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A new iterative algorithm for solving common solutions of generalized mixed equilibrium problems, fixed point problems and variational inclusion problems with minimization problems
Fixed Point Theory and Applications volume 2012, Article number: 111 (2012)
Abstract
In this article, we introduce a new general iterative method for solving a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solutions of generalized mixed equilibrium problems and the set of solutions of the variational inclusion for a β-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006), Su et al. (Nonlinear Anal. 69:2709-2719, 2008), Tan and Chang (Fixed Point Theory Appl. 2011:915629, 2011) and some authors.
MSC:46C05, 47H09, 47H10.
1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm , respectively. A mapping is said to be nonexpansive if . If C is bounded closed convex and S is a nonexpansive mapping of C into itself, then is nonempty [1]. A mapping is said to be a k-strictly pseudo-contraction[2] if there exists such that , where I denotes the identity operator on C. We denote weak convergence and strong convergence by notations ⇀ and →, respectively. A mapping A of C into H is called monotone if . A mapping A is called α-inverse-strongly monotone if there exists a positive real number α such that . A mapping A is called α-strongly monotone if there exists a positive real number α such that . It is obvious that any α-inverse-strongly monotone mappings A is a monotone and -Lipschitz continuous mapping. A linear bounded operator A is called strongly positive if there exists a constant with the property . A self mapping is called contraction on C if there exists a constant such that .
Let be a single-valued nonlinear mapping and be a set-valued mapping. The variational inclusion problem is to find such that
where θ is the zero vector in H. The set of solutions of (1.1) is denoted by . The variational inclusion has been extensively studied in the literature. See, e.g.[3–10] and the reference therein.
A set-valued mapping is called monotone if , and imply . A monotone mapping M is maximal if its graph of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for , for all imply .
Let B be an inverse-strongly monotone mapping of C into H and let be normal cone to C at i.e., and define
Then M is a maximal monotone and if and only if (see [11]).
Let be a set-valued maximal monotone mapping, then the single-valued mapping defined by
is called the resolvent operator associated with M, where λ is any positive number and I is the identity mapping. In the worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone and that a solution of problem (1.1) is a fixed point of the operator for all (see [12]).
Let F be a bifunction of into , where is the set of real numbers, be a mapping and be a real-valued function. The generalized mixed equilibrium problem for finding such that
The set of solutions of (1.3) is denoted by , that is
If and , the problem (1.3) is reduced into the equilibrium problem (see also [13]) for finding such that
The set of solutions of (1.4) is denoted by , that is
This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization and economics. Some methods have been proposed to solve the equilibrium problem, please consult [14–16].
If and , the problem (1.3) is reduced into the Hartmann-Stampacchia variational inequality[17] for finding such that
The set of solutions of (1.5) is denoted by . The variational inequality has been extensively studied in the literature [18].
If and , the problem (1.3) is reduced into the minimize problem for finding such that
The set of solutions of (1.6) is denoted by . Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
where A is a linear bounded operator, is the fixed point set of a nonexpansive mapping S and y is a given point in H[19].
In 2000, Moudafi [20] introduced the viscosity approximation method for nonexpansive mapping and prove that if H is a real Hilbert space, the sequence defined by the iterative method below, with the initial guess is chosen arbitrarily,
where satisfies certain conditions, converge strongly to a fixed point of S (say ) which is the unique solution of the following variational inequality:
In 2005, Iiduka and Takahashi [21] introduced following iterative process
where and for some a b with . They proved that under certain appropriate conditions imposed on and , the sequence generated by (1.10) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say ) which solve some variational inequality
In 2006, Marino and Xu [19] introduced a general iterative method for nonexpansive mapping. They defined the sequence generated by the algorithm
where and A is a strongly positive linear bounded operator. They proved that if and the sequence satisfies appropriate conditions, then the sequence generated by (1.12) converge strongly to a fixed point of S (say ) which is the unique solution of the following variational inequality:
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e. for ).
In 2008, Su et al.[22] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space:
for all , where and satisfy some appropriate conditions. Furthermore, they proved and converge strongly to the same point where .
