# On the equivalence between some kinds of implicit and explicit iterations with applications

- Shih-sen Chang
^{1}, - Jong Kyu Kim
^{2}Email author, - HW Joseph Lee
^{3}and - Chi Kin Chan
^{3}

**2012**:55

https://doi.org/10.1186/1687-1812-2012-55

© Chang et al; licensee Springer. 2012

**Received: **4 September 2011

**Accepted: **3 April 2012

**Published: **3 April 2012

## Abstract

The purpose of this article is to study the equivalence between some kinds of implicit and explicit iterative approximations. As applications, we utilize our results to study the approximation problems arising in nonexpansive semigroup, variational inclusions and equilibrium problem.

## Keywords

## 1 Introduction

**Definition 1.1**Let (

*X, d*) be a metric space and let

*S*:

*X*→

*X*be a mapping.

*S*is said to be

*ψ- weakly contractive (weak contraction, for short)*[1], if there exists a continuous and strictly increasing function

*ψ*: [0, +

*∞*)

*→*[0, +

*∞*) with

*ψ*(

*t*)

*>*0, ∀

*t*∈ (0, +

*∞*) and

*ψ*(0) = 0 such that

From Definition 1.1, it is easy to see that if *ψ* (*t*) = (1 *- h*)*t, h* ∈ (0, 1), then *S* is a *contractive mapping* with the contractive constant *h*.

In 2001, Rhoades [2] proved the following interesting fixed point theorem:

**Theorem 1.2** (Rhoades [2]) Let (*X, d*) be a complete metric space and let *T*: *X* → *X* be a weakly contractive mapping. Then *T* has a unique fixed point in *X*. Moreover, for any given *x* ∈ *X*, the iterative sequence {*T*^{
n
}*x*} converges strongly to this fixed point.

This theorem is one of the generalization of *Banach contraction Theorem*.

*E*be a Banach space and let

*C*be a nonempty, closed and convex subset of

*E*. A mapping

*T*:

*C*→

*C*is said to be

*nonexpansive*, if

Denote by *F*(*T*) the fixed point set of *T*. It is well-known that if *E* is uniformly smooth *C* is bounded and *T*: *C* → *C* is nonexpansive then *F*(*T*) is nonempty (see, G*ö* hde [3]).

Recently, many convergence theorems to fixed points of nonexpansive mappings have been established.

In 1967, Browder [4] and Halpern [5] proved some strong convergence theorems in Hilbert spaces for implicit and explicit iteration, respectively. Later these theorems have been extended in several directions. In 1980, Reich [6] proved the following:

**Theorem 1.3**(Reich [6]) Let

*E*be a uniformly smooth Banach space,

*C*be a bounded, closed and convex subset of

*E*and let

*T*:

*C*→

*C*be a nonexpansive mapping. Fix

*u*∈

*C*and define a net {

*y*

_{ t }} in

*C*by

Then {*y*_{
t
}} converges strongly to *Pu* as *t* → 0+, where *P* is the unique sunny nonexpansive retraction from *C* onto *F*(*T*). (The relative definitions are given in Section 2.)

In the sequel, this theorem is called *Reich's convergence theorem*.

In 2004 Xu [7] proved the following

**Theorem 1.4**(Xu [7]) Let

*E*be a uniformly smooth Banach space and let

*C*be a bounded, closed and convex subset of

*E*. Let

*T*:

*C*→

*C*be a nonexpansive mapping with

*F*(

*T*) ≠ Ø,

*P*be the unique sunny nonexpansive retraction from

*C*onto

*F*(

*T*) and let

*f: C*→

*C*be a contractive mapping with a contractive constant

*h*∈ (0, 1). Let {

*x*

_{ t }} be the net in

*C*which is defined by

Then as *t* → 0+, {*x*_{
t
}} converges strongly to the unique point *z* ∈ *C* satisfying *P* ∇ *f*(*z*) = *z*.

On the other hand, in 2002, Xu proved the following.

**Theorem 1.5**(Xu [8, 9]) Let

*E, C, T, P*be as in Theorem 1.3. Let {

*y*

_{ n }} be the sequence in

*C*defined by

*y*

_{1}∈

*C*and

where {*α*_{
n
}} is a sequence in (0, 1) which satisfies the following conditions:

(C1) lim_{n→∞}*α*_{
n
} = 0;

(C2) ${\sum}_{n=1}^{\infty}{\alpha}_{n}=\infty ;$

(C3) ${\mathsf{\text{lim}}}_{n\to \infty}\frac{{{\alpha}_{n}}_{+1}}{{\alpha}_{n}}=1.$

Then {*y*_{
n
}} converges strongly to *Pu*.

