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# Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-*ϕ*-nonexpansive mappings

- Jong Kyu Kim
^{1}Email author

**2011**:10

https://doi.org/10.1186/1687-1812-2011-10

© Kim; licensee Springer. 2011

**Received: **29 January 2011

**Accepted: **24 June 2011

**Published: **24 June 2011

## Abstract

We consider a hybrid projection method for finding a common element in the fixed point set of an asymptotically quasi-*ϕ*-nonexpansive mapping and in the solution set of an equilibrium problem. Strong convergence theorems of common elements are established in a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property.

**2000 Mathematics subject classification**: 47H05, 47H09, 47H10, 47J25

## Keywords

- Asymptotically quasi-
*ϕ*-nonexpansive mapping - Relatively non-expansive mapping
- Generalized projection
- Equilibrium problem
- Lower semi-continuous

## 1. Introduction and Preliminaries

Let *E* be a real Banach space, *E** the dual space of *E* and *C* a nonempty closed convex subset of *E*. Let *f* be a bifunction from *C* × *C* to ℝ, where ℝ denotes the set of real numbers.

*p*∈

*EP*(

*f*) if and only if

*p*is a solution of the following variational inequality problem. Find

*p*such that

Numerous problems in physics, optimization and economics reduce to find a solution of (1.1) (see [1–4]). Let *T* : *C* → *C* be a mapping.

then *Tx*_{0} = *y*_{0}.

A point *x* ∈ *C* is a fixed point of *T* provided *Tx* = *x*. In this paper, we denote *F*(*T*) the fixed point set of *T* and denote → and ⇀ the strong convergence and weak convergence, respectively.

*T*is said to be asymptotically nonexpansive if there exists a sequence {

*k*

_{ n }} ⊂ [1, ∞) with

*k*

_{ n }→ 1 as

*n*→ ∞ such that

*T*is said to be asymptotically quasi-nonexpansive if

*F*(

*T*) ≠ Ø and there exists a sequence {

*k*

_{ n }} ⊂ [1, ∞) with

*k*

_{ n }→ 1 as

*n*→ ∞ such that

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [5] in 1972. They proved that if *C* is nonempty bounded closed and convex then every asymptotically nonexpansive self-mapping *T* on *C* has a fixed point in uniformly convex Banach spaces. Further, the fixed point set of *T* is closed and convex.

Recently, many authors considered the problem of finding a common element in the set of fixed points of a nonexpansive mapping and in the set of solutions of the equilibrium problem (1.1) based on iterative methods in the framework of real Hilbert spaces; see, for instance [4, 6–14] and the references therein. However, there are few results presented in Banach spaces.

In this paper, we will consider the problem in a Banach space. Before proceeding further, we give some definitions and propositions in Banach spaces.

*E*be a Banach space with the dual

*E**. We denote by

*J*the normalized duality mapping from

*E*to 2

^{E*}defined by

where 〈•,•〉 denotes the generalized duality pairing.

*E*is said to be strictly convex if for all

*x*,

*y*∈

*E*with ||

*x*|| = ||

*y*|| = 1 and

*x*≠

*y*. It is said to be uniformly convex if lim

_{n→∞}||

*x*

_{ n }-

*y*

_{ n }|| = 0 for any two sequences {

*x*

_{ n }} and {

*y*

_{ n }} in

*E*such that ||

*x*

_{ n }|| = ||

*y*

_{ n }|| = 1 and

*U*

_{ E }= {

*x*∈

*E*: ||

*x*|| = 1} be the unit sphere of

*E*. Then the Banach space

*E*is said to be smooth provided

exists for each *x*, *y* ∈ *U*_{
E
} . It is said to be uniformly smooth if the limit (1.3) is attained uniformly for *x*, *y* ∈ *U*_{
E
} . It is well known that if *E* is uniformly smooth, then *J* is uniformly norm-to-norm continuous on each bounded subset of *E*. It is also well known that if *E* is uniformly smooth if and only if *E** is uniformly convex.