In 2011, Tan and Chang [10] introduced following iterative process for is a sequence of nonexpansive mappings. Let be the sequence defined by
where and . The sequence defined by (1.16) converges strongly to a common element of the set of fixed points of nonexpansive mappings, the set of solutions of the variational inequality and the generalized equilibrium problem.
In this article, we mixed and modified the iterative methods (1.12), (1.15) and (1.16) by purposing the following new general viscosity iterative method: and
where , such that , with and with satisfy some appropriate conditions. The purpose of this article, we show that under some control conditions the sequence converges strongly to a common element of the set of fixed points of nonexpansive mappings, the common solutions of the generalized mixed equilibrium problem and the set of solutions of the variational inclusion in a real Hilbert space.
2 Preliminaries
Let H be a real Hilbert space with the inner product and the norm , respectively. Let C be a nonempty closed convex subset of H. Recall that the metric (nearest point) projection from H onto C assigns to each , the unique point in satisfying the property
The following characterizes the projection . We recall some lemmas which will be needed in the rest of this article.
Lemma 2.1 The functionis a solution of the variational inequality (1.5) if and only ifsatisfies the relationfor all.
Lemma 2.2 For a given, , , .
It is well known that is a firmly nonexpansive mapping of H onto C and satisfies
Moreover, is characterized by the following properties: and for all, ,
Lemma 2.3 ([23])
Letbe a maximal monotone mapping and letbe a monotone and Lipshitz continuous mapping. Then the mappingis a maximal monotone mapping.
Lemma 2.4 ([24])
Each Hilbert space H satisfies Opial’s condition, that is, for any sequencewith, the inequality, hold for eachwith.
Lemma 2.5 ([25])
Assume is a sequence of nonnegative real numbers such that
where and is a sequence in such that
-
(i)
;
-
(ii)
or .
Then.
Lemma 2.6 ([26])
Let C be a closed convex subset of a real Hilbert space H and letbe a nonexpansive mapping. Thenis demiclosed at zero, that is,
implies.
For solving the generalized mixed equilibrium problem, let us assume that the bifunction , the nonlinear mapping is continuous monotone and satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for any ;
(A3) for each fixed , is weakly upper semicontinuous;
(A4) for each fixed , is convex and lower semicontinuous;
(B1) for each and , there exist a bounded subset and such that for any ,
(B2) C is a bounded set.
Lemma 2.7 ([27])
Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a bifunction mapping satisfies (A 1)-(A 4) and letis convex and lower semicontinuous such that. Assume that either (B 1) or (B 2) holds. Forand, then there existssuch that
Define a mappingas follows:
for all. Then, the following hold:
-
(i)
is single-valued;
-
(ii)
is firmly nonexpansive, i.e., for any ,
-
(iii)
;
-
(iv)
is closed and convex.
Lemma 2.8 ([19])
Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficientand, then.
Lemma 2.9 ([28])
Let H be a real Hilbert space anda mapping.
-
(i)
If A is δ-strongly monotone and μ-strictly pseudo-contraction with , then is contraction with constant .
-
(ii)
If A is δ-strongly monotone and μ-strictly pseudo-contraction with , then for any fixed number , is contraction with constant .
3 Strong convergence theorems
In this section, we show a strong convergence theorem which solves the problem of finding a common element of , , and of an inverse-strongly monotone mapping in a real Hilbert space.
Theorem 3.1 Let H be a real Hilbert space, C be a closed convex subset of H. Let, be two bifunctions ofintosatisfying (A 1)-(A 4) andbe-inverse-strongly monotone mappings, be convex and lower semicontinuous function, be a contraction with coefficient α (), be a maximal monotone mapping and A be a δ-strongly monotone and μ-strictly pseudo-contraction mapping with, γ is a positive real number such that. Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of H into itself such that
Supposeis a sequence generated by the following algorithmarbitrarily:
where, such that, withandwithsatisfy the following conditions:
(C1): , , ,
(C2): , ,
(C3): ,
(C4): .