In the sequel, this theorem is called *Xu's convergence theorem*.

Moreover, in 2000, Moudafi [10] introduced the concept of *viscosity approximation* and proved the following result

**Theorem 1.6**(Moudafi [10]) Let

*H*be a real Hilbert space,

*C*be a nonempty closed convex subset of

*H, T*:

*C*→

*C*be a nonexpansive mapping with

*F*(

*T*) ≠ Ø and

*f*:

*C*→

*C*be a contractive mapping. Let {

*x*

_{ n }} be the sequence defined by:

Then {*x*_{
n
}} converge strongly to a fixed point of *T* in Hilbert space *H*.

In the sequel, this theorem is called *Moudafi's viscosity convergence theorem*.

Moudafi's viscosity convergence theorem is very important, because it can be applied to convex optimization, linear programming, monotone inclusions and elliptic differential equations (see [11]).

In 2004, Xu [7] extended Moudafi's results to a uniformly smooth Banach space. Song and Chen [12–14] also obtained some strong convergence theorems about viscosity approximations in more general Banach space. More recently, Suzuki [11] and Petrusel and Yao [15] replace a contraction *f* in (1.6) by a Meir-Keeler contraction [16] and a generalized contraction [15], respectively, to approximate a common fixed point for an infinite family of nonexpansive mappings.

The purpose of this article is to study the equivalence between some kinds of implicit and explicit iterative approximations approximations. As applications, we utilize our results to study some approximation problems arising in nonexpansive semigroup, variational inclusions and equilibrium problems.

## 2 Preliminaries

In this section, we give some preliminaries which will be needed in proving our main results.

Throughout this article, we assume that *E* is a real Banach space, *E** is the dual of *E*. The space *E* is said to be *smooth* or said to have a *Gâteaux differentiable norm*, if the limit ${\mathsf{\text{lim}}}_{t\to 0}\frac{\parallel x+ty\parallel -\parallel x\parallel}{t}$ exists for each *x, y* ∈ *U* = {*x* ∈ *E*: ||*x*|| = 1}. The space *E* is said to have a *uniformly Gâteaux differentiable norm*, if for each *y* ∈ *U*, the limit is attained uniformly in *x* ∈ *U*. The space *E* is said to be *uniformly smooth* or said to have a *uniformly Fréchet differentiable norm*, if the limit is attained uniformly in *x, y* ∈ *U*.

Let *C* and *K* be subsets of *E*. Recall that A mapping *P*: *C* → *K* is called *sunny*, if *P*(*Px* + *t*(*x - Px*)) = *Px* for any *x* ∈ *C* with *Px* + *t*(*x - Px*) ∈ *C* and *t ≥* 0.

Let *C* be a convex subset of *E* and *K* be a subset of *C*. A mapping *P*: *C* → *K* is called *retraction from C onto K*, if *P*^{2} = *P* .

**Lemma 2.1**(Reich [17]) Let

*E*be a smooth Banach space, and

*J*:

*E*→

*E** be the

*normalized duality mapping*. Let

*C*be a convex subset of

*E, K*be a subset of

*C*and let

*P*be a retraction from

*C*onto

*K*. Then the following are equivalent:

- (1)
〈

*x*-*Px, J*(*Px*- y)〉*≥*0, ∀*x*∈*C*and*y*∈*K*; - (2)
*P*is sunny and nonexpansive.

Hence there is at most one sunny nonexpansive retraction from *C* onto *K*.

**Remark 2.2** We note that if *E* is a real Hilbert space and *K* is a closed and convex subset of *C*, then the metric projection and the sunny nonexpansive retraction from *C* onto *K* coincide, i.e., when *T: C* → *C* is a nonexpansive mapping, the sunny nonexpansive retraction from *C* onto *F*(*T*) is the metric projection.

