 Research
 Open Access
 Published:
A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixedpoint problems in Banach space
Fixed Point Theory and Applications volume 2011, Article number: 17 (2011)
Abstract
In this paper, the common solution problem (P1) of generalized equilibrium problems for a system of inversestrongly monotone mappings and a system of bifunctions satisfying certain conditions, and the common fixedpoint problem (P2) for a family of uniformly quasiϕasymptotically nonexpansive and locally uniformly Lipschitz continuous or uniformly Hölder continuous mappings are proposed. A new iterative sequence is constructed by using the generalized projection and hybrid method, and a strong convergence theorem is proved on approximating a common solution of (P1) and (P2) in Banach space.
2000 MSC: 26B25, 40A05
1. Introduction
Recently, common solution problems (i.e., to find a common element of the set of solutions of equilibrium problems and/or the set of fixed points of mappings and/or the set of solutions of variational inequalities) with their applications have been discussed. Some authors such as in references [1–7] presented various iterative schemes and showed some strong or weak convergence theorems on common solution problems in Hilbert spaces. In 20082009, Takahashi and Zembayashi [8, 9] introduced several iterative sequences on finding a common solution of an equilibrium problem and a fixedpoint problem for a relatively nonexpansive mapping, and established some strong or weak convergence theorems. In 2010, Chang et al. [10] discussed the common solution of a generalized equilibrium problem and a common fixedpoint problem for two relatively nonexpansive mappings, and established a strong convergence theorem on the common solution problem. The frameworks of spaces in [8–10] are the uniformly smooth and uniformly convex Banach spaces. Chang et al. [11] established a strong convergence theorem on solving the common fixedpoint problem for a family of uniformly quasiϕasymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the KadecKlee property. Some other problems such as optimization problems (e.g. see [1, 4, 6]) and common zeropoint problems (e.g. see [10]) are closely related to common solution problems.
Throughout this paper, unless other stated, ℝ and are denoted by the set of the real numbers and the set {1, 2,..., N}, respectively, where N is any given positive integer. Let E be a real Banach space with the norm  · , E* be the dual of E, and 〈·,·〉 be the pairing between E and E*. Suppose that C is a nonempty closed convex subset of E.
Let be N mappings and be N bifunctions. For each , the generalized equilibrium problem for f_{ k } and A_{ k } is to seek such that
The common solution problem (P1) of generalized equilibrium problems for and is to seek an element in , where and G(k) is the set of solutions of (1.1). We write G instead of in the case of N = 1.
Let be a family of mappings. The common fixedpoint problem (P2) for is to seek an element in , where and F (S_{ i }) is the set of fixed points of S_{ i }.
Motivated by the works in [8–11], in this paper we will produce a new iterative sequence approximating a common solution of (P1) and (P2) (i.e., some point belonging to ), and show a strong convergence theorem in a uniformly smooth and strictly convex Banach space with the KadecKlee property, where in (P2) is a family of uniformly quasiϕasymptotically nonexpansive mappings and for each i ≥ 1, S_{ i } is locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ_{ i } .
2. Preliminaries
Let E be a real Banach space, and {x_{ n }} be a sequence in E. We denote by x_{ n } → x and x_{ n } ⇀ x the strong convergence and weak convergence of {x_{ n }}, respectively. The normalized duality mapping J : E → 2^{E*}is defined by
By the HahnBanach theorem, Jx ≠ ∅ for each x ∈ E.
A Banach space E is said to be strictly convex if for all x, y ∈ U = {u ∈ E : u = 1} with x ≠ y; to be uniformly convex if for each ε ∈ (0, 2], there exists γ > 0 such that for all x, y ∈ U with x  y ≥ ε; to be smooth if the limit
exists for every x, y ∈ U; to be uniformly smooth if the limit (2.1) exists uniformly for all x, y ∈ U.
Remark 2.1. The basic properties below hold (see [12]).

(i)
If E is a real uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.

(ii)
If E is a strictly convex reflexive Banach space, then J^{1} is hemicontinuous, that is, J^{1} is normtoweak*continuous.

(iii)
If E is a smooth and strictly convex reflexive Banach space, then J is singlevalued, onetoone and onto.

(iv)
Each uniformly convex Banach space E has the KadecKlee property, that is, for any sequence {x_{ n }} ⊂ E, if x_{ n } ⇀ x ∈ E and x_{ n } → x, then x_{ n } → x.

(v)
A Banach space E is uniformly smooth if and only if E* is uniformly convex.