Thenconverges strongly to, wherewhich solves the following variational inequality:
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., for).
Proof Since B is β-inverse-strongly monotone mappings, we have
And , are -inverse-strongly monotone mappings, we have
In similar way, we can obtain
It is clear that if , , then , , are all nonexpansive. We will divide the proof into six steps.
Step 1. We will show is bounded. Put , . It follows that
By Lemma 2.7, we have for all . Then, we note that
In similar way, we can obtain
Put for all . From (3.1) and Lemma 2.9(ii), we deduce that
It follows from induction that
Therefore is bounded, so are , , , , and .
Step 2. We claim that . From (3.1), we have
We will estimate . On the other hand, from and , it follows that
and
Substituting in (3.11) and in (3.12), we get
and
From (A2), we obtain
and then
so
It follows that
Without loss of generality, let us assume that there exists a real number e such that , for all . Then, we have
and hence
where . Substituting (3.13) into (3.10) that
We note that
Since be nonexpansive, we have
On the other hand, from and , it follows that
and
Substituting in (3.17) and in (3.18), we get
and
From (A2), we obtain
and then
so
It follows that
Without loss of generality, let us assume that there exists a real number c such that , for all . Then, we have
and hence
where . Substituting (3.19) into (3.16), we have
Substituting (3.20) into (3.15), we obtain that
And substituting (3.13), (3.21) into (3.10), we get
where is a constant satisfying
This together with (C1)-(C4) and Lemma 2.5, imply that
From (3.20), we also have as .
Step 3. We show the followings:
-
(i)
;
-
(ii)
;
-
(iii)
.
For and , then we get
It follows that
By the convexity of the norm , we have
Substituting (3.8), (3.25) into (3.26), we obtain
So, we obtain
where . Since conditions (C1)-(C3) and , then we obtain that as . We consider this inequality in (3.25) that
Substituting (3.6) and (3.8) into (3.27), we have
Substituting (3.7) and (3.28) into (3.26), we obtain
So, we also have
where . Since conditions (C1)-(C3), , then we obtain that as . Substituting (3.24) into (3.27), we have
Substituting (3.8) and (3.30) into (3.26), we obtain
So, we also have
where . Since conditions (C1), (C2), (C4), and , then we obtain that as .
Step 4. We show the followings:
-
(i)
;
-
(ii)
;
-
(iii)
.
Since is firmly nonexpansive, we observe that
Hence, we have
Since is 1-inverse-strongly monotone, we compute
which implies that
Substitute (3.32) into (3.34), we have
Substitute (3.35) into (3.27), we have
Since is firmly nonexpansive, we observe that
Hence, we have
Substitute (3.36) and (3.37) into (3.26), we obtain
Then, we derive
By conditions (C1)-(C4), , , and . So, we have , , as . We note that . From , , and hence
It follows that
Since
So, by (3.40) and , we obtain
Therefore, we observe that
By condition (C1), we have as . Next, we observe that
By (3.42) and (3.43), we have as .
Step 5. We show that and . It is easy to see that is a contraction of H into itself. In fact, from Lemma 2.9, we have
Hence H is complete, there exists a unique fixed point such that . By Lemma 2.2 we obtain that for all .
Next, we show that , where is the unique solution of the variational inequality , . We can choose a subsequence of such that
As is bounded, there exists a subsequence of which converges weakly to w. We may assume without loss of generality that . We claim that . Since and by Lemma 2.6, we have .
Next, we show that . Since , we know that
It follows by (A2) that
Hence,
For and , let . From (3.44), we have
From , we have . Further, from (A4) and the weakly lower semicontinuity of , and , we have
From (A1), (A4) and (3.45), we have
and hence
Letting , we have, for each ,
This implies that . By following the same arguments, we can show that .