Recall that a Banach space *E* has the Kadec-Klee property [15–17], if for any sequence {*x*_{
n
} } ⊂ *E* and *x* ∈ *E* with *x*_{
n
} ⇀ *x* and ||*x*_{
n
} || → ||*x*||, then ||*x*_{
n
} - *x*|| → 0 as *n* → ∞. It is well known that if *E* is a uniformly convex Banach space, then *E* has the Kadec-Klee property.

As we all know that if *C* is a nonempty closed convex subset of a Hilbert space *H* and *P*_{
C
} : *H* → *C* is the metric projection of *H* onto *C*, then *P*_{
C
} is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [18] recently introduced a generalized projection operator Π _{
C
} in a Banach space *E* which is an analogue of the metric projection in Hilbert spaces.

*H*, (1.4) is reduced to

*ϕ*(

*x*,

*y*) = ||

*x-y*||

^{2},

*x*,

*y*∈

*H*. The generalized projection Π

_{ C }:

*E*→

*C*is a mapping that assigns to an arbitrary point

*x*∈

*E*the minimum point of the functional

*ϕ*(

*x*,

*y*), that is, , where is the solution to the minimization problem

_{ C }follows from the properties of the functional

*ϕ*(

*x*,

*y*) and strict monotonicity of the mapping

*J*(see, for example, [15, 17–19]). We know that Π

_{ C }=

*P*

_{ C }in Hilbert spaces. It is obvious from the definition of function

*ϕ*that

**Remark 1.1**. Let *E* be a reflexive, strictly convex and smooth Banach space. Then for *x*, *y* ∈ *E*, *ϕ*(*x*, *y*) = 0 if and only if *x* = *y*. It is sufficient to show that if *ϕ*(*x*, *y*) = 0 then *x* = *y*. From (1.5), we have ||*x*|| = ||*y*||. This implies that 〈*x*, *Jy*〉 = ||*x*||^{2} = ||*Jy*||^{2}. From the definition of *J*, we have *Jx* = *Jy*. Therefore, we have *x* = *y* (see [15, 17]).

*C*be a nonempty closed convex subset of

*E*and

*T*a mapping from

*C*into itself. A point

*p*in

*C*is said to be an asymptotic fixed point of

*T*[20] if

*C*contains a sequence {

*x*

_{ n }} which converges weakly to

*p*such that

The set of asymptotic fixed points of *T* will be denoted by
.

for all *x* ∈ *C* and *p* ∈ *F*(*T*).

*T*is said to be relatively asymptotically nonexpansive [24] if and there exists a sequence {

*k*

_{ n }} ⊂ [1, ∞) with

*k*

_{ n }→ 1 as

*n*→ ∞ such that

for all *x* ∈ *C*, *p* ∈ *F*(*T*) and *n* ≥ 1. The asymptotic behavior of a relatively nonexpansive mapping was studied in [21–23].

for all *x*, *y* ∈ *C*.

for all *x* ∈ *C* and *p* ∈ *F*(*T*).

*T*is said to be asymptotically

*ϕ*-nonexpansive if there exists a sequence {

*k*

_{ n }} ⊂ [1, ∞) with

*k*

_{ n }→ 1 as

*n*→ ∞ such that

for all *x*, *y* ∈ *C*.

*T*is said to be asymptotically quasi-

*ϕ*-nonexpansive [27, 28] if

*F*(

*T*) ≠ ∅ and there exists a sequence {

*k*

_{ n }} ⊂ [0, ∞) with

*k*

_{ n }→ 1 as

*n*→ ∞ such that

for all *x* ∈ *C*, *p* ∈ *F*(*T*) and *n* ≥ 1.

**Remark 1.2**. The class of (asymptotically) quasi-*ϕ*-nonexpansive mappings is more general than the class of relatively (asymptotically) nonexpansive mappings which requires the restriction:
. In the framework of Hilbert spaces, (asymptotically) quasi-*ϕ*-nonexpansive mappings is reduced to (asymptotically) quasi-nonexpansive mappings (cf. [29–32]).