**Proposition 2.3**(Suzuki [11]) Let

*E*be a smooth Banach space and let

*C*be a closed and convex subset of

*E*. Let

*K*be a subset of

*C*and let

*P*be the unique sunny nonexpansive retraction from

*C*onto

*K*. Let

*f*:

*C*→

*C*be a mapping and

*z*∈

*K*be a point. Then the following statements are equivalent:

- (1)
*z*is a fixed point of*P*∇*f*; - (2)
*z*is a solution to the following variational inequality:$\u27e8fz-z,J\left(z-y\right)\u27e9\ge 0,\phantom{\rule{1em}{0ex}}\forall v\in K.$

**Proposition 2.4**(Song [14]) Let

*E*be a Banach space and let

*C*be a convex subset of

*E*. Let

*T: C*→

*C*be a nonexpansive mapping and let

*g: C*→

*C*be a weakly contractive mapping. Then the following conclusions hold:

- (1)
*T*∇*g*is weakly contractive; - (2)for each
*t*∈ (0, 1), the mapping*x*α*tg*(*x*) + (1*- t*)*Tx*is weakly contractive and the net {*x*_{ t }} defined by${x}_{t}=tg\left({x}_{t}\right)+\left(1-t\right)T{x}_{t}$

is well-defined.

**Lemma 2.5**(Alber and Guerre-Delabriere [1]) Let {λ

_{ n }} and {

*β*

_{ n }} be two sequences of nonnegative real numbers and let {

*α*

_{ n }} be a sequence of positive numbers satisfying the conditions ${\sum}_{n=1}^{\infty}{\alpha}_{n}=\infty $ and ${\mathsf{\text{lim}}}_{n\to \infty}\frac{{\beta}_{n}}{{\alpha}_{n}}=0.$ Let the recursive inequality

be given where *ψ*: [0, +*∞*) *→* [0, +*∞*) is a continuous and strictly increasing function with *ψ*(0) = 0. Then {*λ*_{
n
}} converges to 0 as *n → ∞*.

## 3. Equivalence between some implicit and explicit iterative schemes

We first give the following result.

**Theorem 3.1**Let

*E*be a uniformly smooth Banach space and let

*C*be a bounded, closed and convex subset of

*E*. Let

*T*:

*C*→

*C*be a nonexpansive mapping with

*F*(

*T*) ≠ Ø,

*P*be the unique sunny nonexpansive retraction from

*C*onto

*F*(

*T*) and let

*g*:

*C*→

*C*be a

*ψ-*weakly contractive mapping. Let {

*x*

_{ t }} be the net in

*C*defined by

Then {*x*_{
t
}} converges strongly to the unique point *z* ∈ *C* satisfying *P* ∇ *g*(*z*) = *z* as *t* → 0+.

**Proof**. Since *P* is nonexpansive and *g* is *ψ-* weakly contractive, by Proposition 2.4 (1), *P* ∇ *g: C* → *C* is *ψ-* weakly contractive. From Theorem 1.2, there exists a unique *z* ∈ *C* such that *z* = *P* ∇ *g*(*z*).

*y*

_{ t }} as follows:

*y*

_{ t }

*→ P*∇

*g*(

*z*) =

*z*, as

*t*→ 0+. Moreover, for each

*t*∈ (0, 1), we have

*ψ*is continuous and strictly increasing, we have

This completes the proof of Theorem 3.1.

**Theorem 3.2** Let *E, C, T, P, g* be the same as in Theorem 3.1. Then Theorems 1.3, 1.4, and 3.1 are equivalent.

**Proof**. Theorem 1.3 ⇒ Theorem 3.1 has been proved in Theorem 3.1. Now we proved that Theorem 3.1 ⇒ Theorem 1.4 ⇒ Theorem 1.3.

In fact, in Theorem 3.1 take *g*: *C* → *C* to be a contractive mapping, then the conclusion of Theorem 1.4 can be obtained from Theorem 3.1. As well as in Theorem 1.4 take *f* as a constant valued mapping, then the conclusion Theorem 1.3 can be obtained from Theorem 1.4 immediately. This completes the proof of Theorem 3.2.

**Remark 3.3** From Theorems 3.1 and 3.2, it follows that the convergence of the implicit iterative sequences defined by (1.3), (1.4), and (3.1) are equivalent.

Next we consider the equivalence between some explicit iterative sequences

In 2004, Xu [7] proved the following result:

**Theorem 3.4**(Xu [7]) Let

*E*be a uniformly smooth Banach space and let

*C*be a bounded, closed and convex subset of

*E*. Let

*T*:

*C*→

*C*be a nonexpansive mapping with

*F*(

*T*) ≠ Ø,

*P*be the unique sunny nonexpansive retraction from

*C*onto

*F*(

*T*) and let

*f: C*→

*C*be a contractive mapping. Let {

*x*

_{ n }} be the sequence in

*C*which is defined by

where {*α*_{
n
}} is a sequence in (0, 1) satisfying the conditions (C1), (C2) and (C3) in Theorem 1.5. Then {*x*_{
n
}} converges strongly to the unique point *z* ∈ *C* satisfying *P* ∇ *f*(*z*) = *z*.