Lastly, we show that . In fact, since B is a β-inverse-strongly monotone, B is monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that is a maximal monotone. Let , since . Again since , we have , that is, . By virtue of the maximal monotonicity of , we have
and hence
It follows from , we have and that
It follows from the maximal monotonicity of that , that is, . Therefore, . It follows that
Step 6. We prove . By using (3.1) and together with Schwarz inequality, we have
Since is bounded, where for all . It follows that
where . By , we get . Applying Lemma 2.5, we can conclude that . This completes the proof. □
Corollary 3.2 Let H be a real Hilbert space, C be a closed convex subset of H. Let, be two bifunctions ofintosatisfying (A 1)-(A 4) andbe-inverse-strongly monotone mappings, be convex and lower semicontinuous function, be a contraction with coefficient α (), be a maximal monotone mapping. Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of H into itself such that
Supposeis a sequence generated by the following algorithmarbitrarily:
where, such that, withandwithsatisfy the conditions (C 1)-(C 4).
Thenconverges strongly to, wherewhich solves the following variational inequality:
Proof Putting and in Theorem 3.1, we can obtain desired conclusion immediately. □
Corollary 3.3 Let H be a real Hilbert space, C be a closed convex subset of H. Let, be two bifunctions ofintosatisfying (A 1)-(A 4) andbe-inverse-strongly monotone mappings, be convex and lower semicontinuous function. Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of H into itself such that
Supposeis a sequence generated by the following algorithmarbitrarily:
where, such that, withandwithsatisfy the conditions (C 1)-(C 4).
Thenconverges strongly to, wherewhich solves the following variational inequality:
Proof Putting is a constant in Corollary 3.2, we can obtain desired conclusion immediately. □
Corollary 3.4 Let H be a real Hilbert space, C be a closed convex subset of H. Let, be two bifunctions ofintosatisfying (A 1)-(A 4) andbe-inverse-strongly monotone mappings, be convex and lower semicontinuous function, be a contraction with coefficient α () and A is δ-strongly monotone and μ-strictly pseudo-contraction with, γ is a positive real number such that. Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of C into itself such that
Supposeis a sequence generated by the following algorithmarbitrarily:
where, such that, withandwithsatisfy the conditions (C 1)-(C 4).
Thenconverges strongly to, wherewhich solves the following variational inequality:
Proof Taking in Theorem 3.1, we can obtain desired conclusion immediately. □
Corollary 3.5 Let H be a real Hilbert space, C be a closed convex subset of H. Letbe a contraction with coefficient α (), A is δ-strongly monotone and μ-strictly pseudo-contraction with, γ is a positive real number such that. Let S be a nonexpansive mapping of C into itself such that
Supposeis a sequence generated by the following algorithmarbitrarily:
whereand satisfy the condition. Thenconverges strongly to, wherewhich solves the following variational inequality:
Proof Taking , and in Corollary 3.4, we can obtain desired conclusion immediately. □
Remark 3.6 Corollary 3.5 generalizes and improves the result of Marino and Xu [19].
Corollary 3.7 Let H be a real Hilbert space, C be a closed convex subset of H. Let, be two bifunctions ofintosatisfying (A 1)-(A 4) andbe-inverse-strongly monotone mappings, be convex and lower semicontinuous function, be a contraction with coefficient α (). Assume that either (B 1) or (B 2) holds. Let S be a nonexpansive mapping of C into itself such that
Supposeis a sequence generated by the following algorithmarbitrarily:
where, withandwithsatisfy the conditions (C 1)-(C 4).
Thenconverges strongly to, wherewhich solves the following variational inequality:
Proof Taking , , and in Theorem 3.1, we can obtain desired conclusion immediately. □
Remark 3.8 Corollary 3.7 generalizes and improves the result of Yao and Liou [29].
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Acknowledgements
This work was supported by the Higher Education Research Promotion and the National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC Project No. 55000613). Furthermore, the authors are grateful for the reviewers for the careful reading of the article and for the suggestions which improved the quality of this study.
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Jitpeera, T., Kumam, P. A new iterative algorithm for solving common solutions of generalized mixed equilibrium problems, fixed point problems and variational inclusion problems with minimization problems. Fixed Point Theory Appl 2012, 111 (2012). https://doi.org/10.1186/1687-1812-2012-111
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DOI: https://doi.org/10.1186/1687-1812-2012-111