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Strong convergence theorems for a generalized mixed equilibrium problem and variational inequality problems
Fixed Point Theory and Applications volume 2013, Article number: 65 (2013)
Abstract
In this paper, a new iterative scheme based on the extragradient-like method for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a strongly positive linear bounded operator and the set of solutions of a mixed equilibrium problem is proposed. A strong convergence theorem for this iterative scheme in Hilbert spaces is established. Our results extend recent results announced by many others.
MSC:49J30, 49J40, 47J25, 47H09.
1 Introduction
Let H be a real Hilbert space with the inner product and the norm . Let C be a nonempty closed convex subset of H. Recall that a mapping is said to be nonexpansive if
We denote by the set of fixed points of T. Let be the projection of H onto the convex subset C. Moreover, we also denote by ℝ the set of all real numbers.
Peng and Yao [1] considered the generalized mixed equilibrium problem of finding such that
where is a nonlinear mapping and is a function and is a bifunction. The set of solutions of problem (1.1) is denoted by GMEP.
In the case of , problem (1.1) reduces to the mixed equilibrium problem of finding such that
which was considered by Ceng and Yao [2]. GMEP is denoted by MEP.
In the case of , problem (1.1) reduces to the generalized equilibrium problem of finding such that
which was studied by Takahashi and Takahashi [3] and many others, for example, [4–10].
In the case of and , problem (1.1) reduces to the equilibrium problem of finding such that
The set of solutions of (1.2) is denoted by .
In the case , and , problem (1.1) reduces to the classical variational inequality problem of finding such that
The set of solutions of problem (1.3) is denoted by .
Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, for instance, [2, 3, 11]. Peng and Yao [1] considered iterative methods for finding a common element of the set of solutions of problem (1.1), the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping.
Let be two bifunctions and let be two nonlinear mappings. We consider the generalized equilibrium problem such that
where and are two constants.
In the case , problem (1.4) reduces to the general system of variational inequalities of finding such that
where and are two constants, which was considered by Ceng, Wang and Yao [12]. In particular, if , then problem (1.5) reduces to the system of variational inequalities of finding such that
which was studied by Verma [13].
If in (1.6), then (1.6) reduces to the classical variational inequality (1.3). Further, problem (1.6) is equivalent to the following projection formulas:
Recently, Ceng et al. [12] introduced and studied a relaxed extragradient method for finding solutions of problem (1.5).
Let be an infinite family of nonexpansive mappings of C into itself and be real sequences such that for every . For any , we define a mapping of C into itself as follows:
Such a mapping is called the W-mapping generated by and . Nonexpansivity of each ensures the nonexpansivity of . Moreover, in [1], it is shown that .
Throughout this article, let us assume that a bifunction and a convex function satisfy the following conditions:
-
(H1)
for all ;
-
(H2)
Θ is monotone, i.e., for all ;
-
(H3)
for each , is weakly upper semicontinuous;
-
(H4)
for each , is convex and lower semicontinuous;
-
(A1)
for each and , there exists a bounded subset and such that for any ,
-
(A2)
C is a bounded set.
Recently, Qin et al. [8] studied the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. More precisely, they proved the following theorem.
Theorem 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a bifunction from to ℝ which satisfies (H 1)-(H 4). Let be a finite family of nonexpansive mappings of C into H and let B be a μ-Lipschitz, relaxed -cocoercive mapping of C into H such that . Let f be a contraction of H into itself with a coefficient α () and let A be a strongly positive linear bounded operator with a coefficient such that . Assume that . Let and be sequences generated by and
where and satisfy
-
(i)
and ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
for some a, b with , ;
-
(v)
for all .
Then, both and converge strongly to , where , which solves the following variational inequality:
In this paper, motivated by Takahashi and Takahashi [3], Ceng, Wang and Yao [12], Peng and Yao [1] and Qin, Shang and Su [8], we introduce the general iterative scheme for finding a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of the generalized mixed equilibrium problem (1.1) and the set of solutions of the generalized equilibrium problem (1.4), which solves the variational inequality
where and Ω is the set of solutions of the generalized equilibrium problem (1.4). The results obtained in this paper improve and extend the recent results announced by Qin et al. [8], Chen et al. [14], Combetters and Hirstoaga [15], Iiduka and Takahashi [16], Marino and Xu [17], Takahashi and Takahashi [18], Wittmann [19] and many others.
2 Preliminaries
Let C be a nonempty closed convex subset of a real Hilbert space H. For every point , there exists a unique nearest point of C, denoted by , such that for all . Such a is called the metric projection of H onto C. We know that is a firmly nonexpansive mapping of H onto C, i.e.,
Further, for any and , if and only if
It is also known that H satisfies Opial’s condition [20] if for each sequence in H which converges weakly to a point , we have
Moreover, we assume that A is a bounded strongly positive operator on H with a constant , that is, there exists such that
A mapping is called β-inverse strongly monotone if there exists such that
It is obvious that any inverse strongly monotone mapping is Lipschitz continuous.