It is easy to see that Theorem 3.4 is a generalization of Moudafi's Theorem 1.6.

Now we give the following result.

**Theorem 3.5**Let

*E, C, T , P*be the same as in Theorem 3.4. Let

*g*:

*C*→

*C*be a

*ψ-*weakly contractive mapping. Let {

*x*

_{ n }} be the sequence in

*C*which is defined by

where {*α*_{
n
}} is a sequence in (0, 1) satisfying the conditions (C1), (C2), and (C3) in Theorem 1.5. Then {*x*_{
n
}} converges strongly to the unique point *z* ∈ *C* satisfying *P* ∇ *g*(*z*) = *z*;

**Proof**. Since *P* is nonexpansive and *g* is a *ψ-* weak contraction, by Proposition 2.4 (1), *P* ∇ *g: C* → *C* is weakly contractive. From Theorem 1.2, there exists a unique *z* ∈ *C* such that *z* = *P* ∇ *g*(*z*).

*y*

_{ n }} as follows:

*y*

_{ n }

*→ P*∇

*g*(

*z*) =

*z*, as

*n*→

*∞*. Moreover, for each

*n ≥*1, we have

*λ*

_{ n }=

*||x*

_{ n }

*-y*

_{ n }

*||, β*

_{ n }=

*α*

_{ n }{

*||y*

_{ n }

*-z||-ψ*(

*||y*

_{ n }

*-z||*)} in Lemma 2.5, from the assumptions, we know that all the conditions in Lemma 2.5 are satisfied. It follows from Lemma 2.5 and (3.5) that

The conclusion of Theorem 3.5 is proved.

**Theorem 3.6** Let *E, C, T, P, g* be the same as in Theorem 3.5. Then Theorems 3.5, 3.4, and 1.5 are equivalent.

Indeed, the conclusion that Theorem 1.5 ⇒ Theorem 3.5 has been proved in Theorem 3.5. Now we proved that Theorem 3.5 ⇒ Theorem 3.4 ⇒ Theorem 1.5. In fact, in Theorem 3.5 take *g*: *C* → *C* to be a contractive mapping, the conclusion of Theorem 3.4 can be obtained from Theorem 3.5 immediately. Again in Theorem 3.4 take *f*: *C* → *C* to be a constant-valued mapping, the conclusion of Theorem 1.5 can be obtained from Theorem 3.4 immediately.

**Remark 3.7** From Theorem 3.5, it follows that the convergence of the explicit iterative sequences defined by (1.5), (3.4), and (3.3) are equivalent.

## 4. Equivalence between some more general implicit and explicit iterative schemes

In this section we study the equivalence between some more general implicit and explicit iterative schemes. Following Suzuki [11], we introduce the following concepts.

**Definition 4.1**(1) Let

*E*be a Banach space and let

*C*be a nonempty, closed and convex subset of

*E*. Let {

*S*

_{ n }}:

*C*→

*C*be an infinite family of nonexpansive mappings and {

*α*

_{ n }} be a sequence in (0, 1) satisfying the condition (C1) in Theorem 1.5. The tuple (

*E, C*, {

*S*

_{ n }}, {

*α*

_{ n }}) is said to have

*Browder's property*, if for each

*u*∈

*C*the implicit iterative sequence {

*y*

_{ n }} defined by

- (2)Let {
*α*_{ n }} be a sequence in (0, 1) satisfying the conditions (C1) and (C2) in Theorem 1.4 Then the tuple (*E, C*, {*S*_{ n }}, {*α*_{ n }}) is said to have*Halpern's property*, if for each*u*∈*C*the explicit iterative sequence {*y*_{ n }} defined by*y*_{1}∈*C*and${{y}_{n}}_{+\mathsf{\text{1}}}={\alpha}_{n}u+\mathsf{\text{}}\left(\mathsf{\text{1}}-{\alpha}_{n}\right){S}_{n}{y}_{n},\phantom{\rule{1em}{0ex}}n\ge \mathsf{\text{1}},$(4.2)

converges strongly.

It is well known that if *E* is a Hilbert space, *C* is a bounded subset of *E* and {*S*_{
n
}} = {*S*} is a constant sequence of nonexpansive mappings, then the tuple (*E, C*, {*S*_{
n
}}, $\left\{\frac{1}{n}\right\}$) has both Browder's and Halpern's property.