We assume that *f* satisfies the following conditions for studying the equilibrium problem (1.1).

(A1): *f*(*x*, *x*) = 0∀*x* ∈ *C*;

(A2): *f* is monotone, i.e., *f*(*x*, *y*) + *f*(*y*, *x*) ≤ 0∀*x*, *y* ∈ *C*;

(A3): lim sup_{t↓0}*f* (*tz* + (1 - *t*)*x*, *y*) ≤ *f*(*x*, *y*)∀*x*, *y*, *z* ∈ *C*;

(A4): for each *x* ∈ *C*, *y* α *f*(*x*, *y*) is convex and weakly lower semi-continuous.

Recently, Takahashi and Zembayshi [33] considered the problem of finding a common element in the fixed point set of a relatively nonexpansive mapping and in the solution set of the equilibrium problem (1.1) (cf. [32]).

**Theorem TZ**. ([33])

*Let E be a uniformly smooth and uniformly convex Banach space and let C be a nonempty closed convex subset of E. Let f be a bifunction from C*×

*C to*ℝ

*satisfying*(

*A*1)

*-*(

*A*4)

*and let T be a relatively nonexpansive mapping from C into itself such that F*(

*T*) ∩

*EP*(

*f*) ≠ Ø.

*Let*{

*x*

_{ n }}

*be a sequence generated by*

*and* {*r*_{
n
}} ⊂ [*a*, ∞) *for some a* > 0. *Then* {*x*_{
n
}} *converges strongly to* ∏_{F(T)∩EP(f)}*x*, *where* ∏_{F(T)∩EP(f)}*is the generalized projection of E onto F* (*T*) ∩ *EP* (*f* ).

Very recently, Qin et al. [25] further improved Theorem TZ by considering shrinking projection methods which were introduced in [34] for quasi-*ϕ*-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.

**Theorem QCK**. [25]

*Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Ban ach space E. Let f be a bifunction from C*×

*C to*ℝ

*satisfying*(

*A*1)-(

*A*4)

*and let T*:

*C*→

*C be a closed quasi-ϕ-nonexpansive mappings such that*.

*Let*{

*x*

_{ n }}

*be a sequence generated in the following manner:*

*and* {*r*_{
n
} } ⊂ [*a*, ∞) *for some a* > 0. *Then* {*x*_{
n
} } *converges strongly to*
.

In this paper, we considered the problem of finding a common element in the fixed point set of an asymptotically quasi-*ϕ*-nonexpansive mapping which is an another generalization of asymptotically nonexpansive mappings in Hilbert spaces and in the solution set of the equilibrium problem (1.1). The results presented this paper mainly improve the corresponding results announced in [33].

In order to prove our main results, we need the following lemmas.

**Lemma 1.3**. [18]

*Let C be a nonempty closed convex subset of a smooth Banach space E and x*∈

*E. Then x*

_{0}= ∏

_{ C }

*x if and only if*

**Lemma 1.4**. [18]

*Let E be a reflexive, strictly convex and smooth Banach space, C a nonempty closed convex subset of E and x*∈

*E*.

*Then*

**Lemma 1.5**. *Let E be a strictly convex and smooth Banach space, C a nonempty closed convex subset of E and T* : *C* → *C a quasi-* *ϕ* *-nonexpansive mapping. Then F*(*T*) *is a closed convex subset of C*.

*Proof*. Let {

*p*

_{ n }} be a sequence in

*F*(

*T*) with

*p*

_{ n }→

*p*as

*n*→ ∞. Then we have to prove that

*p*∈

*F*(

*T*) for the closedness of

*F*(

*T*). From the definition of

*T*, we have

Letting *n* → ∞ in the above equality, we see that *ϕ*(*p*, *Tp*) = 0. This shows that *p* = *Tp*.