(vi)
A Banach space E is strictly convex if and only if J is strictly monotone, that is,

(vii)
Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.
Now let E be a smooth and strictly convex reflexive Banach space. As Alber [13] and Kamimura and Takahashi [14] did, the Lyapunov functional ϕ : E × E → ℝ^{+} is defined by
It follows from [15] that ϕ(x, y) = 0 if and only if x = y, and that
Further suppose that C is a nonempty closed convex subset of E. The generalized projection (see [13]) Π_{ C }: E→C is defined by for each x ∈ E,
A mapping A : C → E* is said to be δinversestrongly monotone, if there exists a constant δ > 0 such that
A mapping S : C → C is said to be closed if for each {x_{ n }} ⊂ C, x_{ n } → x and Sx_{ n } → y imply Sx = y; to be quasiϕasymptotically nonexpansive (see [16]) if F(S) ≠ ∅, and there exists a sequence {l_{ n }} ⊂ [1, ∞) with l_{ n } → 1 such that
It is easy to see that if A : C → E* is δinversestrongly monotone, then A is Lipschitz continuous. The class of quasiϕasymptotically nonexpansive mappings contains properly the class of relatively nonexpansive mappings (see [17]) as a subclass.
Definition 2.1 (see [11]). Let be a sequence of mappings. is said to be a family of uniformly quasi ϕ asymptotically nonexpansive mappings, if and there exists a sequence {l_{ n }} ⊂ [1, ∞) with l_{ n } → 1 such that for each i ≥ 1,
Now we introduce the following concepts.
Definition 2.2. A mapping S : C → C is said

(1)
to be locally uniformly Lipschitz continuous if for any bounded subset D in C, there exists a constant L_{ D }> 0 such that

(2)
to be uniformly Hölder continuous with order Θ (Θ > 0) if there exists a constant L > 0 such that
Remark 2.2. It is easy to see that any uniformly Lipschitz continuous mapping (see [11]) is locally uniformly Lipschitz continuous, and is also uniformly Hölder continuous with order Θ = 1. However, the converse is not true.
Example 2.1. Suppose that S : ℝ → ℝ is defined by
Then S is locally uniformly Lipschitz continuous. In fact, for any bounded subset D in ℝ, setting M = 1 + sup{x : x ∈ D}, we have S^{n}x  S^{n}y ≤ 2M x  y, x, y ∈ D, ∀n ≥ 1. But S fails to be uniformly Lipschitz continuous.
Example 2.2. Suppose that S : ℝ  ℝ is defined by
S is uniformly Hölder continuous with order , since , ∀x, y ∈ ℝ, ∀n ≥ 1. But S fails to be uniformly Lipschitz continuous.
Lemma 2.1 (see [13, 14]). If C is a nonempty closed convex subset of a smooth and strictly convex reflexive Banach space E, then

(1)
ϕ(x, Π_{ C }(y)) + ϕ(Π_{ C }(y), y) ≥ ϕ(x, y), ∀x ∈ C, y ∈ E;

(2)
for × ∈ E and u ∈ C, one has
□
Lemma 2.2. Let E be a uniformly smooth and strictly convex Banach space with the KadecKlee property, {x_{ n }} and{y_{ n }} be two sequences of E, and . If and ϕ(x_{ n }, y_{ n }) → 0, then .
Proof. We complete this proof by two steps.
Step 1. Show that there exists a subsequence of {y_{ n }} such that .
In fact, since ϕ(x_{ n }, y_{ n }) → 0, by (2.2) we have x_{ n }  y_{ n } → 0. It follows from that
and so
Then {Jy_{ n }} is bounded in E*. It follows from Remark 2.1(v) and (vii) that E* is reflexive. Hence there exist a point f_{0} ∈ E* and a subsequence of {Jy_{ n }} such that
It follows from Remark 2.1(vii) and (iii) that there exists a point x ∈ E such that Jx = f_{0}. By the definition of ϕ, we obtain
By weak lower semicontinuity of norm  · , we have
which implies that and . It follows from Remark 2.1(iv) and (v) that E* has the KadecKlee property, and so by (2.4) and (2.5). By Remark 2.1(vii) and (ii), we have , which implies that by (2.3) and the KadecKlee property of E.
Step 2. Show that .
In fact, suppose that . For some given number ε_{0} > 0, there exists a positive integer sequence {n_{ k }} with n_{1} < n_{2} < · · · < n_{ k }< · · ·, such that
Replacing {y_{ n }} by in Step 1, there exists a subsequence of such that , which contradicts (2.6). □
Lemma 2.3. Let C be a nonempty closed convex subset of a smooth and strictly convex reflexive Banach space E, and let A : C → E* be a δinversestrongly monotone mapping and f : C × C → ℝ be a bifunction satisfying the following conditions
(B_{1}) f(z, z) = 0, ∀z ∈ C;
(B_{2}) ;
(B_{3}) for any z ∈ C, the function y α f(z, y) is convex and lower semicontinuous;
(B_{4}) for some β ≥ 0 with β ≤ δ,
Then the following conclusions hold:

(1)
For any r > 0 and u ∈ E, there exists a unique point z ∈ C such that
(2.7) 
(2)
For any given r > 0, define a mapping K_{ r } : E → C as follows: ∀u ∈ E,
We have (i) F(K_{ r }) = G and G is closed convex in C, where
(ii) ϕ(z, K_{ r }u) + ϕ(K_{ r }u, u) ≤ ϕ(z, u), ∀z ∈ F(K_{ r }).
(3) For each n ≥ 1, r_{ n }> a > 0 and u_{ n } ∈ C with , we have
Proof. (1) We consider the bifunction instead of f. It follows from the proof of Lemma 2.5 in [10] that satisfies (B_{1})(B_{3}). Since A is δinversestrongly monotone, by (B_{4}), we have
which implies is monotone. By Blum amd Oettli [18], for any r > 0 and u ∈ E, there exists z ∈ C such that (2.7) holds. Next we show that (2.7) has a unique solution. If for any given r > 0 and u ∈ E, z_{1} and z_{2} are two solutions of (2.7), then
and
Adding these two inequalities, we have
It follows from (2.8) that
which implies that z_{1} = z_{2} by Remark 2.1(vi).
(2) Since satisfies (B_{1})(B_{3}) and is monotone, the conclusion (2) follows from Lemmas 2.8 and 2.9 in [9].
(3) Since
we have
by the monotonicity of . It follows from . r_{ n }> a > 0 and Remark 2.1(i) that
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting n → ∞ in (2.9), we have , ∀y ∈ C. For any t ∈ (0, 1] and y ∈ C, setting , we have y_{ t } ∈ C and , which together with (B_{1}) implies that
Thus f(y_{ t }, y) + 〈y  y_{ t }, Ay_{ t }〉 ≥ 0, ∀y ∈ C, ∀t ∈ (0, 1]. Letting t ↓ 0, since z α f(z, y) + 〈y  z, Az〉 satisfies (B_{2}), we have , ∀y ∈ C.
Remark 2.3. If β = 0 in (B_{4}), that is, f is monotone, then the conclusions (1) and (2) in Lemma 2.3 reduce to the relating results of Lemmas 2.5 and 2.6 in [10], respectively.
Next we give an example to show that there exist the mapping A and the bifunction f satisfying the conditions of Lemma 2.3. However, f is not monotone.
Example 2.3. Define A : ℝ → ℝ and f : ℝ × ℝ → ℝ by ∈ ∀x ∈ ℝ and , ∀(x, y) ∈ ℝ × ℝ, respectively. It is easy to see that A is inversestrongly monotone, f satisfies (B_{1})(B_{3}), and , ∀(x, y) : ℝ × ℝ with .
Lemma 2.4 (see [12]). Let C be a nonempty closed convex subset of a real uniformly smooth and strictly convex Banach space E with the KadecKlee property, S : C → C be a closed and quasi ϕ asymptotically nonexpansive mapping with a sequence {l_{ n }} ⊂ [1, ∞), l_{ n } → 1. Then F(S) is closed convex in C.
Lemma 2.5 (see [11]). Let E be a uniformly convex Banach space, η > 0 be a positive number and B_{ η }(0) be a closed ball of E. Then, for any given sequence and for any given with , there exists a continuous, strictly increasing and convex function g : [0, 2η) → [0, ∞) with g(0) = 0 such that for any positive integers i, j with i < j,
□
3. Strong convergence theorem
In this section, let C be a nonempty closed convex subset of a real uniformly smooth and strictly convex Banach space E with the KadecKlee property.
Theorem 3.1. Suppose that
(C_{1}) for each , the mapping A_{ k } : C → E* is δ_{ k }inversestrongly monotone, the bifunction f_{ k } : C × C → ℝ satisfies (B_{1})(B_{3}), and for some β_{ k } ≥ 0 with β_{ k } ≤ δ_{ k },
(C_{2}) is a family of closed and uniformly quasi ϕ asymptotically nonexpansive mappings with a sequence {l_{ n }} ⊂ [1, ∞), l_{ n } → 1;
(C_{3}) for each i ≥ 1, S_{ i } is either locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ_{ i } (Θ_{ i }> 0), and is bounded in C.
(C_{4}) . Take the sequence generated by
where for each , with some a > 0, , and . If , ∀n ≥ 0 and lim inf_{n→∞}α_{n,0}α_{ n, i }> 0, ∀i ≥ 1, then .
Proof. We shall complete this proof by seven steps below.
Step 1. Show that , , H_{ n } and W_{ n } for all n ≥ 0 are closed convex.