**Lemma 4.2**(Suzuki [11])

- (1)
Let (

*E, C*, {*S*_{ n }}, {*α*_{ n }}) have the "Browder's property". For each*u*∈*C*, let {*y*_{ n }} be the sequence in*C*defined by (4.1). Denote*Pu*= lim_{n→∞}*y*_{ n }, then*P*is a nonexpansive mapping from*C*to*C*; - (2)
Let (

*E, C*, {*S*_{ n }}, {*α*_{ n }}) have the "Halpern's property". For each*u*∈*C*, let {*y*_{ n }} be the sequence in*C*defined by (4.2). Denote*Pu*= lim_{n→∞}*y*_{ n }, then the following hold: - (i)
*Pu*does not depend on the initial point*y*_{1}∈*C*; - (ii)
*P*is a nonexpansive mapping from*C*into*C*.

**Theorem 4.3**Let (

*E, C*, {

*S*

_{ n }}, {

*α*

_{ n }},

*P*) has the "Browder's property". Let

*g*:

*C*→

*C*be a

*ψ*-weakly contractive mapping. Define a sequence {

*x*

_{ n }} in

*C*by

Then {*x*_{
n
}} converges strongly to the unique point *z* ∈ *C* satisfying *P* ∘ *gz* = *z*.

**Proof**. By Proposition 2.4, the existence and uniqueness of {

*x*

_{ n }} and

*z*=

*P*∘

*gz*are assured. Define a sequence {

*y*

_{ n }} in

*C*by

*y*

_{ n }→

*P*∘

*fz*which equals to

*z*. And for each

*n ≥*1, we have

*ψ*, this implies that lim

_{n→∞}||

*x*

_{ n }

*- y*

_{ n }|| = 0. Hence we have

This completes the proof of Theorem 4.3.

**Remark 4.4** It follows from Theorem 4.3 that the implicit iterative sequences (4.1) and (4.3) are equivalent.

**Theorem 4.5**Let (

*E, C*, {

*S*

_{ n }}, {

*α*

_{ n }},

*P*) have the "Halpern's property". Let

*g*:

*C*→

*C*be a

*ψ*-weakly contractive mapping. Define a sequence {

*x*

_{ n }} in

*C*by

*x*

_{1}∈

*C*and

Then {*x*_{
n
}} converges strongly to the unique point *z ∈ C* satisfying *P* ∘ *gz* = *z*.

**Proof**. By Proposition 2.4 (1),

*P*∘

*g*is a

*ψ-*weak contraction from

*C*to

*C*. Hence there exists a unique

*z*∈

*C*such that

*z*=

*P*∘

*gz*. Now we define a sequence {

*y*

_{ n }} by

*λ*

_{ n }= ||

*x*

_{ n }

*- y*

_{ n }||,

*β*

_{ n }=

*α*

_{ n }{||

*y*

_{ n }

*- z*||

*- ψ*(||

*y*

_{ n }

*- z*||)} in Lemma 2.5 and from the assumptions, we know that all the conditions in Lemma 2.5 are satisfied, and therefore we have

This completes the proof of Theorem 4.5.

**Remark 4.6** From the above arguments, we get that the explicit iterative sequences (4.2) and (4.4) are equivalent.

## 5 Applications

In this section, we will utilize Theorems 4.3 and 4.5 to study some approximation problems and to state several deduced theorems.

### 5.1 Application to nonexpansive semigroup problems

**Theorem 5.1**Let

*E*be a uniformly convex Banach space having a weakly continuous normalized duality mapping

*J*:

*E*→

*E**,

*C*be a closed and convex subset of

*E*and let $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}:C\to C$ be a nonexpansive semigroup with $F\left(\mathcal{T}\right):={\bigcap}_{t\ge 0}F\left(T\left(t\right)\right)\ne \varnothing $. Let

*g*:

*C*→

*C*be a

*ψ-*weakly contractive mapping and let {

*x*

_{ n }} be the implicit iterative sequence defined by:

where {*α*_{
n
}} and {*t*_{
n
}} are sequences of real numbers such that *α*_{
n
} ∈ (0, 1) and *t*_{
n
} *>* 0 for all *n ≥* 1 and ${\mathsf{\text{lim}}}_{n\to \infty}{t}_{n}={\mathsf{\text{lim}}}_{n\to \infty}\left(\frac{{\alpha}_{n}}{{t}_{n}}\right)=0$. Then *x*_{
n
} → *z* = *P* ∘ *gz*, where *P* is the sunny nonexpansive retraction from *C* onto $F\left(\mathcal{T}\right)$.

