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An iterative approach to mixed equilibrium problems and fixed points problems
Fixed Point Theory and Applications volume 2013, Article number: 183 (2013)
Abstract
In the present paper, an iterative algorithm for solving mixed equilibrium problems and fixed points problems has been constructed. It is shown that under some mild conditions, the sequence generated by the presented algorithm converges strongly to the common solution of mixed equilibrium problems and fixed points problems. As an application, we can find the minimum norm element without involving projection.
MSC:47J05, 47J25, 47H09.
1 Introduction
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. For a nonlinear mapping and a bifunction , the mixed equilibrium problem is to find such that
The solution set of (1.1) is denoted by MEP. If , then (1.1) reduces to the following equilibrium problem of finding such that
The solution set of (1.2) is denoted by EP. If , then (1.1) reduces to the variational inequality problem of finding such that
The solution set of (1.3) is denoted by VI. Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others. See, e.g., [1–22].
For solving mixed equilibrium problem (1.1), Moudafi [9] introduced an iterative algorithm and proved a weak convergence theorem. Further, Takahashi and Takahashi [15] introduced the following iterative algorithm for finding an element of :
for all , where is a nonexpansive mapping. They proved that the sequence generated by (1.4) converges strongly to .
Recently, Yao and Shahzad [19] gave the following iteration process for nonexpansive mappings with perturbation: and
where and are sequences in , and the sequence in H is a small perturbation for the n-step iteration satisfying as . In fact, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate.
Using the ideas in [19], Chuang et al. [4] introduced the following iteration process for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points for a quasi-nonexpansive mapping with perturbation: and
for all . They showed that the sequence converges strongly to .
Motivated and inspired by the above works, in the present paper, we construct an iterative algorithm for solving mixed equilibrium problems and fixed points problems. It is shown that under some mild conditions the sequence generated by the presented algorithm converges strongly to the common solution of mixed equilibrium problems and fixed points problems. As an application, we can find the minimum norm element without involving projection.
2 Preliminaries
Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping is called α-inverse-strongly monotone if there exists a positive real number such that
It is clear that any α-inverse-strongly monotone mapping is monotone and -Lipschitz continuous. A mapping is said to be nonexpansive if for all . And a mapping is said to be strictly pseudo-contractive if there exists a constant such that
For such a case, we also say that S is a κ-strictly pseudo-contractive mapping.
Throughout this paper, we assume that a bifunction satisfies the following conditions:
-
(H1)
for all ;
-
(H2)
F is monotone, i.e., for all ;
-
(H3)
for each , ;
-
(H4)
for each , is convex and lower semicontinuous.
We need the following lemmas for proving our main results.
Lemma 2.1 [7]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction which satisfies conditions (H1)-(H4). Let and . Then there exists such that
Further, if , then we have
-
(i)
is single-valued and is firmly nonexpansive, i.e., for any , ;
-
(ii)
EP is closed and convex and .
Lemma 2.2 [19]
Let C, H, F and be as in Lemma 2.1. Then we have
for all and .
Lemma 2.3 [19]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping be α-inverse strongly monotone and be a constant. Then we have
In particular, if , then is nonexpansive.
Lemma 2.4 [23]
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose that for all and . Then .
Lemma 2.5 [24]
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a λ-strict pseudo-contraction. Then we have
-
(i)
is closed convex;
-
(ii)
for is nonexpansive.
Lemma 2.6 [25]
Let C be a nonempty closed and convex of a real Hilbert space H. Let be a κ-strictly pseudo-contractive mapping. Then is demi-closed at 0, i.e., if and , then .
Lemma 2.7 [16]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(1)
;
-
(2)
or .
Then .
3 Main results
In this section, we prove our main results.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Let be an α-inverse-strongly monotone mapping and let be a κ-strictly pseudo-contractive mapping. Suppose that . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r1)
for some ;
-
(r2)
and ;
-
(r3)
and ;
-
(r4)
, where and .
Then generated by (3.1) converges strongly to .
Proof Note that can be rewritten as for each n. Take . It is obvious that for all . By using the nonexpansivity of and the convexity of , we derive
Since A is α-inverse strongly monotone, we know from Lemma 2.3 that
It follows that
So, we have
Since , is bounded. Therefore, by induction, we deduce that is bounded. Hence, , and are also bounded.
Putting for all n, we have
It follows that
From Lemma 2.3, we know that is nonexpansive for all . Thus, we have is nonexpansive for all n due to the fact that . Then we get
By Lemma 2.2, we have
From (3.3)-(3.5), we obtain
Then
Therefore,
Since , and , we obtain
This together with Lemma 2.4 implies that
Consequently, we obtain
From (3.1) and (3.2), we have
Then we obtain
Since , and , we have
Next, we show . By using the firm nonexpansivity of , we have
We note that
Thus,
That is,
It follows that
Hence,
Since , , and , we deduce
This implies that
Put . We will finally show that .
Setting for all n. Taking in (3.7) to get . First, we prove . We take a subsequence of such that
It is clear that is bounded due to the boundedness of and . Then there exists a subsequence of which converges weakly to some point . Hence, also converges weakly to w. At the same time, from (3.6) and (3.8), we have
By the demi-closedness principle (see Lemma 2.6) and (3.9), we deduce .