In order to prove our main results in the next section, we need the following lemmas and proposition.
Lemma 2.1 [2]
Let C be a nonempty closed convex subset of H. Let be a bifunction satisfying conditions (H1)-(H4) and let be a lower semicontinuous and convex function. For and , define a mapping
for all . Assume that either (A1) or (A2) holds. Then the following results hold:
-
(i)
for each and is single-valued;
-
(ii)
is firmly nonexpansive, i.e., for any ,
-
(iii)
;
-
(iv)
is closed and convex.
Remark 2.1 If , then is rewritten as .
By a similar argument as that in the proof of Lemma 2.1 in [12], we have the following result.
Lemma 2.2 Let C be a nonempty closed convex subset of H. Let be two bifunctions satisfying conditions (H1)-(H4) and let the mappings be -inverse strongly monotone and -inverse strongly monotone, respectively. Then, for given , is a solution of (1.4) if and only if is a fixed point of the mapping defined by
where .
The set of fixed points of the mapping Γ is denoted by Ω.
Proposition 2.1 [3]
Let C, H, Θ, φ and be as in Lemma 2.1. Then the following holds:
for all and .
Lemma 2.3 [21]
Assume that T is a nonexpansive self-mapping of a nonempty closed convex subset C of H. If T has a fixed point, then is demi-closed; that is, when is a sequence in C converging weakly to some and the sequence converges strongly to some y, it follows that .
Lemma 2.4 [22]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(i)
;
-
(ii)
or .
Then .
Lemma 2.5 [17]
Assume A is a strong positive linear bounded operator on a Hilbert space H with a coefficient and . Then .
The following lemma is an immediate consequence of an inner product.
Lemma 2.6 In a real Hilbert space H, the following inequality holds:
for all .
3 Main results
Now we state and prove our main results.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be three bifunctions which satisfy assumptions (H1)-(H4) and be a lower semicontinuous and convex function with assumption (A1) or (A2). Let the mappings be ζ-inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively. Let be a finite family of nonexpansive mappings of C into H such that . Let f be a contraction of C into itself with a constant α () and let A be a strongly positive linear bounded operator with a coefficient such that . Assume that . Let and let be a sequence defined by
where , , and satisfy the following conditions:
-
(i)
, and ;
-
(ii)
and ;
-
(iii)
and for all .
Then converges strongly to , which solves the following variational inequality:
and is a solution of problem (1.4), where .
Proof We divide the proof into several steps.
Step 1. is bounded.
Indeed, take arbitrarily. Since , F is ζ-inverse strongly monotone and , we obtain that for any ,
Putting and , we have
And since , we know that for any ,
Furthermore, from (3.1), we have
By induction, we obtain that for all ,
Hence is bounded. Consequently, we deduce immediately that , , and are bounded.
Step 2. .
It follows from the definition of that
Since is bounded and , are nonexpansive, .
Step 3. .
We estimate , and . From (3.1) we have
and
It follows from (3.5) and (3.6) that
Without loss of generality, let us assume that there exists a real number a such that for all n. Utilizing Proposition 2.1, we have
It follows from the definition of that
where is a constant such that . Next, we consider
where is a constant that . In a similar way, we obtain
where is an appropriate constant. Substituting (3.10) into (3.9), we have that
where is a constant such that . Similarly, we have
where is an appropriate constant. Hence it follows from (3.1), (3.7), (3.8), (3.11) and (3.12) that
where is a constant such that . By Lemma 2.4, we get .
Step 4. , and .
Indeed, from (3.1)-(3.4) we get
Therefore
Since and as , we have , and .
Step 5. , and .
Indeed, from (3.2), (3.3) and Lemma 2.1, we have
and
which imply that
and
It follows from (3.14) that
which gives that
Since , and as , we have
Also, from (3.4) and (3.13), we have
So, we have
Note that as . Then we have
In addition, from the firm nonexpansivity of , we have
which implies that
From (3.1), (3.4) and (3.17), we have
It follows that
Since as , we obtain
Thus, from (3.15), (3.16) and (3.18), we obtain that
and
Step 6. .
Indeed, observe that
From Step 3 and as , we have . Consequently,
Step 7. , where .
Indeed, take a subsequence of such that
Correspondingly, there exists a subsequence of . Since is bounded, there exists a subsequence of which converges weakly to w. Without loss of generality, we can assume that . Next we show . First, we prove that . Utilizing Lemma 2.1, we have for all
This shows that is nonexpansive. Note that
According to Lemma 2.2 and Lemma 2.3, we obtain .
