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# Uniformly closed replaced AKTT or ∗AKTT condition to get strong convergence theorems for a countable family of relatively quasi-nonexpansive mappings and systems of equilibrium problems

## Abstract

The purpose of this paper is to construct a new iterative scheme and to get a strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but does not satisfy condition AKTT and AKTT. Our results can be applied to solve a convex minimization problem. In addition, this paper clarifies an ambiguity in a useful lemma. The results of this paper modify and improve many other important recent results.

MSC:47H05, 47H09, 47H10.

## 1 Introduction and preliminaries

Let E be a real Banach space and C be a nonempty closed convex subset of E. A mapping $T:C→C$ is called nonexpansive if

$∥Tx−Ty∥≤∥x−y∥,∀x,y∈C.$

Let E be a real Banach space and C be a nonempty closed convex subset of E. A point $p∈C$ is said to be an asymptotic fixed point of T if there exists a sequence ${ x n } n = 0 ∞ ⊂C$ such that $x n ⇀p$ and $lim n → ∞ ∥ x n −T x n ∥=0$. The set of asymptotic fixed point is denoted by $F ˆ (T)$. We say that a mapping T is relatively nonexpansive (see [14]) if the following conditions are satisfied:

1. (I)

$F(T)≠∅$;

2. (II)

$ϕ(p,Tx)≤ϕ(p,x)$, $∀x∈C$, $p∈F(T)$;

3. (III)

$F(T)= F ˆ (T)$.

If T satisfies (I) and (II), then T is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings.

Let E be a real Banach space. The modulus of smoothness of E is the function $ρ E :[0,∞)→[0,∞)$ defined by

$ρ E (τ)=sup { 1 2 ( ∥ x + y ∥ + ∥ x − y ∥ ) − 1 : ∥ x ∥ ≤ 1 , ∥ y ∥ ≤ τ } .$

E is uniformly smooth if and only if

$lim τ → 0 ρ E τ τ =0.$

Let $dimE≥2$. The modulus of convexity of E is the function $δ E (ϵ):=inf{1−∥ x + y 2 ∥:∥x∥=∥y∥=1;ϵ=∥x−y∥}$. E is uniformly convex if for any $ϵ∈(0,2]$, there exists $δ=δ(ϵ)>0$ such that if $x,y∈E$ with $∥x∥≤1$, $∥y∥≤1$ and $∥x−y∥≥ϵ$, then $∥ 1 2 (x+y)∥≤1−δ$. Equivalently, E is uniformly convex if and only if $δ E (ϵ)>0$ for all $ϵ∈(0,2]$. A normed space E is called strictly convex if for all $x,y∈E$, $x≠y$, $∥x∥=∥y∥=1$, we have $∥λx+(1−λ)y∥<1$, $∀λ∈(0,1)$.

Let $E ∗$ be the dual space of E. We denote by J the normalized duality mapping from E to $2 E ∗$ defined by

$J(x)= { f ∈ E ∗ : 〈 x , f 〉 = ∥ x ∥ 2 = ∥ f ∥ 2 } .$

The following properties of J are well known (see [57] for more details):

1. (1)

If E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E.

2. (2)

If E is reflexive, then J is a mapping from E onto $E ∗$.

3. (3)

If E is smooth, then J is single valued.

Throughout this paper, we denote by ϕ the functional on $E×E$ defined by

$ϕ(x,y)= ∥ x ∥ 2 −2 〈 x , J ( y ) 〉 + ∥ y ∥ 2 ,∀x,y∈E.$
(1.1)

Let E be a smooth, strictly convex, and reflexive real Banach space and let C be a nonempty closed convex subset of E. Following Alber [8], the generalized projection $Π C$ from E onto C is defined by

$Π C (x)= arg min y ∈ C ϕ(y,x),∀x∈E.$

The existence and uniqueness of $Π C$ follows from the property of the functional $ϕ(x,y)$ and strict monotonicity of the mapping J. It is obvious that

$( ∥ x ∥ − ∥ y ∥ ) 2 ≤ϕ(x,y)≤ ( ∥ x ∥ + ∥ y ∥ ) 2 ,∀x,y∈E.$
(1.2)

Next, we recall the notion of generalized f-projection operator and its properties. Let $G:C× E ∗ →R∪{+∞}$ be a functional defined as follows:

$G(ξ,φ)= ∥ ξ ∥ 2 −2〈ξ,φ〉+ ∥ φ ∥ 2 +2ρf(ξ),$
(1.3)

where $ξ∈C$, $φ∈ E ∗$, ρ is a positive number and $f:C→R∪{+∞}$ is proper, convex, and lower semi-continuous. From the definitions of G and f, it is easy to see the following properties:

1. (i)

$G(ξ,φ)$ is convex and continuous with respect to φ when ξ is fixed;

2. (ii)

$G(ξ,φ)$ is convex and lower semi-continuous with respect to ξ when φ is fixed.

Definition 1.1 [9]

Let E be a real Banach space with its dual $E ∗$. Let C be a nonempty, closed, and convex subset of E. We say that $Π C f : E ∗ → 2 C$ is a generalized f-projection operator if

$Π C f φ= { u ∈ C : G ( u , φ ) = inf ξ ∈ C G ( ξ , φ ) } ,∀φ∈ E ∗ .$

For the generalized f-projection operator, Wu and Huang [9] proved in the following theorem some basic properties.

Lemma 1.2 [9]

Let E be a real reflexive Banach space with its dual $E ∗$. Let C be a nonempty, closed, and convex subset of E. Then the following statements hold:

1. (i)

$Π C f$ is a nonempty closed convex subset of C for all $φ∈ E ∗$.