*F*(

*T*) is convex. To end this, for arbitrary

*p*

_{1},

*p*

_{2}∈

*F*(

*T*),

*t*∈ (0, 1), putting

*p*

_{3}=

*tp*

_{1}+ (1 -

*t*)

*p*

_{2}, we prove that

*Tp*

_{3}=

*p*

_{3}. Indeed, from the definition of

*ϕ*, we see that

This implies that *p*_{3} ∈ *F* (*T* ). This completes the proof.

Now we will improve the above Lemma 1.6 as follows.

**Lemma 1.6**. *Let E be a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property, C a nonempty closed convex subset of E and T* : *C* → *C a closed and asymptotically quasi-* *ϕ* *-nonexpansive mapping. Then F*(*T*) *is a closed convex subset of C*.

*Proof*. It is easy to check that the closedness of *F*(*T*) can be deduced from the closedness of *T*. We mainly show that *F*(*T*) is convex. To end this, for arbitrary *p*_{1}, *p*_{2} ∈ *F*(*T*), *t* ∈ (0, 1), putting *p*_{3} = *tp*_{1} + (1 - *t*)*p*_{2}, we prove that *Tp*_{3} = *p*_{3}.

*J*(

*T*

^{ n }

*p*

_{3})}is bounded. Note that

*E** is reflexive; we may, without loss of generality, assume that

*J*(

*T*

^{ n }

*p*

_{3}) ⇀

*e** ∈

*E**. In view of the reflexivity of

*E*, we have

*J*(

*E*) =

*E**. This shows that there exists an element

*e*∈

*E*such that

*Je*=

*e**. It follows that

This implies that *p*_{3} = *e*, that is, *Jp*_{3} = *e**. It follows that *J*(*T*^{
n
}*p*_{3}) ⇀ *Jp*_{3} ∈ *E**.

*J*

^{-1}:

*E** →

*E*is demi-continuous, we see that

*T*

^{ n }

*p*

_{3}⇀

*p*

_{3}. By virtue of the Kadec-Klee property of

*E*and (1.8), we have

*T*

^{ n }

*p*

_{3}→

*p*

_{3}as

*n*→ ∞. Hence

as *n* → ∞. In view of the closedness of *T*, we can obtain that *p*_{3} ∈ *F* (*T*). This shows that *F*(*T*) is convex. This completes of proof

**Lemma 1.7**. [35, 36]

*Let E be a smooth and uniformly convex Banach space and let r >*0.

*Then there exists a strictly increasing, continuous and convex function g*: [0, 2

*r*] →

*R such that g*(0) = 0

*and*

*for all x*, *y* ∈ *B*_{
r
} = {*x* ∈ *E* : ||*x*|| ≤ *r*} *and t* ∈ [0, 1].

**Lemma 1.8**. *Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let f be a bifunction from C* × *C to* ℝ *satisfying* (*A* 1)*-*(*A* 4). *Let r >* 0 *and x* ∈ *E. Then we have the followings*.

*Then the following conclusions hold:*

(1): *S*_{
r
} *is single-valued;*

(3): *F*(*S*_{
r
} ) = *EP*)(*f*);

(4): *S*_{
r
} *is quasi-* *ϕ* *-nonexpansive;*

(5): *ϕ*(*q*, *S*_{
r
}*x*) + *ϕ*(*S*_{
r
}*x, x*) ≤ *ϕ* (*q*, *x*), ∀*q* ∈ *F*(*S*_{
r
} );

(6): *EP*(*f*) *is closed and convex*.