**Proof**. It follows from [[18], Theorem 3.3] that (*E, C*, {*S*_{
n
}}, {*α*_{
n
}}, *P*) has the "Browder's property", where *S*_{
n
} = *T* (*t*_{
n
}), for any *n ≥* 1. Hence the conclusion of Theorem 5.1 can be obtained immediately from Theorem 4.3.

**Theorem 5.2**Let

*H*be a real Hilbert space,

*C*be a nonempty, closed and convex subset of

*H,*$\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}:C\to C$ be a nonexpansive semigroup with $F\left(\mathcal{T}\right):={\bigcap}_{t\ge 0}F\left(T\left(t\right)\right)\ne \varnothing $ and the mapping

*t*α ||

*T*(

*t*)

*x - y*||

^{2}is measurable for each

*x*∈

*C*and

*y*∈

*H*. Let {

*σ*

_{ n }}

*:*[0,

*∞*) → [0,

*∞*) be a sequence of measurable functions such that ${\int}_{0}^{\infty}{\sigma}_{n}\left(t\right)dt=1$ for each

*n ≥*1, lim

_{ n→∞ }

*σ*

_{ n }(

*t*) = 0 for almost every

*t ≥*0, ${\mathsf{\text{lim}}}_{n\to \infty}{\int}_{0}^{\infty}|{\sigma}_{n}\left(t+s\right)-{\sigma}_{n}\left(t\right)|dt=0$, for all

*s ≥*0 and there exists $\beta \in \phantom{\rule{0.3em}{0ex}}{L}_{\mathsf{\text{loc}}}^{1}\phantom{\rule{0.3em}{0ex}}\left[0,\phantom{\rule{0.3em}{0ex}}\infty \right)$ such that sup

_{ n }

*σ*

_{ n }(

*t*)

*≤ β*(

*t*) for almost every

*t ≥*0, where $\beta \phantom{\rule{0.3em}{0ex}}\in \phantom{\rule{0.3em}{0ex}}{L}_{\mathsf{\text{loc}}}^{1}\phantom{\rule{0.3em}{0ex}}\left[0,\phantom{\rule{0.3em}{0ex}}\infty \right)$ means a restriction of

*β*on [0,

*s*] belongs to

*L*

^{1}[0,

*s*] for each

*s >*0. Let

*g: C*→

*C*be a weakly contractive mapping, {

*α*

_{ n }} be a real sequence in (0, 1] such that

*α*

_{ n }

*→*0 and let {

*β*

_{ n }} be a sequence in 0[1] such that

*β*

_{ n }→ 0 and ${\sum}_{n=0}^{\infty}{\beta}_{n}=\infty .$ Let

*x*and

*y*

_{0}be points of

*C*and let {

*x*

_{ n }} and {

*y*

_{ n }} be the sequences which are defined by

respectively. Then both {*x*_{
n
}} and {*y*_{
n
}} converge strongly to *z* = *P* ∘ *gz*, where *P* is the metric projection from *C* onto $F\left(\mathcal{T}\right)$.

**Proof**. By the assumption that $\mathcal{T}:=\left\{T\left(t\right):t\ge 0\right\}:C\to C$ is a nonexpansive semiproup, hence for each *n ≥* 1, ${S}_{n}:={\int}_{0}^{\infty}{\sigma}_{n}\left(t\right)T\left(t\right)dt:C\to C$ is a nonexpansive mapping. From [[19], Theorem 10], we know that (*H, C*, {*S*_{
n
}}, {*α*_{
n
}}, *P*) and (*H, C*, {*S*_{
n
}}, {*β*_{
n
}}, *P*) have the Browder's and Halpern's property, respectively. Hence the conclusions of Theorem 5.2 follow immediately from Theorems 4.3 and 4.5.

**Definition 5.3**Let

*ρ*

_{E}: [0,

*∞*) → [0,

*∞*) be the modulus of smoothness of a Banach space

*E*defined by

where *S*(*E*) = {*x* ∈ *E*: ||*x*|| = 1}.

A Banach space *E* is said to be *uniformly smooth* if $\frac{{\rho}_{E}\left(t\right)}{t}\to 0$ as *t* → 0. Let *q >* 1. A Banach space *E* is said to be *q-uniformly smooth*, if there exists a fixed constant *c >* 0 such that *ρ*_{
E
}(*t*) *≤ ct*^{
q
}. It is well-known that *E* is uniformly smooth if and only if the norm of *E* is uniformly *Fréchet* differentiable. If *E* is *q*-uniformly smooth, then *q ≤* 2 and *E* is uniformly smooth.