2. (ii)

If E is smooth, then for all $φ∈ E ∗$, $x∈ Π C f φ$ if and only if

$〈x−y,φ−Jx〉+ρf(y)−ρf(x)≥0,∀y∈C.$
3. (iii)

If E is strictly convex and $f:C→R∪{+∞}$ is positive homogeneous (i.e., $f(tx)=tf(x)$ for all $t>0$ such that $tx∈C$ where $x∈C$), then $Π C f$ is a single-valued mapping.

Fan et al. [10] showed that the condition f is positive homogeneous which appeared in Lemma 1.2 can be removed.

Lemma 1.3 [10]

Let E be a real reflexive Banach space with its dual $E ∗$ and C a nonempty, closed, and convex subset of E. Then if E is strictly convex, then $Π C f$ is a single-valued mapping.

Recall that J is a single-valued mapping when E is a smooth Banach space. There exists a unique element $φ∈ E ∗$ such that $φ=Jx$ for each $x∈E$. This substitution in (1.3) gives

$G(ξ,Jx)= ∥ ξ ∥ 2 −2〈ξ,Jx〉+ ∥ x ∥ 2 +2ρf(ξ).$
(1.4)

Now, we consider the second generalized f-projection operator in a Banach space.

Definition 1.4 [11]

Let E be a real Banach space and C a nonempty, closed, and convex subset of E. We say that $Π C f :E→ 2 C$ is a generalized f-projection operator if

$Π C f x= { u ∈ C : G ( u , J x ) = inf ξ ∈ C G ( ξ , J x ) } ,∀x∈E.$

Obviously, the definition of relatively quasi-nonexpansive mapping T is equivalent to

1. (1)

$F(T)≠∅$;

2. (2)

$G(p,JTx)≤G(p,Jx)$, $∀x∈C$, $p∈F(T)$.

Lemma 1.5 [12]

Let E be a Banach space and $f:E→R∪{+∞}$ be a lower semi-continuous convex functional. Then there exist $x∈ E ∗$ and $α∈R$ such that

$f(x)≥ 〈 x , x ∗ 〉 +α,∀x∈E.$

We know that the following lemmas hold for operator $Π C f$.

Lemma 1.6 [13]

Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Then the following statements hold:

1. (i)

$Π C f$ is a nonempty, closed, and convex subset of C for all $x∈E$;

2. (ii)

for all $x∈E$, $x ˆ ∈ Π C f x$ if and only if

$〈 x ˆ −y,Jx−J x ˆ 〉+ρf(y)−ρf(x)≥0,∀y∈C;$
3. (iii)

if E is strictly convex, then $Π C f x$ is a single-valued mapping.

Lemma 1.7 [13]

Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Let $x∈E$ and $x ˆ ∈ Π C f x$. Then

$ϕ(y, x ˆ )+G( x ˆ ,Jx)≤G(y,Jx),∀y∈C.$

The fixed points set $F(T)$ of a relatively quasi-nonexpansive mapping is closed convex as given in the following lemma.

Lemma 1.8 [14, 15]

Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let T be a closed relatively quasi-nonexpansive mapping of C into itself. Then $F(T)$ is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 1.9 [16]

Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let ${ x n } n = 0 ∞$ and ${ y n } n = 0 ∞$ be sequences in E such that either ${ x n } n = 0 ∞$ or ${ y n } n = 0 ∞$ is bounded. If $lim n → ∞ ϕ( x n , y n )=0$, then $lim n → ∞ ∥ x n − y n ∥=0$.

Lemma 1.10 [17]

Let $p>1$ and $r>0$ be two fixed real numbers. Then a Banach space X is uniformly convex if and only if there is a continuous, strictly increasing and convex function $g: R + → R +$, $g(0)=0$, such that

$∥ λ x + ( 1 − λ ) y ∥ p ≤λ ∥ x ∥ p +(1−λ) ∥ y ∥ p − W p (λ)g ( ∥ x − y ∥ )$

for all $x,y∈ B r$ and $0≤λ≤1$, where $W p (λ)=λ ( 1 − λ ) p + λ p (1−λ)$.

Remark We can see from the Lemma 1.10 that the function g has no relation with the selection of x, y and λ. However, the key point above, in the process of generalization and application about this lemma, has been ambiguous gradually. For instance, the following lemma states that the function g has something to do with λ, which always leads to failure in the proof.

Lemma (stated in [[11], Lemma 2.10])

Let E be a uniformly convex real Banach space. For arbitrary $r>0$, let $B r (0):={x∈E:∥x∥≤r}$ and $λ∈[0,1]$. Then there exists a continuous strictly increasing convex function

$g:[0,2r]→R,g(0)=0$

such that for every $x,y∈ B r (0)$, the following inequality holds:

$∥ λ x + ( 1 − λ ) y ∥ 2 ≤λ ∥ x ∥ 2 +(1−λ) ∥ y ∥ 2 −λ(1−λ)g ( ∥ x − y ∥ ) .$

Let F be a bifunction of $C×C$ into R. The equilibrium problem is to find $x ∗ ∈C$ such that $F( x ∗ ,y)≥0$, for all $y∈C$. We shall denote the solutions set of the equilibrium problem by $EP(F)$. Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases.

For solving the equilibrium problem for a bifunction $F:C×C→R$, let us assume that F satisfies the following conditions:

(A1) $F(x,x)=0$ for all $x∈C$;

(A2) F is monotone, i.e., $F(x,y)+F(y,x)≤0$ for all $x,y∈C$;

(A3) for each $x,y∈C$, $lim t → 0 F(tz+(1−t)x,y)≤F(x,y)$;

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

Lemma 1.11 [18]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E and let F be a bifunction of $C×C$ into R satisfying (A1)-(A4). Let $r>0$ and $x∈E$. Then there exists $z∈C$ such that

$F(z,y)+ 1 r 〈y−z,Jz−Jx〉≥0,∀y∈K.$

Lemma 1.12 [19]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that $F:C×C→R$ satisfies (A1)-(A4). For $r>0$ and $x∈E$, define a mapping $T r F :E→C$ as follows:

$T r F (x)= { z ∈ C : F ( z , y ) + 1 r 〈 y − z , J z − J x 〉 ≥ 0 , ∀ y ∈ C }$

for all $z∈E$. Then the following hold:

1. (1)

$T r F$ is single valued;

2. (2)

$T r F$ is a firmly nonexpansive-type mapping, i.e., for any $x,y∈E$,

$〈 T r F x − T r F y , J T r F x − J T r F y 〉 ≤ 〈 T r F x − T r F y , J x − J y 〉 ;$
3. (3)

$F( T r F )=EP(F)$;

4. (4)

$EP(F)$ is closed and convex.

