- Open Access
Hybrid methods for common solutions in Hilbert spaces with applications
© Sun; licensee Springer. 2014
Received: 26 February 2014
Accepted: 29 April 2014
Published: 13 May 2014
In this paper, hybrid methods are investigated for treating common solutions of nonlinear problems. A strong convergence theorem is established in the framework of real Hilbert spaces.
1 Introduction and preliminaries
Common solutions to variational inclusion, equilibrium and fixed point problems have been recently extensively investigated based on iterative methods; see [1–33] and the references therein. The motivation for this subject is mainly to its possible applications to mathematical modeling of concrete complex problems, which use more than one constraint. The aim of this paper is to investigate a common solution of variational inclusion, equilibrium and fixed point problems. The organization of this paper is as follows. In Section 1, we provide some necessary preliminaries. In Section 2, a hybrid method is introduced and analyzed. Strong convergence theorems are established in the framework of Hilbert spaces. In Section 3, applications of the main results are discussed.
In what follows, we always assume that H is a real Hilbert space with the inner product and the norm . Let C be a nonempty, closed, and convex subset of H and let be the metric projection from H onto C. Let be a mapping. stands for the fixed point set of S; that is, .
If , then S is said to be nonexpansive. Let be a mapping. If C is nonempty closed and convex, then the fixed point set of S is nonempty.
For such a case, A is also said to be α-inverse-strongly monotone.
Recall that a set-valued mapping is said to be monotone iff, for all , , and imply . M is maximal iff the graph of R is not properly contained in the graph of 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 denote the domain of M. It is well known that is firmly nonexpansive, and , where , and .
In this paper, the set of such an is denoted by .
To study the equilibrium problems (1.1), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A4) for each , is convex and weakly lower semicontinuous.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 
Lemma 1.2 
- (2)is firmly nonexpansive, i.e., for any ,
is closed and convex.
Such a mapping is nonexpansive from C to C and it is called a W-mapping generated by and .
Lemma 1.3 
is nonexpansive and , for each ;
for each and for each positive integer k, the limit exists;
- (3)the mapping defined by(1.3)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 1.4 
Throughout this paper, we always assume that , .
Lemma 1.5 
Let be a mapping and let be a maximal monotone operator. Then .
Lemma 1.6 
Lemma 1.7 
Then D is maximal monotone and if and only if .
2 Main results
, , ;
Then the sequence converges strongly to , where .
Since is bounded, there exists a subsequence of which converges weakly to w. Without loss of generality, we may assume that . Therefore, we see that . We also have .
In view of Lemma 1.4, we obtain that . This implies from (2.22) that . This is a contradiction. Thus, we have .
This implies that . This implies that , that is, .
This implies that . This implies that , that is, .
Using Lemma 1.1, we find that . This completes the proof. □
In this section, we consider some applications of the main results.
In this paper, we use to denote the solution set of the inequality. It is well known that is a solution of the inequality iff x is a fixed point of the mapping , where is a constant, I stands for the identity mapping. If A is α-inverse-strongly monotone and , then the mapping is nonexpansive. It follows that is closed and convex.
From Rockafellar , we know that ∂g is maximal monotone. It is not hard to verify that if and only if .
Since is a proper lower semicontinuous convex function on H, we see that the subdifferential of is a maximal monotone operator. It is clearly that , , and .
, , ;
Then the sequence converges strongly to .
Putting , where is a k-strict pseudo-contraction, we find that A is -inverse-strongly monotone.
Next, we consider fixed points of strict pseudo-contractions.
, , ;
Then the sequence converges strongly to .
Proof Taking , wee see that is a -strict pseudo-contraction with and for . In view of Theorem 3.1, we find the desired conclusion immediately. □
The author is very grateful to the reviewers for useful suggestions which improved the contents of this paper.
- Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
- Wu C: Convergence of algorithms for an infinite family nonexpansive mappings and relaxed cocoercive mappings in Hilbert spaces. Adv. Fixed Point Theory 2014, 4: 125–139.Google Scholar
- Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.MathSciNetMATHGoogle Scholar
- Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618. 10.1016/S0252-9602(12)60127-1View ArticleMathSciNetMATHGoogle Scholar
- Cho SY, Qin X: On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl. Math. Comput. 2014, 235: 430–438.MathSciNetView ArticleGoogle Scholar
- Zhang M: Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 21Google Scholar
- Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10Google Scholar
- Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleMATHGoogle Scholar
- Cho SY, Qin X, Kang SM: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013, 57: 1429–1446. 10.1007/s10898-012-0017-yMathSciNetView ArticleMATHGoogle Scholar
- Luo H, Wang Y: Iterative approximation for the common solutions of a infinite variational inequality system for inverse-strongly accretive mappings. J. Math. Comput. Sci. 2012, 2: 1660–1670.MathSciNetGoogle Scholar
- Zhang Q: Iterative approximation of solutions of monotone quasi-variational inequalities via nonlinear mappings. J. Fixed Point Theory 2014., 2014: Article ID 1Google Scholar
- Jeong JU: Fixed point solutions of generalized mixed equilibrium problems and variational inclusion problems for nonexpansive semigroups. Fixed Point Theory Appl. 2014., 2014: Article ID 57Google Scholar
- Cho SY, Qin X, Wang L: Strong convergence of a splitting algorithm for treating monotone operators. Fixed Point Theory Appl. 2014., 2014: Article ID 94Google Scholar
- Qin X, Cho SY: Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions. Appl. Math. Lett. 2010, 23: 1252–1255. 10.1016/j.aml.2010.06.008MathSciNetView ArticleMATHGoogle Scholar
- Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4Google Scholar
- Qing Y, Cho SY: Proximal point algorithms for zero points of nonlinear operators. Fixed Point Theory Appl. 2014., 2014: Article ID 42Google Scholar
- Zhang M: Strong convergence of a viscosity iterative algorithm in Hilbert spaces. J. Nonlinear Funct. Anal. 2014., 2014: Article ID 1Google Scholar
- Wu C, Lv S: Bregman projection methods for zeros of monotone operators. J. Fixed Point Theory 2013., 2013: Article ID 7Google Scholar
- Lv S, Hao Y: Some results on continuous pseudo-contractions in a reflexive Banach space. J. Inequal. Appl. 2013., 2013: Article ID 538Google Scholar
- Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleMATHGoogle Scholar
- Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi- ϕ -nonexpansive mapping. Numer. Funct. Anal. Optim. 2010, 31: 1072–1089. 10.1080/01630563.2010.501643MathSciNetView ArticleMATHGoogle Scholar
- Kim KS, Kim JK, Lim WH: Convergence theorems for common solutions of various problems with nonlinear mapping. J. Inequal. Appl. 2014., 2014: Article ID 2Google Scholar
- Lv S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 19Google Scholar
- Wang Z, Lou W: A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems. J. Math. Comput. Sci. 2013, 3: 57–72.Google Scholar
- Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008MathSciNetView ArticleMATHGoogle Scholar
- Qin X, Shang M, Zhou H: Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces. Appl. Math. Comput. 2008, 200: 242–253. 10.1016/j.amc.2007.11.004MathSciNetView ArticleMATHGoogle Scholar
- Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035MathSciNetView ArticleMATHGoogle Scholar
- Qin X, Cho SY, Wang L: Iterative algorithms with errors for zero points of m -accretive operators. Fixed Point Theory Appl. 2013., 2013: Article ID 148Google Scholar
- Song J, Chen M: On generalized asymptotically quasi- ϕ -nonexpansive mappings and a Ky Fan inequality. Fixed Point Theory Appl. 2013., 2013: Article ID 237Google Scholar
- Zhang QN: Common solutions of equilibrium and fixed point problems. J. Inequal. Appl. 2013., 2013: Article ID 425Google Scholar
- Qin X, Cho SY, Wang L: A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl. 2014., 2014: Article ID 75Google Scholar
- Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 2006, 128: 191–201. 10.1007/s10957-005-7564-zMathSciNetView ArticleMATHGoogle Scholar
- Qin X, Cho SY, Kang SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 2011, 49: 679–693. 10.1007/s10898-010-9556-2MathSciNetView ArticleMATHGoogle Scholar
- Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289MathSciNetView ArticleMATHGoogle Scholar
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetMATHGoogle Scholar
- Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 2001, 5: 387–404.MathSciNetMATHGoogle Scholar
- Cho SY: Strong convergence of an iterative algorithm for sums of two monotone operators. J. Fixed Point Theory 2013., 2013: Article ID 6Google Scholar
- Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals. J. Math. Anal. Appl. 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
- Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar
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