In fact, is closed convex since for each i ≥ 1, F(S_{ i }) is closed convex by (C_{2}) and Lemma 2.4. is closed convex since for each , G(k) is closed convex by (C_{1}) and Lemma 2.3(2)(i). H_{0} = C is closed convex. Since ϕ(v,u_{N,n}) ≤ ϕ(v,x_{ n }) + ξ_{ n }is equivalent to
we know that H_{ n }(n ≥ 0) are closed convex. Finally, W_{ n } is closed convex by its definition. Thus and are well defined.
Step 2. Show that {x_{ n }} and are bounded.
From , ∀n ≥ 0 and Lemma 2.1(1), we have
which implies that {ϕ(x_{ n }, x_{0})} is bounded, and so is {x_{ n }} by (2.2). It follows from (C_{2}) that for all , i ≥ 1, n ≥ 1,
Hence for all i ≥ 1, is uniformly bounded, and so is by (2.2). Obviously,
Step 3. Show that , ∀n ≥ 0.
Since Banach space E is uniformly smooth, E* is uniformly convex, by Remark 2.1(v). For any given , any n ≥ 1 and any positive integer j, by (C_{2}) and Lemma 2.5, we have
Put , , ∀n ≥ 0. It follows from (3.3) and Lemma 2.3(2)(ii) that
which implies that if , then p ∈ H_{ n }, ∀n ≥ 0. Hence, , ∀n ≥ 0. By induction, now we prove that , ∀n ≥ 0. In fact, it follows from W_{0} = C that . Suppose that for some m ≥ 0. By the definition of and Lemma 2.1(2), we have
and so
which shows z ∈ W_{m+1}, so .
Step 4. Show that there exists such that .
Without loss of generalization, we can assume that , since {x_{ n }} is bounded and E is reflexive. Moreover, it follows that , ∀n ≥ 0 from H_{n+1}∩ W_{n+1}⊂ H_{ n } ∩ W_{ n } and the closeness and convexity of H_{ n } ∩ W_{ n }. Noting that
we have
by (3.1). It follows that
and so by . Hence,
by the KadecKlee property of E, and so
by Remark 2.1(i).
Step 5. Show that .
Since x_{n+1}∈ C, setting u = x_{n+1}in (3.1), we have
By (3.5),
By x_{n+1}∈ H_{n+1}, (3.2) and (3.8), we have
which together with (3.6) and Lemma 2.2 implies that
For any j ≥ 1 and any given , it follows from (3.2)(3.4) and (3.9) that
which implies that
since , ∀i ≥ 1. We obtain
since g(0) = 0 and g is strictly increasing and continuous. By (3.7) and (3.11), we have and for all j ≥ 1. It follows from Remark 2.1(ii) that , which implies
by the uniform boundedness of and the KadecKlee property of E. Thus
By (C_{3}) and (3.6), we have
Hence, for each j ≥ 1,
By (3.12) and the closeness of S_{ j }, we have for all j ≥ 1 and so .
Step 6. Show that .
In fact, it is easy to see that for each , and , the sequence {ϕ(p, u_{k,n})} is bounded by (3.2), (3.4) and the boundedness of {x_{ n }} and , which implies that {u_{k,n}} is bounded in C by (2.2). Since , by (3.2), (3.3), (3.5) and (3.10), we have
It follows from Lemma 2.2 that
Furthermore, it follows from (3.4) and Lemma 2.3(2)(ii) that for any given ,
which implies
by Remark 2.1(i), (3.9) and (3.13). Then by (3.13) and Lemma 2.2. Similarly, we also obtain . Hence, together with (3.9) and (3.13), for each ,
For each , since , we have
which together with (3.14) and Lemma 2.3(3) implies that , ∀y ∈ C. Therefore and so .
Step 7. Show that .
In fact, letting , by and , we have
It follows from (3.6) that
Hence, , and so . □
Setting N = 1, u_{0},_{ n } = y_{ n }and u_{N,n}= u_{ n }in Theorem 3.1, we can obtain the following result.
Corollary 3.1 Suppose that
(D_{1}) the mapping A : C → E* is a mapping with δ inversestrongly monotone, the bifunction f : C × C → ℝ satisfies (B_{1})(B_{3}) and for some β > 0 with β ≤ δ,
(D_{2}) both (C_{2}) and (C_{3}) hold, and Take the sequence generated by
where , for some a > 0 and. If , ∀_{ n } ≥ 0 and lim inf_{n→∞}α_{n,0}α_{n,i}> 0, ∀i ≥ 1, then . □
Furthermore, if S_{ i } = S, i ≥ 1 in Corollary 3.1, the following corollary can be obtained immediately.
Corollary 3.2. Suppose that, besides (D1),
(E_{1}) S : C → C is closed and quasi ϕ asymptotically nonexpansive with {l_{ n }} ⊂ [1, ∞), l_{ n } → 1;