Lemma 1.13 [19]

Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that $F:C×C→R$ satisfies (A1)-(A4) and let $r>0$. Then for each $x∈E$ and $q∈F( T r F )$,

$ϕ ( q , T r F x ) +ϕ ( T r F x , x ) ≤ϕ(q,x).$

Let ${ T n }$ be a sequence of mappings from C into E, where C is a nonempty closed convex subset of a real Banach space E. For a subset B of C, we say that

1. (i)

$({ T n },B)$ satisfies condition AKTT (see [15]) if

$∑ n = 1 ∞ sup { ∥ T n + 1 x − T n x ∥ : x ∈ B } <∞;$
2. (ii)

$({ T n },B)$ satisfies condition AKTT (see [15]) if

$∑ n = 1 ∞ sup { ∥ J T n + 1 x − J T n x ∥ : x ∈ B } <∞.$

Recently, Shehu [11] proved strong convergence theorems for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The author obtained the following theorem.

Theorem 1.14 [11]

Let E be a uniformly convex real Banach space which is also uniformly smooth. Let C be a nonempty closed convex subset of E. For each $k=1,2,…,m$, let $F k$ be a bifunction from $C×C$ satisfying (A1)-(A4) and let ${ T n } n = 1 ∞$ be an infinite family of relatively quasi-nonexpansive mappings of C into itself such that $F:=( ⋂ n = 1 ∞ F( T n ))∩( ⋂ k = 1 m EP( F k ))≠∅$. Let $f:E→R$ be a convex and lower semi-continuous mapping with $C⊂int(D(f))$ and suppose ${ x n } n = 0 ∞$ is iteratively generated by $x 0 ∈C$, $C 1 =C$, $x 1 = Π C 1 f x 0$,

${ y n = J − 1 ( α n J x n + ( 1 − α n ) J T n x n ) , u n = T r m , n F m T r m − 1 , n F m − 1 ⋯ T r 2 , n F 2 T r 1 , n F 1 y n , C n + 1 = { w ∈ C n : G ( w , J u n ) ≤ G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n ≥ 1 ,$
(1.5)

where J is the duality mapping on E. Suppose ${ α n } n = 1 ∞$ is a sequence in $(0,1)$ such that $lim inf n → ∞ α n (1− α n )>0$ ${ r k , n } n = 1 ∞ ⊂(0,∞)$ ($k=1,2,…,m$) satisfying $lim inf n → ∞ r k , n >0$ ($k=1,2,…,m$). Suppose that for each bounded subset B of C, the ordered pair $({ T n },B)$ satisfies either condition AKTT or condition AKTT. Let T be the mapping from C into E defined by $Tx= lim n → ∞ T n x$ for all $x∈C$ and suppose that T is closed and $F(T)= ⋂ n = 1 ∞ F( T n )$. Then ${ x n } n = 0 ∞$ converges strongly to $Π F f x 0$.

In this paper we will construct a new iterative scheme and will get strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but not satisfy condition AKTT and AKTT.

## 2 Main results

Now, we shall first introduce the notion of uniformly closed mappings and give an example which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of G. Another example shall be given which is uniformly closed but not satisfy condition AKTT and AKTT.

Definition 2.1 Let E be a Banach space, C be a nonempty closed convex subset of E. Let ${ T n } n = 1 ∞ :C→E$ be a sequence of mappings of C into E such that $⋂ n = 1 ∞ F( T n )$ is nonempty. ${ T n } n = 1 ∞$ is said to be uniformly closed, if $p∈ ⋂ n = 1 ∞ F( T n )$, whenever ${ x n }⊂C$ converges strongly to p and $∥ x n − T n x n ∥→0$ as $n→∞$.

Example 1 Let $E= l 2$, where

$l 2 = { ξ = ( ξ 1 , ξ 2 , ξ 3 , … , ξ n , … ) : ∑ n = 1 ∞ | ξ n | 2 < ∞ } , ∥ ξ ∥ = ( ∑ n = 1 ∞ | ξ n | 2 ) 1 2 , ∀ ξ ∈ l 2 , 〈 ξ , η 〉 = ∑ n = 1 ∞ ξ n η n , ∀ ξ = ( ξ 1 , ξ 2 , ξ 3 , … , ξ n , … ) , η = ( η 1 , η 2 , η 3 , … , η n , … ) ∈ l 2 .$

It is well known that $l 2$ is a Hilbert space, so that $( l 2 ) ∗ = l 2$. Let ${ x n }⊂E$ be a sequence defined by

$x 0 = ( 1 , 0 , 0 , 0 , … ) , x 1 = ( 1 , 1 , 0 , 0 , … ) , x 2 = ( 1 , 0 , 1 , 0 , 0 , … ) , x 3 = ( 1 , 0 , 0 , 1 , 0 , 0 , … ) , ⋯ x n = ( ξ n , 1 , ξ n , 2 , ξ n , 3 , … , ξ n , k , … ) ⋯ ,$

where

for all $n≥1$.

Define a countable family of mappings $T n :E→E$ as follows:

for all $n≥0$.

