad. Sci. USA 1965, 54:1041–1044. 10.1073/pnas.54.4.1041
Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinearmappings in Hilbert spaces.J. Optim. Theory Appl. 2010, 147:27–41. 10.1007/s1095701097132
Baillon JB, Haddad G: Quelques propriétés des opérateurs anglebornés etcycliquement monotones.Isr. J. Math. 1977, 26:137–150. 10.1007/BF03007664
Acknowledgements
The author is very grateful to the editor and anonymous reviewers’suggestions which improved the contents of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhang, Y. Some results on a viscosity splitting algorithm in Hilbert spaces. J Inequal Appl 2015, 2 (2015). https://doi.org/10.1186/1029242X20152
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X20152
Keywords
 zero point
 fixed point
 variational inclusion
 nonexpansive mapping