(E_{2}) S is either locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ (Θ > 0), F(S) is bounded in C and F(S) ∩ G ≠ ∅. Take the sequence generated by
where , for some a > 0 and ξ = sup_{u∈F(S)}(l_{ n }1)ϕ(u, x_{ n }) . If lim inf_{n→∞}α_{ n }(1 α_{ n }) > 0, then . □
References
Zhang F, Su YF: A general iterative method of fixed points for equilibrium problems and optimization problems. J Syst Sci Complex 2009, 22: 503–517. 10.1007/s1142400991826
Ceng LC, AlHomidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudocontraction mappings. J Comput Appl Math 2009, 223: 967–974. 10.1016/j.cam.2008.03.032
Zhang SS, Rao RF, Huang JL: Strong convergence theorem for a generalized equilibrium problem and a k strict pseudocontraction in Hilbert spaces. Appl Math Mech English edition. 2009,30(6):685–694. 10.1007/s104830090602z
Peng JW, Yao JC: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems. Math Comput Model 2009, 49: 1816–1828. 10.1016/j.mcm.2008.11.014
Peng JW, Yao JC: A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings. Nonlinear Anal Theory Methods Appl 2009, 71: 6001–6010. 10.1016/j.na.2009.05.028
Cianciaruso F, Marino G, Muglia L: Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces. J Optim Theory Appl 2010, 146: 491–509. 10.1007/s109570099628y
Qin XL, Chang SS, Cho YJ: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal Real World Appl 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017
Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory Appl 2008, 2008: 11. (Article ID 528476)
Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal Theory Methods Appl 2009,70(1):45–57. 10.1016/j.na.2007.11.031
Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications. Nonlinear Anal Theory Methods Appl 2010, 73: 2260–2270. 10.1016/j.na.2010.06.006
Chang SS, Kim JK, Wang XR: Modified block iterative algorithm for solving convex feasibility problems in Banach spacesm. J Inequal Appl 2010, 2010: 14. (Article ID 869684)
Cioranescu I: Geometry of Banach spaces, Duality Mappings and Nonlinear Problems. In Mathematics and Its Applications. Volume 62. Edited by: Hazewinkel M. Kluwer Academic Publishers, Dordecht; 1990.
Alber YI: Metric and generalized projection operators in Banach spaces: properities and applications. In Theory and Applications of Nonlinear operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartosator AG. Marcel Dekker, New York; 1996:15–50.
Kamimura S, Takahashi W: Strong convergence of a proxiamltype algorithm in a Banach space. SIAM J Optim 2002,13(3):938–945. 10.1137/S105262340139611X
Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansivetype mappings in Banach spaces. SIAM J Optim 2008,19(2):824–835. 10.1137/070688717
Zhou HY, Gao GL, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi ϕ asymptotically nonexpansive mappings. J Appl Math Comput 2010, 32: 453–464. 10.1007/s1219000902634
Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2004,2004(1):37–47. 10.1155/S1687182004310089
Blum E, Oettli W: From optimization and variational inequalities and equilibrium problems. Math Student 1994, 63: 123–145.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no completing interests.
Authors' contributions
All the authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Qu, Dn., Cheng, Cz. A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixedpoint problems in Banach space. Fixed Point Theory Appl 2011, 17 (2011). https://doi.org/10.1186/16871812201117
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16871812201117
Keywords
 Common solution
 Equilibrium problem
 Fixedpoint problem
 Iterative sequence
 Strong convergence