Conclusion 2.2 ${ T n } n = 0 ∞$ has a unique fixed point 0, that is, $F( T n )={0}≠∅$, $∀n≥0$.

Proof The conclusion is obvious. □

Let ${ T n } n = 1 ∞$ be a countable family of quasi-relatively quasi-nonexpansive mappings, if

$⋂ n = 0 ∞ F( T n )= F ˆ ( { T n } n = 0 ∞ ) ,$

the ${ T n } n = 1 ∞$ is said to be a countable family of relatively nonexpansive mappings in the sense of functional G, where

$F ˆ ( { T n } n = 0 ∞ ) = { p ∈ C : ∃ x n ⇀ p , ∥ x n − T n x n ∥ → 0 , x n ∈ C }$

is said to be the asymptotic fixed point set of ${ T n } n = 1 ∞$.

Conclusion 2.3 ${ T n } n = 0 ∞$ is a countable family of relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional G.

Proof By Conclusion 2.2, we only need to show that $G(0,J T n x)≤G(0,Jx)$, $∀x∈E$. Note that $E= l 2$ is a Hilbert space, for any $n≥0$ we can derive

$G ( 0 , J T n x ) ≤ G ( 0 , J x ) ∀ x ∈ E ⇔ ϕ ( 0 , T n x ) ≤ ϕ ( 0 , x ) ⇔ ∥ 0 − T n x ∥ 2 ≤ ∥ 0 − x ∥ 2 ⇔ ∥ T n x ∥ 2 ≤ ∥ x ∥ 2 .$

It is obvious that ${ x n }$ converges weakly to $x 0 =(1,0,0,…)$, and

$∥ x n − T n x n ∥= ∥ n n + 1 x n − x n ∥ = 1 n + 1 ∥ x n ∥→0,$

as $n→∞$, so $x 0$ is an asymptotic fixed point of ${ T n } n = 0 ∞$. Joining with Conclusion 2.2, we can obtain $⋂ n = 0 ∞ F( T n )≠ F ˆ ( { T n } n = 0 ∞ )$.

Thus, ${ T n } n = 0 ∞$ is a countable family of relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of G. □

Conclusion 2.4 ${ T n } n = 0 ∞$ is a countable family of uniformly closed relatively quasi-nonexpansive mappings in the sense of functional G.

Proof In fact, for any strong convergent sequence ${ z n }⊂E$ such that $z n → z 0$ and $∥ z n − T n z n ∥→0$ as $n→∞$, there exists a sufficiently large nature number N, such that $z n ≠ x m$ for any $n,m>N$ (since $x n$ is not a Cauchy sequence it cannot converge to any element in E). Then $T n z n =− z n$ for $n>N$, it follows from $∥ z n − T n z n ∥→0$ that $2 z n →0$ and hence $z n → z 0 =0$.

Therefore, ${ T n } n = 0 ∞$ is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of functional G. □

Now, we give an example which is a countable family of uniformly closed quasi-nonexpansive mappings but not satisfied condition AKTT and AKTT.

Example 2 Let $X= ℜ 2$. For any complex number $x=r e i θ ∈X$, define a countable family of quasi-nonexpansive mappings as follows:

$T n :r e i θ →r e i ( θ + n π 2 ) ,n=1,2,3,….$

Proof It is easy to see that $⋂ n = 1 ∞ F( T n )={0}$. We first prove that ${ T n }$ is uniformly closed. In fact, for any strong convergent sequence ${ x n }⊂X$ such that $x n → x 0$ and $∥ x n − T n x n ∥→0$ as $n→∞$, there must be $x 0 =0∈ ⋂ n = 1 ∞ F( T n )$. Otherwise, if $x n → x 0 ≠0$, and

$∥ x 4 n + 1 − T 4 n + 1 x 4 n + 1 ∥→0,$

since $T 1$ is continuous, we have

$∥ x 4 n + 1 − T 4 n + 1 x 4 n + 1 ∥ = ∥ x 4 n + 1 − T 1 x 4 n + 1 ∥ → ∥ x 0 − T 1 x 0 ∥ ≠ 0 .$

This is a contradiction. Therefore, ${ T n }$ is uniformly closed.

Besides, take any $x=r e i θ ≠0$. For any n by the definition of $T n$, we have

$∥ T n x− T n + 1 x∥= ∥ r e π i 2 ∥ =r>0$

and

$∥J T n x−J T n + 1 x∥= ∥ r e π i 2 ∥ =r>0.$

That is to say, ${ T n }$ does not satisfied condition AKTT and AKTT. □

Now we are in a position to present our main theorems.

Theorem 2.5 Let ${ T n } n = 1 ∞$ be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself and other conditions are the same as Theorem 1.14 except for condition AKTT, AKTT and condition ‘Let T be the mapping from C into E defined by $Tx= lim n → ∞ T n x$ for all $x∈C$ and suppose that T is closed and $F(T)= ⋂ n = 1 ∞ F( T n )$ . Then the sequence ${ x n } n = 0 ∞$ generated by (1.5) converges strongly to $Π F f x 0$.

Proof We first show that $C n$, $∀n≥1$, is closed and convex. It is obvious that $C 1 =C$ is closed and convex. Suppose that $C n$ is closed convex for some $n>1$. From the definition of $C n + 1$, we have $z∈ C n + 1$ implies $G(z,J u n )≤G(z,J x n )$. This is equivalent to

$2 ( 〈 z , J x n 〉 − 〈 z , J u n 〉 ) ≤ ∥ x n ∥ 2 − ∥ u n ∥ 2 .$

This implies that $C n + 1$ is closed convex for the same $n>1$. Hence, $C n$ is closed and convex for all $n≥1$. This shows that $Π C n + 1 f x 0$ is well defined for all $n≥0$.