Further, we show that w is also in MEP. From (3.1), we have
From (H2), we have
Put for all and . Then we have . So, from (3.10), we have
Since , we have . Further, from monotonicity of A, we have . So, from (H4), we have
From (H1), (H4) and (3.11), we also have
and hence
Letting , we have, for each ,
This implies . Hence, we have . This implies that
Note that . Then , . Therefore,
Since , we have
From (3.1), we have
It is clear that and . We can therefore apply Lemma 2.7 to conclude that . This completes the proof. □
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Let be an α-inverse-strongly monotone mapping and let be a nonexpansive mapping. Suppose that . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r1)
for some ;
-
(r2)
and ;
-
(r3)
and ;
-
(r4)
, where and .
Then generated by (3.12) converges strongly to .
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Let be a κ-strictly pseudo-contractive mapping. Suppose that . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r1)
for some ;
-
(r2)
and ;
-
(r3)
and ;
-
(r4)
, where and .
Then generated by (3.13) converges strongly to .
Corollary 3.4 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Let be a nonexpansive mapping. Suppose that . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r1)
for some ;
-
(r2)
and ;
-
(r3)
and ;
-
(r4)
, where and .
Then generated by (3.14) converges strongly to .
Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Let be an α-inverse-strongly monotone mapping and let be a κ-strictly pseudo-contractive mapping. Suppose that . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r2)
and ;
-
(r3)
and ;
-
(r4)
, where and .
Then generated by (3.15) converges strongly to , which is the minimum norm element in .
Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Let be a κ-strictly pseudo-contractive mapping. Suppose that . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r2)
and ;
-
(r3)
and ;
-
(r4)
, where and .
Then generated by (3.16) converges strongly to , which is the minimum norm element in .
Corollary 3.7 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Let be an α-inverse-strongly monotone mapping. Suppose that . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r2)
and ;
-
(r3)
;
-
(r4)
, where and .
Then generated by (3.17) converges strongly to , which is the minimum norm element in MEP.
Corollary 3.8 Let C be a nonempty closed convex subset of a real Hilbert space H and let be a bifunction satisfying conditions (H1)-(H4). Suppose . Let , and be sequences in C generated by
for all , where , and satisfy
-
(r2)
and ;
-
(r3)
;
-
(r4)
, where and .
Then generated by (3.18) converges strongly to , which is the minimum norm element in EP.
References
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Ceng LC, Al-Homidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009, 223: 967–974. 10.1016/j.cam.2008.03.032
Ceng LC, Schaible S, Yao JC: Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. J. Optim. Theory Appl. 2008, 139: 403–418. 10.1007/s10957-008-9361-y
Chuang CS, Lin LJ, Takahashi W: Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points. J. Glob. Optim. 2013. 10.1007/s10898-012-9911-6
Colao V, Acedo GL, Marino G: An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 2009, 71: 2708–2715. 10.1016/j.na.2009.01.115
Colao V, Marino G, Xu HK: An iterative method for finding common solutions of equilibrium and fixed point problems. J. Math. Anal. Appl. 2008, 344: 340–352. 10.1016/j.jmaa.2008.02.041
Combettes PL, Hirstoaga A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.
Konnov IV, Schaible S, Yao JC: Combined relaxation method for mixed equilibrium problems. J. Optim. Theory Appl. 2005, 126: 309–322. 10.1007/s10957-005-4716-0
Moudafi A: Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 2008, 9: 37–43.
Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. 477. In Lecture Notes in Economics and Mathematical Systems. Springer, Berlin; 1999:187–201.
Peng JW, Yao JC: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwan. J. Math. 2008, 12: 1401–1432.
Plubtieng S, Punpaeng R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 336: 455–469. 10.1016/j.jmaa.2007.02.044
Qin X, Cho SY, Kang SM: Some results on generalized equilibrium problems involving a family of nonexpansive mappings. Appl. Math. Comput. 2010, 217: 3113–3126. 10.1016/j.amc.2010.08.043
Qin X, Cho YJ, Kang SM: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 72: 99–112. 10.1016/j.na.2009.06.042
Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042
Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589
Yao Y, Liou YC: Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems. Abstr. Appl. Anal. 2010., 2010: Article ID 763506
Yao Y, Liou YC, Yao JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. 2007., 2007: Article ID 64363
Yao Y, Shahzad N: New methods with perturbations for non-expansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 79
Zegeye H, Ofoedu EU, Shahzad N: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput. 2010, 216: 3439–3449. 10.1016/j.amc.2010.02.054
Zegeye H, Shahzad N: A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems. Nonlinear Anal. 2011, 74: 263–272. 10.1016/j.na.2010.08.040
Zhang SS, Joseph HWL, Chan CK: Algorithms of common solutions for quasi variational inclusion and fixed point problems. Appl. Math. Mech. 2008, 29: 571–581. 10.1007/s10483-008-0502-y
Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 2005: 103–123.
Zhou H: Convergence theorems of fixed points for λ -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2008, 69: 456–462. 10.1016/j.na.2007.05.032
Marino C, Xu HK: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336–349. 10.1016/j.jmaa.2006.06.055
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Yao, Y., Liou, YC. & Kang, S.M. An iterative approach to mixed equilibrium problems and fixed points problems. Fixed Point Theory Appl 2013, 183 (2013). https://doi.org/10.1186/1687-1812-2013-183
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DOI: https://doi.org/10.1186/1687-1812-2013-183