## 2. Main Results

**Theorem 2.1**.

*Let E be a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property and C a nonempty closed convex subset of E. Let f be a bifunction from C × C to*ℝ

*satisfying*(

*A*1)-(

*A*4)

*and T*:

*C*→

*C a closed and asymptotically quasi-ϕ-nonexpansive mapping. Assume that T is asymptotically regular on C and*

*is nonempty and bounded. Let*{

*x*

_{ n }}

*be a sequence generated in the following manner:*

*where*
*for each n* ≥ 1, {α_{
n
}} *is a real sequence in* [0, 1] *such that* lim inf_{n→ ∞}*α*_{
n
}(1 - *α*_{
n
}) > 0, {*r*_{
n
}} *is a real sequence in* [*a*, ∞), *where a is some positive real number and J is the duality mapping on E. Then the sequence* {*x*_{
n
}} *converges strongly to*
, *where*
*is the generalized projection from E onto*
.

*Proof*. First, we show that

*C*

_{ n }is closed and convex by induction on

*n*≥ 1. It is obvious that

*C*

_{1}=

*C*is closed and convex. Suppose that

*C*

_{ m }is closed and convex for some integer

*m*. For

*z*∈

*C*

_{ m }, we see that

*ϕ*(

*z*,

*u*

_{ m }) ≤

*ϕ*(

*z*,

*x*

_{ m }) + (

*k*

_{ m }-1)

*M*

_{ m }is equivalent to

It is easy to see that *C*_{m+1}is closed and convex. This proves that *C*_{
n
} is closed and convex for each *n* ≥ 1. This in turn shows that
*x*_{0} is well defined. Putting
, we from Lemma 1.8 see that
is quasi-*ϕ*-nonexpansive.

*n*≥ 1. Indeed, is obvious. Suppose that for some positive integer

*m*. Then, , we have

which shows that *w* ∈ *C*_{m+1}. This implies that
for each *n* ≥ 1.

*n*≥ 1. This shows that the sequence

*ϕ*(

*x*

_{ n },

*x*

_{0}) is bounded. From (1.5), we see that the sequence {

*x*

_{ n }} is also bounded. Since the space is reflexive, we may, without loss of generality, assume that

*x*

_{ n }⇀

*p*. Not that

*C*

_{ n }is closed and convex for each

*n*≥ 1. It is easy to see that

*p*∈

*C*

_{ n }for each

*n*≥ 1. Note that

Hence, we have ||*x*_{
n
} || → ||*p*|| as *n* → ∞. In view of the Kadec-Klee property of *E*, we obtain that *x*_{
n
} → *p* as *n* → ∞.

*p*∈

*F*(

*T*). By the construction of

*C*

_{ n }, we have that

*C*

_{n+1}⊂

*C*

_{ n }and . It follows that

*Ju*

_{ n }} is bounded. Note that

*E*is reflexive and

*E** is also reflexive. We may assume that

*Ju*

_{ n }→

*x** ∈

*E**. In view of the reflexivity of

*E*, we see that

*J*(

*E*) =

*E**. This shows that there exists an

*x*∈

*E*such that

*Jx*=

*x**. It follows that

*p*=

*x*, which in turn implies that

*x** =

*Jp*. It follows that

*Ju*

_{ n }→

*Jp*∈

*E**. From (2.4) and

*E** has the Kadec-Klee property, we obtain that

*J*

^{-1}:

*E** →

*E*is demi-continuous. It follows that

*u*

_{ n }→

*p*. From (2.3) and

*E*has the Kadec-Klee property, we obtain that

*r*= sup

_{n≥0}{||

*x*

_{ n }||, ||

*T*

^{ n }

*x*

_{ n }||}. Since

*E*is uniformly smooth, we know that

*E** is uniformly convex. In view of Lemma 1.7, we see that

_{n→∞}(

*k*

_{ n }-1)

*M*

_{ n }= 0 and (2.8) and the assumption lim inf

_{n→∞}

*α*

_{ n }(1 -

*α*

_{ n }) > 0, we see that

*x*

_{ n }→

*p*as

*n*→∞ and

*J*:

*E*→

*E** is demi-continuous, we obtain that

*Jx*

_{ n }→

*Jp*∈

*E**. Note that

*Jx*

_{ n }|| → ||

*Jp*|| as

*n*→ ∞. Since

*E** has the Kadec-Klee property, we see that

*J*

^{-1}:

*E** →

*E*is demi-continuous. It follows that

*T*

^{ n }

*x*

_{ n }→

*p*. On the other hand, we have

*T*

^{ n }

*x*

_{ n }|| → ||

*p*|| as

*n*→ ∞. Since

*E*has the Kadec-Klee property, we obtain that

That is, *TT*^{
n
}*x*_{
n
} - *p* → 0 as *n* → ∞*:* It follows from the closedness of *T* that *Tp* = *p:*

*u*

_{ n }||

*-*||

*y*

_{ n }|| → 0 as

*n*→ ∞. In view of

*u*

_{ n }→

*p*as

*n*→ ∞, we have

*E** is reflexive, we may assume that

*Jy*

_{ n }→

*q**∈

*E**: In view of

*J*(

*E*) =

*E**, we see that there exists

*q*∈

*E*such that

*Jq*=

*q**. It follows that

*p*=

*q*, which in turn implies that

*q** =

*Jp*. It follows that

*Jy*

_{ n }→

*Jp*∈

*E**. From (2.16) and

*E** has the Kadec-Klee property, we obtain that

*J*

^{-1}:

*E** →

*E*is demi-continuous. It follows that

*y*

_{ n }→

*p*. From (2.15) and

*E*has the Kadec-Klee property, we obtain that

*n*→ ∞ in the above inequality, we from conditions (

*A*4) and (2.19) obtain that

*< t <*1 and

*y*∈

*C*, define

*y*

_{ t }=

*ty*+ (1 -

*t*)

*p*. It follows that

*y*

_{ t }∈

*C*, which yields that

*f*(

*y*

_{ t }

*, p*) ≤ 0. It follows from conditions (

*A*1) and (

*A*4) that

Letting *t* ↓ 0, from condition (*A* 3), we obtain that *f*(*p*, *y*) ≥ 0 ∀*y* ∈ *C:* This implies that *p* ∈ *EP*(*f*). This shows that
.

In view of Lemma 1.3, we can obtain that . This completes the proof.

**Remark 2.2**. Theorem 2.1 improves Theorem QCK in the following aspects:

- (a)
From a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property;

- (b)
From a quasi-

*ϕ*-nonexpansive mapping to an asymptotically quasi-*ϕ*-non-expansive mapping.

From the definition of quasi-*ϕ*-nonexpansive mappings, we see that every quasi-*ϕ*-nonexpansive mapping is asymptotically quasi-*ϕ*-nonexpansive with the constant sequence {1}. From the proof of Theorem 2.1, we have the following results immediately.

**Corollary 2.3**. *Let E be a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property and C a nonempty closed convex subset of E. Let f be a bifunction from C* × *C to* ℝ *satisfying* (*A* 1)*-*(*A* 4) *and T* : *C* → *C a closed and quasi-* *ϕ* *-nonexpansive mapping. Assume that*
*is nonempty*.

*where* {*α*_{
n
}} *is a real sequence in* [0, 1] *such that* lim inf_{n→∞}*α*_{
n
}(1 - *α*_{
n
}) > 0, {*r*_{
n
}} *is a real sequence in* [*a*, ∞), *where a is some positive real number and J is the duality mapping on E. Then the sequence* {*x*_{
n
}} *converges strongly to*
, *where*
*is the generalized projection from E onto*
.

**Remark 2.4**. Corollary 2.3 improves Theorem TZ in the following aspects.

- (a)
For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property (note that every uniformly convex Banach space has the Kadec-Klee property).

- (b)
For the mappings, we extend the mapping from a relatively nonexpansive mapping to a quasi-

*ϕ*-nonexpansive mapping (we remove the restriction , where denotes the asymptotic fixed point set). - (c)
For the algorithms, we remove the set "

*W*_{ n }" in Theorem TZ.

## Declarations

### Acknowledgements

This work was supported by the Kyungnam University Foundation Grant 2010.

## Authors’ Affiliations

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