By taking $θ n k = T r k , n F k T r k − 1 , n F k − 1 ⋯ T r 2 , n F 2 T r 1 , n F 1$, $k=1,2,…,m$ and $θ n 0 =I$ for all $n≥1$, we obtain $u n = θ n m y n$.

We next show that $F⊂ C n$, $∀n≥1$. From Lemma 1.12, one sees that $T r k , n F k$, $k=1,2,…,m$, is relatively nonexpansive mapping. For $n=1$, we have $F⊂C= C 1$. Now, assume that $F⊂ C n$ for some $n≥2$. Then for each $x ∗ ∈F$, we obtain

$G ( x ∗ , J u n ) = G ( x ∗ , J θ n m y n ) ≤ G ( x ∗ , J y n ) = G ( x ∗ , ( α n J x n + ( 1 − α n ) J T n x n ) ) = ∥ x ∗ ∥ 2 − 2 α n 〈 x ∗ , J x n 〉 − 2 ( 1 − α n ) 〈 x ∗ , J T n x n 〉 + ∥ α n J x n + ( 1 − α n ) J T n x n ∥ 2 + 2 ρ f ( x ∗ ) ≤ ∥ x ∗ ∥ 2 − 2 α n 〈 x ∗ , J x n 〉 − 2 ( 1 − α n ) 〈 x ∗ , J T n x n 〉 + α n ∥ J x n ∥ 2 + ( 1 − α n ) ∥ J T n x n ∥ 2 + 2 ρ f ( x ∗ ) = α n G ( x ∗ , J x n ) + ( 1 − α n ) G ( x ∗ , J T n x n ) ≤ G ( x ∗ , J x n ) .$
(2.1)

So, $x ∗ ∈ C n$. This implies that $F⊂ C n$, $∀n≥1$ and the sequence ${ x n } n = 0 ∞$ generated by (1.5) is well defined.

We now show that $lim n → ∞ G( x n ,J x 0 )$ exists. Since $f:E→R$ is a convex and lower semi-continuous, applying Lemma 1.5, we see that there exist $u ∗ ∈ E ∗$ and $α∈R$ such that

$f(y)≥ 〈 y , u ∗ 〉 +α,∀y∈E.$

It follows that

$G ( x n , J x 0 ) = ∥ x n ∥ 2 − 2 〈 x n , J x 0 〉 + ∥ x 0 ∥ 2 + 2 ρ f ( x n ) ≥ ∥ x n ∥ 2 − 2 〈 x n , J x 0 〉 + ∥ x 0 ∥ 2 + 2 ρ 〈 x n , u ∗ 〉 + 2 ρ α = ∥ x n ∥ 2 − 2 〈 x n , J x 0 − ρ u ∗ 〉 + ∥ x 0 ∥ 2 + 2 ρ α ≥ ∥ x n ∥ 2 − 2 ∥ x n ∥ ∥ J x 0 − ρ u ∗ ∥ + ∥ x 0 ∥ 2 + 2 ρ α = ( ∥ x n ∥ − ∥ J x 0 − ρ u ∗ ∥ ) 2 + ∥ x 0 ∥ 2 − ∥ J x 0 − ρ u ∗ ∥ 2 + 2 ρ α .$
(2.2)

Since $x n = Π C n f x 0$, it follows from (2.2) that

$G ( x ∗ , J x 0 ) ≥G( x n ,J x 0 )≥ ( ∥ x n ∥ − ∥ J x 0 − ρ u ∗ ∥ ) 2 + ∥ x 0 ∥ 2 − ∥ J x 0 − ρ u ∗ ∥ 2 +2ρα$

for each $x ∗ ∈F(T)$. This implies that ${ x n } n = 1 ∞$ is bounded and so is ${ G ( x n , J x 0 ) } n = 0 ∞$. By the construction of $C n$, we have $C m ⊂ C n$ and $x m = Π C m f x 0 ∈ C n$ for any positive integer $m≥n$. It then follows from Lemma 1.7 that

$ϕ( x m , x n )+G( x n ,J x 0 )≤G( x m ,J x 0 ).$
(2.3)

It is obvious that

$ϕ( x m , x n )≥ ( ∥ x m ∥ − ∥ x n ∥ ) 2 ≥0.$

In particular,

$ϕ( x n + 1 , x n )+G( x n ,J x 0 )≤G( x n + 1 ,J x 0 )$

and

$ϕ( x n + 1 , x n )≥ ( ∥ x n + 1 ∥ − ∥ x n ∥ ) 2 ≥0,$

and so ${ G ( x n , J x 0 ) } n = 0 ∞$ is nondecreasing. It follows that the limit of ${ G ( x n , J x 0 ) } n = 0 ∞$ exists.

By the fact that $C m ⊂ C n$ and $x m = Π C m f x 0 ∈ C n$ for any positive integer $m≥n$, we obtain

$ϕ( x m , u n )≤ϕ( x m , x n ).$

Now, (2.3) implies that

$ϕ( x m , u n )≤ϕ( x m , x n )≤G( x m ,J x 0 )−G( x n ,J x 0 ).$
(2.4)

Taking the limit as $m,n→∞$ in (2.4), we obtain

$lim n → ∞ ϕ( x m , x n )=0.$

It then follows from Lemma 1.9 that $∥ x m − x n ∥→0$ as $m,n→∞$. Hence, ${ x n } n = 0 ∞$ is a Cauchy sequence. Since E is a Banach space and C is closed and convex, there exists $p∈C$ such that $x n →p$ as $n→∞$.