**Lemma 5.4**(Xu [20]) Let

*E*be a real 2-uniformly smooth Banach space with the best smooth constant

*K*. Then the following inequality holds:

for any *x, y E*, where *K* is a positive constant. The minimal positive constant *K* satisfying the above inequality is said to be *the best smooth constant*.

### 5.2 Application to a system of variational inclusions

**Theorem 5.5**Let

*E*be a uniformly convex and 2-uniformly smooth Banach space with the best smooth constant

*K*. Let

*M*

_{ i }:

*E*→ 2

^{ E }be maximal monotone mappings and let

*A*

_{ i }:

*E*→

*E*be

*γ*

_{ i }

*-*inverse-strongly accretive mappings,

*i*= 1, 2. Let

*T: E*→

*E*be a

*λ-*strict pseudocontraction with a fixed point. Define a mapping

*S*by $Sx=\left(1-\frac{\lambda}{{K}^{2}}\right)x+\frac{\lambda}{{K}^{2}}Tx$, for all

*x*∈

*E*. Let

*g*:

*E*→

*E*be a

*ψ-*weak concraction. Assume that Ω =

*F*(

*T*) ∩

*F*(

*Q*) ≠ Ø, where

*Q*:

*E*→

*E*is a mapping which is defined by

*M*

_{ i }defined by ${J}_{\left({M}_{i},\phantom{\rule{0.3em}{0ex}}{\rho}_{i}\right)}\left(x\right)={\left(I+{\rho}_{i}M\right)}^{-1}\left(x\right)$,

*x*∈

*E*, where

*ρ*is any positive number and

*I*is the identity mapping. Let

*x*

_{1}=

*u*and {

*x*

_{ n }} be a sequence defined by

*μ*∈ (0, 1), ${\rho}_{i}\in \left(0,\frac{{\gamma}_{i}}{{K}^{2}}\right)$,

*i*= 1, 2, and {

*α*

_{ n }} and {

*β*

_{ n }} are sequences in (0, 1). If the following conditions are satisfied:

- (i)
0

*<*lim inf_{n→∞}*β*_{ n }*≤*lim sup_{n→∞}*β*_{ n }*<*1; - (ii)
lim

_{n→∞}*α*_{ n }= 0*and*${\sum}_{n=1}^{\infty}{\alpha}_{n}=\infty ,$

*x*

_{ n }} converges strongly to

*x** =

*P*

_{Ω}

*g*(

*x**), where

*P*

_{Ω}is the sunny nonexpansive retraction from

*E*onto Ω and (

*x**,

*y**), where ${y}^{*}={J}_{\left({M}_{2},{\rho}_{2}\right)}\left({x}^{*}-{\rho}_{2}{A}_{2}{x}^{*}\right)$ is a solution of the following system of quasivariational inclusions:

**Proof**. By the assumptions, we know that the mappings

*S*and

*Q*are nonexpansive on

*E*. From [21], the mapping

*U*: =

*μS*+ (1

*-μ*)

*Q*is also nonexpansive on

*E*. Since

*C*(see [21]), hence (4.4) can be rewritten as follows:

*E, C*, {

*S*

_{ n }}, {

*α*

_{ n }}),

*P*

_{Ω}) has the "Halpern's property", where

*u*∈

*E*the sequence {

*y*

_{ n }} defined by

*y*

_{1}=

*u*∈

*E*and

converges strongly to *x** = *P*_{Ω}*u* and (*x**, *y**), where ${y}^{*}={J}_{{M}_{2},{\rho}_{2}}\left(I-{\rho}_{2}{A}_{2}\right){x}^{*}$, is a solution to the system of quasivariational inclusions (5.5). Hence from Theorem 4.3, we know that *x*_{
n
} *→ x** = *P*_{Ω}*g*(*x**), and (*x**, *y**), where ${y}^{*}={J}_{{M}_{2},{\rho}_{2}}\left(I-{\rho}_{2}{A}_{2}\right){x}^{*}$, is a solution to the system of quasivariational inclusions (5.5).

This completes the proof of Theorem 5.5.

### 5.3 Application to equilibrium problem

Let *H* be a real Hilbert space, *C* be a nonempty, closed and convex subset of *H* and let $G:C\times C\to \mathcal{R}$ be an *equilibrium function*, i.e., *G*(*u, u*) = 0, for any *u* ∈ *C*. The *equilibrium problem* is to find *x** ∈ *C* such that *G*(*x**, *y*) *≥* 0, for all *y* ∈ *C*. The solution set of equilibrium problem is denoted by *EP*(*G*).