Next, let us show that . From , we obtain
It follows from (H2) that
Replacing n by , we have
Let for all and . Then we have . It follows from (3.19) that
Since , we have . From the monotonicity of F, we have
From (H4), the weakly lower semicontinuity of φ, and , we have
as . By (H1), (H4) and (3.20), we obtain
Hence we obtain
Putting , we have
This implies that .
Since Hilbert spaces satisfy Opial’s condition, it follows from Step 5 that
which derives a contraction. This implies that . It follows from that .
Since , we have
Step 8. as .
Indeed, from Lemma 2.6 and (3.4), we have
which implies that
Put
and
Then we can write the last inequality as
It follows from condition (i) and Step 6 that
and
Hence, applying Lemma 2.4, we immediately obtain that as . This completes the proof. □
As corollaries of Theorem 3.1, we have the following results.
Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be three bifunctions which satisfy assumptions (H1)-(H4) and be a lower semicontinuous and convex function satisfying (A1) or (A2). Let the mappings be -inverse strongly monotone and -inverse strongly monotone, respectively. Let be a finite family of nonexpansive mappings of C into H such that . Let f be a contraction of H into itself with a constant α () and let A be a strongly positive linear bounded operator with a coefficient such that . Assume that . Let and let be a sequence defined by
where , , and satisfy the following conditions:
-
(i)
, and ;
-
(ii)
, for all ,
-
(iii)
and .
Then converges strongly to and is a solution of problem (1.4), where , which solves the following variational inequality:
Proof In Theorem 3.1, for all , is equivalent to
Putting , we obtain
□
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H and let be two bifunctions with satisfy assumptions (H1)-(H4). Let the mappings be ζ-inverse strongly monotone, -inverse strongly monotone and -inverse strongly monotone, respectively. Let be a finite family of nonexpansive mappings of C into H such that . Let f be a contraction of H into itself with a constant α () and let A be a strongly positive linear bounded operator with a coefficient such that . Assume that . Let and let be a sequence defined by
where , , and satisfy the following conditions:
-
(i)
, and ;
-
(ii)
and for all ;
-
(iii)
and .
Then converges strongly to and is a solution of problem (1.4), where , which solves the following variational inequality:
Proof Put and in Theorem 3.1. Then we have from (3.21) that
That is,
It follows that for all . We can obtain the desired conclusion easily. □
Remark 3.1 We can see easily that Takahashi and Takahashi [18], Peng and Yao’s [1] results are special cases of Theorem 3.1.
References
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.
Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 2008, 214: 186–201. 10.1016/j.cam.2007.02.022
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
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.2008.04.035
Ding XP: Iterative algorithm of solutions for a system of generalized mixed implicity equilibrium problems in reflexive Banach spaces. Appl. Math. Comput. 2012, 218: 4953–4961. 10.1016/j.amc.2011.10.060
Jaiboon C, Kumam P: A general iterative method for addressing mixed equilibrium problems and optimization problems. Nonlinear Anal. 2010, 73: 1180–1202. 10.1016/j.na.2010.04.041
Moudafi A: Mixed equilibrium problems: sensitivity analysis and algorithmic aspect. Comput. Math. Appl. 2002, 44: 1099–1108. 10.1016/S0898-1221(02)00218-3
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.2007.12.008
Yang S, Li W: Iterative solutions of a system of equilibrium problems in Hilbert spaces. Adv. Fixed Point Theory 2011, 1: 15–26.
Ye J, Huang J: Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. J. Math. Comput. Sci. 2011, 1: 1–18.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Ceng LC, Wang CY, Yao JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 2008, 67: 375–390. 10.1007/s00186-007-0207-4
Verma RU: On a new system of nonlinear variational inequalities and associated iterative algorithms. Math. Sci. Res. Hot-Line 1999, 3: 65–68.
Chen JM, Zhang LJ, Fan TG: Viscosity approximation methods for nonexpansive mappings and monotone mappings. J. Math. Anal. Appl. 2007, 334: 1450–1461. 10.1016/j.jmaa.2006.12.088
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.
Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings. Nonlinear Anal. 2005, 61: 341–350. 10.1016/j.na.2003.07.023
Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036
Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486–491. 10.1007/BF01190119
Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 595–597.
Goebel K, Kirk WA: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
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Jeong, J.U. Strong convergence theorems for a generalized mixed equilibrium problem and variational inequality problems. Fixed Point Theory Appl 2013, 65 (2013). https://doi.org/10.1186/1687-1812-2013-65
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DOI: https://doi.org/10.1186/1687-1812-2013-65