Now since $ϕ( x m , x n )→0$ as $m,n→∞$ we have in particular that $ϕ( x n + 1 , x n )→0$ as $n→∞$ and this further implies that $lim n → ∞ ∥ x n + 1 − x n ∥=0$. Since $x n + 1 = Π C n = 1 f x 0 ∈ C n + 1$ we have

$ϕ( x n + 1 , u n )≤ϕ( x n + 1 , x n ),∀n≥0.$

Then we obtain

$lim n → ∞ ϕ( x n + 1 , u n )=0.$

Since E is uniformly convex and smooth, we have from Lemma 1.9

$lim n → ∞ ∥ x n + 1 − x n ∥=0= lim n → ∞ ∥ x n + 1 − u n ∥.$

So,

$∥ x n − u n ∥≤∥ x n + 1 − x n ∥+∥ x n + 1 − u n ∥.$

Hence,

$lim n → ∞ ∥ x n − u n ∥=0.$
(2.5)

Since J is uniformly norm-to-norm continuous on bounded sets and $lim n → ∞ ∥ x n − u n ∥=0$, we obtain

$lim n → ∞ ∥J x n −J u n ∥=0.$
(2.6)

Let $r= sup n ≥ 1 {∥ x n ∥,∥ T n x n ∥}$. Since E is uniformly smooth, we know that $E ∗$ is uniformly convex. Then from Lemma 1.10, we have

$G ( x ∗ , J u n ) = G ( x ∗ , J θ n m y n ) ≤ G ( x ∗ , J y n ) = G ( x ∗ , ( α n J x n + ( 1 − α n ) J T n x n ) ) = ∥ x ∗ ∥ 2 − 2 α n 〈 x ∗ , J x n 〉 − 2 ( 1 − α n ) 〈 x ∗ , J T n x n 〉 + ∥ α n J x n + ( 1 − α n ) J T n x n ∥ 2 + 2 ρ f ( x ∗ ) ≤ ∥ x ∗ ∥ 2 − 2 α n 〈 x ∗ , J x n 〉 − 2 ( 1 − α n ) 〈 x ∗ , J T n x n 〉 + α n ∥ J x n ∥ 2 + ( 1 − α n ) ∥ J T n x n ∥ 2 − α n ( 1 − α n ) g ( ∥ J x n − J T n x n ∥ ) + 2 ρ f ( x ∗ ) = α n G ( x ∗ , J x n ) + ( 1 − α n ) G ( x ∗ , J T n x n ) − α n ( 1 − α n ) g ( ∥ J x n − J T n x n ∥ ) ≤ G ( x ∗ , J x n ) − α n ( 1 − α n ) g ( ∥ J x n − J T n x n ∥ ) .$

It then follows that

$α n (1− α n )g ( ∥ J x n − J T n x n ∥ ) ≤G ( x ∗ , J x n ) −G ( x ∗ , J u n ) .$

But

$G ( x ∗ , J x n ) − G ( x ∗ , J u n ) = ∥ x n ∥ 2 − ∥ u n ∥ 2 − 2 〈 x ∗ , J x n − J u n 〉 ≤ ∥ x n ∥ 2 − ∥ u n ∥ 2 + 2 | 〈 x ∗ , J x n − J u n 〉 | ≤ | ∥ x n ∥ − ∥ u n ∥ | ( ∥ x n ∥ + ∥ u n ∥ ) + 2 ∥ x ∗ ∥ ∥ J x n − J u n ∥ ≤ ∥ x n − u n ∥ ( ∥ x n ∥ + ∥ u n ∥ ) + 2 ∥ x ∗ ∥ ∥ J x n − J u n ∥ .$

From (2.5) and (2.6), we obtain

$G ( x ∗ , J x n ) −G ( x ∗ , J u n ) →0,n→∞.$

Using the condition $lim inf n → ∞ α n (1− α n )>0$, we have

$lim n → ∞ g ( ∥ J x n − J T n x n ∥ ) =0.$

By the properties of g, we have $lim n → ∞ ∥J x n −J T n x n ∥=0$. Since $J − 1$ is also uniformly norm-to-norm continuous on bounded sets, we have

$lim n → ∞ ∥ x n − T n x n ∥=0.$

Since ${ T n } n = 1 ∞$ are uniformly closed, and ${ x n } n = 1 ∞$ is a Cauchy sequence. Then $p∈F(T)= ⋂ n = 1 ∞ F( T n )$.

Next, we show that $p∈ ⋂ k = 1 m EP( F k )$. From (2.1), we obtain

$ϕ ( x ∗ , u n ) = ϕ ( x ∗ , θ n m y n ) = ϕ ( x ∗ , T r m , n F m θ n m − 1 y n ) ≤ ϕ ( x ∗ , θ n m − 1 y n ) ≤ ϕ ( x ∗ , x n ) .$
(2.7)

Since $x ∗ ∈EP( F m )=F( T r m , n F m )$ for all $n≥1$, it follows from (2.7) and Lemma 1.13 that

$ϕ ( u n , θ n m − 1 y n ) = ϕ ( T r m , n F m θ n m − 1 y n , θ n m − 1 y n ) ≤ ϕ ( x ∗ , θ n m − 1 y n ) − ϕ ( x ∗ , u n ) ≤ ϕ ( x ∗ , x n ) − ϕ ( x ∗ , u n ) .$

From (2.5) and (2.6), we obtain $lim n → ∞ ϕ( θ n m y n , θ n m − 1 y n )= lim n → ∞ ϕ( u n , θ n m − 1 y n )=0$. From Lemma 1.9, we have

$lim n → ∞ ∥ θ n m y n − θ n m − 1 y n ∥ = lim n → ∞ ∥ u n − θ n m − 1 y n ∥ =0.$
(2.8)

Hence, we have from (2.8) that

$lim n → ∞ ∥ J θ n m y n − J θ n m − 1 y n ∥ =0.$
(2.9)

Again, since $x ∗ ∈EP( F m − 1 )=F( T r m − 1 , n F m − 1 )$ for all $n≥1$, it follows from (2.7) and Lemma 1.13 that