For solving equilibrium problem of a function $G:C\times C\to \mathcal{R}$, let us assume that *G* satisfies the following conditions:

(A1) *G*(*x, x*) = 0;

(A2) *G*(*x, y*) + *G*(*y, x*) *≤* 0, ∀*x, y* ∈ *C*;

(A3) for each *x, y, z* ∈ *C*, lim sup_{t→ 0}*G*(*tz* + (1 *- t*)*x, y*) *≤ G*(*x, y*);

(A4) for each *x* ∈ *C, y ↦ G*(*x, y*) is convex and lower semi-continuous.

Let {*T*_{
i
}} be an infinite family of nonexpansive mappings on *C* and let {*λ*_{
i
}} be a real sequence in 0[1]. Following Shimoji and Takahashi [23], for any *n ≥* 1, we define a mapping

*W*

_{ n }

*: C*→

*C*by

**Lemma 5.6**(Combettes and Hirstoaga [24]) Let $G:C\times C\to \mathcal{R}$ be an equilibrium function which satisfies the conditions (A1)-(A4). For

*r >*0 and

*x*∈

*H*define the mapping

*S*

_{ r }

*: H*→

*C*by

then *S*_{
r
} is well defined and the following hold: (1) *S*_{
r
} is single-valued; (2) *S*_{
r
} is firmly nonexpansive, i.e., ||*S*_{
r
}*x - S*_{
r
}||^{2} *≤* 〈*S*_{
r
}*x - S*_{
r
}*, x - y*〉, for all *x, y* ∈ *H*; (3) *F*(*S*_{
r
}) = EP(*G*); (4) EP(*G*) is closed and convex.

**Theorem 5.7**Let

*H*be a real Hilbert space, {

*T*

_{ i }}:

*H*→

*H*be an infinite family of nonexpansive mappings,

*G*

_{ k }:

*H × H*→

*R, k*= 1, 2,...,

*N*be functions which satisfy conditions (A1)-(A4) such that $\mathcal{F}:=\left({\cap}_{k=1}^{N}\mathsf{\text{EP(}}{G}_{k})\cap {\cap}_{n=1}^{\infty}F\left({T}_{n}\right)\right)\ne \varnothing $,

*A*:

*H*→

*H*be a strongly positive bounded linear operator with coefficient $\stackrel{\u0304}{\gamma}>0$ and Φ:

*H*→

*H*be a

*ψ-*weak contraction. Moreover, let ${\left\{{r}_{k,n}\right\}}_{k=1}^{N}$ be positive sequences, {

*α*

_{ n }} be a sequence in 0[1], and γ be a real number such that $0<\gamma <\stackrel{\u0304}{\gamma}$. Assume that

- (i)
lim

_{n→∞}*α*_{ n }= 0; - (ii)
${\mathsf{\text{lim}}}_{n\to \infty}{r}_{k,n}={\widehat{r}}_{k}$ for each

*k*= 1, 2,...,*N*.

*x*

_{ n }} be the sequence defined by

where *W*_{
n
} is the mapping generated by {*T*_{
i
}} and {λ_{
n
}} as in (5.7), and for each *k* = 1, 2,..., *N* and each *n ≥* 1, ${S}_{{r}_{k},n}^{k}$ is the mapping generated by *G*_{
k
} and *r*_{
k, n
}as in Lemma 5.6. Then {*x*_{
n
}} converges strongly ${x}^{*}={P}_{\mathcal{F}}\left(I-\left(A-\gamma \Phi \right)\right){x}^{*}$.

**Proof**. Since for each

*n ≥*1 and for each

*k*= 1, 2,...,

*N*the mappings

*W*

_{ n }and ${S}_{{r}_{k},n}^{k}$ all are nonexpansive on

*H*. Define a mapping

*V*

_{ n }:

*H*→

*H*by:

*n ≥*1,

*V*

_{ n }is a

*ψ-*weakly contractive mapping. Hence there exists a unique

*x*

_{ n }∈

*H*such that

*x*

_{ n }=

*V*

_{ n }

*x*

_{ n }. This shows that the sequence {

*x*

_{ n }} defined by (5.8) is well-defined. By the similar method as in [[25], Theorem 8], we can prove that

*x*

_{ n }→

*x**, where

*x** is the unique solution of the variational inequality

*x** is the unique solution of the minimization problem:

where *h* is the potential function for γΦ.

## Declarations

### Acknowledgements

The authors would like to express their thanks to the referees for their helpful comments and suggestions. This work was supported by the National Research Foundation of Korean Grant funded by the Korean Government (2011-0002581).

## Authors’ Affiliations

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