# A strong convergence theorem on solving common solutions for generalized equilibrium problems and fixed-point problems in Banach space

- De-ning Qu
^{1, 2}and - Cao-zong Cheng
^{1}Email author

**2011**:17

https://doi.org/10.1186/1687-1812-2011-17

© Qu and Cheng; licensee Springer. 2011

**Received: **7 January 2011

**Accepted: **21 July 2011

**Published: **21 July 2011

## Abstract

In this paper, the common solution problem (P1) of generalized equilibrium problems for a system of inverse-strongly monotone mappings
and a system of bifunctions
satisfying certain conditions, and the common fixed-point problem (P2) for a family of uniformly quasi-*ϕ*-asymptotically 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

## Keywords

## 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 2008-2009, Takahashi and Zembayashi [8, 9] introduced several iterative sequences on finding a common solution of an equilibrium problem and a fixed-point 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 fixed-point 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 fixed-point problem for a family of uniformly quasi-*ϕ*-asymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Some other problems such as optimization problems (e.g. see [1, 4, 6]) and common zero-point 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*.

*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 fixed-point 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 Kadec-Klee property, where
in (P2) is a family of uniformly quasi-*ϕ*-asymptotically nonexpansive mappings and for each *i* ≥ 1, *S*_{
i
} is locally uniformly Lipschitz continuous or uniformly Hölder continuous with order Θ_{
i
} .

## 2. Preliminaries

*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 Hahn-Banach theorem, *Jx* ≠ ∅ for each *x* ∈ *E*.

*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 norm-to-weak*-continuous. - (iii)
If

*E*is a smooth and strictly convex reflexive Banach space, then*J*is single-valued, one-to-one and onto. - (iv)
Each uniformly convex Banach space

*E*has the*Kadec-Klee 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)
- (vii)
Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.

*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

*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*:

*C*→

*E** is said to be

*δ-inverse-strongly monotone*, if there exists a constant

*δ*> 0 such that

*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

*quasi*-

*ϕ*-

*asymptotically 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 *δ*-inverse-strongly monotone, then *A* is
-Lipschitz continuous. The class of quasi-*ϕ*-asymptotically 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)
- (2)

**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.

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*| ≤ 2*M* |*x* - *y*|, *x*, *y* ∈ *D*, ∀*n* ≥ 1. But *S* fails to be uniformly Lipschitz continuous.

*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)

□

**Lemma 2.2**. *Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee 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
.

*ϕ*(

*x*

_{ n },

*y*

_{ n }) → 0, by (2.2) we have ||

*x*

_{ n }|| - ||

*y*

_{ n }|| → 0. It follows from that

*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

*x*∈

*E*such that

*Jx*=

*f*

_{0}. By the definition of

*ϕ*, we obtain

which implies that
and
. It follows from Remark 2.1(iv) and (v) that *E** has the Kadec-Klee 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 Kadec-Klee property of *E*.

*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 δ-inverse-strongly monotone mapping and f* : *C* × *C* → ℝ *be a bifunction satisfying the following conditions*

(B_{1}) *f*(*z*, *z*) = 0, ∀*z* ∈ *C*;

(B_{3}) *for any z* ∈ *C*, *the function y* α *f*(*z*, *y*) *is convex and lower semicontinuous*;

*Then the following conclusions hold*:

- (1)
- (2)

(ii) *ϕ*(*z*, *K*_{
r
}*u*) + *ϕ*(*K*_{
r
}*u*, *u*) ≤ *ϕ*(*z*, *u*), ∀*z* ∈ *F*(*K*_{
r
}).

**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

*δ*-inverse-strongly monotone, by (B

_{4}), we have

*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

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].

*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
-inverse-strongly 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 Kadec-Klee 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, 2

*η*) → [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 Kadec-Klee property.

**Theorem 3.1**. *Suppose that*

_{1})

*for each*,

*the mapping A*

_{ k }:

*C*→

*E**

*is δ*

_{ k }-

*inverse-strongly 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*.

*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.

*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.

*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,

*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

*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

**Step 4**. Show that there exists
such that
.

*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

by Remark 2.1(i).

*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 (3.12) and the closeness of *S*_{
j
}, we have
for all *j* ≥ 1 and so
.

*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

which together with (3.14) and Lemma 2.3(3) implies that
, ∀*y* ∈ *C*. Therefore
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*

_{1})

*the mapping A*:

*C*→

*E**

*is a mapping with δ*-

*inverse*-

*strongly monotone*,

*the bifunction f*: C × C → ℝ

*satisfies*(B

_{1})-(B

_{3})

*and for some β*> 0

*with β*≤

*δ*,

*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;

_{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*
. □

## Declarations

## Authors’ Affiliations

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