$ϕ ( θ n m − 1 y n , θ n m − 2 y n ) = ϕ ( T r m − 1 , n F m − 1 θ n m − 2 y n , θ n m − 2 y n ) ≤ ϕ ( x ∗ , θ n m − 2 y n ) − ϕ ( x ∗ , θ n m − 1 y n ) ≤ ϕ ( x ∗ , x n ) − ϕ ( x ∗ , u n ) .$

Again, from (2.5) and (2.6), we obtain $lim n → ∞ ϕ( θ n m − 1 y n , θ n m − 2 y n )=0$. From Lemma 1.9, we have

$lim n → ∞ ∥ θ n m − 1 y n − θ n m − 2 y n ∥ =0$
(2.10)

and hence,

$lim n → ∞ ∥ J θ n m − 1 y n − J θ n m − 2 y n ∥ =0.$
(2.11)

In a similar way, we can verify that

$lim n → ∞ ∥ θ n m − 2 y n − θ n m − 3 y n ∥ =⋯= lim n → ∞ ∥ θ n 1 y n − y n ∥ =0.$
(2.12)

From (2.8), (2.10), and (2.12), we can conclude that

$lim n → ∞ ∥ θ n k y n − θ n k − 1 y n ∥ =0,k=1,2,…,m.$
(2.13)

Since $x n →p$, $n→∞$, we obtain from (2.5) that $u n →p$, $n→∞$. Again, from (2.8), (2.10), (2.12), and $u n →p$, $n→∞$, we have that $θ n k y n →p$, $n→∞$ for each $k=1,2,…,m$. Also, using (2.13), we obtain

$lim n → ∞ ∥ J θ n k y n − J θ n k − 1 y n ∥ =0,k=1,2,…,m.$

Since $lim inf n → ∞ r k , n >0$, $k=1,2,…,m$,

$lim n → ∞ ∥ J θ n k y n − J θ n k − 1 y n ∥ r k , n =0.$
(2.14)

By Lemma 1.12, we have for each $k=1,2,…,m$

$F k ( θ n k y n , y ) + 1 r k , n 〈 y − θ n k y n , J θ n k y n − J θ n k − 1 y n 〉 ≥0,∀y∈C.$

Furthermore, using (A2) we obtain

$1 r k , n 〈 y − θ n k y n , J θ n k y n − J θ n k − 1 y n 〉 ≥ F k ( y , θ n k y n ) .$
(2.15)

By (A4), (2.14), and $θ n k y n →p$, we have for each $k=1,2,…,m$

$F k (y,p)≤0,y∈C.$

For fixed $y∈C$, let $z t =ty+(1−t)p$ for all $t∈(0,1]$. This implies that $z t ∈C$. This yields $F k ( z t ,p)≤0$. It follows from (A1) and (A4) that

$0= F k ( z t , z t )≤t F k ( z t ,y)+(1−t) F k ( z t ,p)≤t F k ( z t ,y)$

and hence

$0≤ F k ( z t ,y).$

From condition (A3), we obtain

$F k (p,y)≥0,y∈C.$

This implies that $p∈EP( F k )$, $k=1,2,…,m$. Thus, $p∈ ⋂ k = 1 m EP( F k )$. Hence, we have $p∈F= ⋂ k = 1 m EP( F k )∩( ⋂ n = 1 ∞ F( T n ))$.

Finally, we show that $p= Π F f x 0$. Since $F= ⋂ k = 1 m EP( F k )∩( ⋂ n = 1 ∞ F( T n ))$ is a closed and convex set, from Lemma 1.6, we know that $Π F f x 0$ is single valued and denote $w= Π F f x 0$. Since $x n = Π c n f x 0$ and $w∈F⊂ C n$, we have

$G( x n ,J x 0 )≤G(w,J x 0 ),∀n≥0.$

We know that $G(ξ,Jφ)$ is convex and lower semi-continuous with respect to ξ when φ is fixed. This implies that

$G(p,J x 0 )≤ lim inf n → ∞ G( x n ,J x 0 )≤ lim sup n → ∞ G( x n ,J x 0 )≤G(w,J x 0 ).$

From the definition of $Π F f x 0$ and $p∈F$, we see that $p=w$. This completes the proof. □

Corollary 2.6 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. For each $k=1,2,…,m$, let $F k$ be a bifunction from $C×C$ satisfying (A1)-(A4) and let ${ T n } n = 1 ∞$ be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that $F:=( ⋂ n = 1 ∞ F( T n ))∩( ⋂ k = 1 m EP( F k ))≠∅$. Suppose ${ x n } n = 0 ∞$ is iteratively generated by $x 0 ∈C$, $C 1 =C$, $x 1 = Π C 1 f x 0$,

${ y n = J − 1 ( α n J x n + ( 1 − α n ) J T n x n ) , u n = T r m , n F m T r m − 1 , n F m − 1 ⋯ T r 2 , n F 2 T r 1 , n F 1 y n , C n + 1 = { w ∈ C n : ϕ ( w , u n ) ≤ ϕ ( w , x n ) } , x n + 1 = Π C n + 1 x 0 , n ≥ 1 ,$

where J is the duality mapping on E. Suppose ${ α n } n = 1 ∞$ is a sequence in $(0,1)$ such that $lim inf n → ∞ α n (1− α n )>0$, and ${ r k , n } n = 1 ∞ ⊂(0,∞)$ ($k=1,2,…,m$) satisfying $lim inf n → ∞ r k , n >0$ ($k=1,2,…,m$). Then ${ x n } n = 0 ∞$ converges strongly to $Π F x 0$.

Proof Take $f(x)=0$ for all $x∈E$ in Theorem 2.5, then $G(ξ,Jx)=ϕ(ξ,x)$ and $Π C f x 0 = Π C x 0$. Then Corollary 2.6 holds. □

Take $F k ≡0$ ($k=1,2,…,m$), it is obvious that the following holds.

Corollary 2.7 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. Let ${ T n } n = 1 ∞$ be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that $F=( ⋂ n = 1 ∞ F( T n ))≠∅$. Let $f:E→R$ be a convex and lower semi-continuous mapping with $C⊂int(D(f))$ and suppose ${ x n } n = 0 ∞$ is iteratively generated by $x 0 ∈C$, $C 1 =C$, $x 1 = Π C 1 f x 0$,

${ y n = J − 1 ( α n J x n + ( 1 − α n ) J T n x n ) , C n + 1 = { w ∈ C n : G ( w , J y n ) ≤ G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n ≥ 1 ,$

where J is the duality mapping on E. Suppose ${ α n } n = 1 ∞$ is a sequence in $(0,1)$ such that $lim inf n → ∞ α n (1− α n )>0$, and ${ r k , n } n = 1 ∞ ⊂(0,∞)$ ($k=1,2,…,m$) satisfying $lim inf n → ∞ r k , n >0$ ($k=1,2,…,m$). Then ${ x n } n = 0 ∞$ converges strongly to $Π F x 0$.

## 3 Applications

Let $φ:C→R$ be a real-valued function. The convex minimization problem is to find $x ∗ ∈C$ such that

$φ ( x ∗ ) ≤φ(y),$
(3.1)

$∀y∈C$. The set of solutions of (3.1) is denoted by $CMP(φ)$. For each $r>0$ and $x∈E$, define the mapping

$T r φ (x)= { z ∈ C : φ ( y ) + 1 r 〈 y − z , J z − J x 〉 ≥ φ ( z ) , ∀ y ∈ C } .$

Theorem 3.1 Let E be a uniformly convex and uniformly smooth real Banach space, and let C be a nonempty closed convex subset of E. For each $k=1,2,…,m$, let $φ k$ be a bifunction from $C×C$ satisfying (A1)-(A4) and let ${ T n } n = 1 ∞$ be a countable family of uniformly closed relatively quasi-nonexpansive mappings of C into itself such that $F:=( ⋂ n = 1 ∞ F( T n ))∩( ⋂ k = 1 m CMP( φ k ))≠∅$. Let $f:E→R$ be a convex and lower semi-continuous mapping with $C⊂int(D(f))$ and suppose ${ x n } n = 0 ∞$ is iteratively generated by $x 0 ∈C$, $C 1 =C$, $x 1 = Π C 1 f x 0$,

${ y n = J − 1 ( α n J x n + ( 1 − α n ) J T n x n ) , u n = T r m , n φ m T r m − 1 , n φ m − 1 ⋯ T r 2 , n φ 2 T r 1 , n φ 1 y n , C n + 1 = { w ∈ C n : G ( w , J u n ) ≤ G ( w , J x n ) } , x n + 1 = Π C n + 1 f x 0 , n ≥ 1 ,$

where J is the duality mapping on E. Suppose ${ α n } n = 1 ∞$ is a sequence in $(0,1)$ such that $lim inf n → ∞ α n (1− α n )>0$ and ${ r k , n } n = 1 ∞ ⊂(0,∞)$ ($k=1,2,…,m$) satisfying $lim inf n → ∞ r k , n >0$ ($k=1,2,…,m$). Then ${ x n } n = 0 ∞$ converges strongly to $Π F f x 0$.

Proof Define $F k (x,y)= φ k (y)− φ k (x)$, $x,y∈C$ and $k=1,2,…,m$. Then $F( T r k F k )=EP( F k )=CMP( φ k )=F( T r k φ k )$ for each $k=1,2,…,m$, and therefore ${ F k } k = 1 m$ satisfies conditions (A1) and (A2). Furthermore, one can easily show that ${ F k } k = 1 m$ satisfies (A3) and (A4). Therefore, from Theorem 2.5, we obtain Theorem 3.1. □

## References

1. 1.

Butnariu D, Reich S, Zaslavski AJ: Asymptotic behaviour of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 2001, 7: 151–174.

2. 2.

Butnariu D, Reich S, Zaslavski AJ: Weak convergence of orbits of nonlinear operator in reflexive Banach spaces. Numer. Funct. Anal. Optim. 2003, 24: 489–508. 10.1081/NFA-120023869

3. 3.

Censor Y, Reich S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996, 37: 323–339. 10.1080/02331939608844225

4. 4.

Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in Banach spaces. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007

5. 5.

Chidume CE Lecture Notes in Mathematics 1965. In Geometric Properties of Banach Spaces and Nonlinear Iterations. Springer, Berlin; 2009. xviii+326 pp.

6. 6.

Takahashi W: Nonlinear Functional Analysis-Fixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2000. (in Japanese)

7. 7.

Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.

8. 8.

Alber YI: Metric and generalized projection operator in Banach spaces: properties and applications. Lecture Notes in Pure and Applied Mathematics 178. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Dekker, New York; 1996:15–50.

9. 9.

Wu KQ, Huang NJ: The generalized f -projection operator with application. Bull. Aust. Math. Soc. 2006, 73: 307–317. 10.1017/S0004972700038892

10. 10.

Fan JH, Liu X, Li JL: Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces. Nonlinear Anal. 2009, 70: 3997–4007. 10.1016/j.na.2008.08.008

11. 11.

Shehu Y: Strong convergence theorems for infinite family of relatively quasi-nonexpansive mappings and systems of equilibrium problems. Appl. Math. Comput. 2012, 218: 5146–5156. 10.1016/j.amc.2011.11.001

12. 12.

Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.

13. 13.

Li X, Huang N, O’Regan D: Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications. Comput. Math. Appl. 2010, 60: 1322–1331. 10.1016/j.camwa.2010.06.013

14. 14.

Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011

15. 15.

Nilsrakoo W, Saejung S: Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 312454

16. 16.

Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611X

17. 17.

Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16(2):1127–1138.

18. 18.

Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.

19. 19.

Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031

## Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

## Author information

Correspondence to Yongfu Su.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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