Research  Open  Published:
Uniformly closed replaced AKTT or ^{∗}AKTT condition to get strong convergence theorems for a countable family of relatively quasinonexpansive mappings and systems of equilibrium problems
Fixed Point Theory and Applicationsvolume 2014, Article number: 103 (2014)
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 quasinonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized fprojection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasinonexpansive 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 $T:C\to C$ is called nonexpansive if
Let E be a real Banach space and C be a nonempty closed convex subset of E. A point $p\in C$ is said to be an asymptotic fixed point of T if there exists a sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset C$ such that ${x}_{n}\rightharpoonup p$ and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0$. The set of asymptotic fixed point is denoted by $\stackrel{\u02c6}{F}(T)$. We say that a mapping T is relatively nonexpansive (see [1–4]) if the following conditions are satisfied:

(I)
$F(T)\ne \mathrm{\varnothing}$;

(II)
$\varphi (p,Tx)\le \varphi (p,x)$, $\mathrm{\forall}x\in C$, $p\in F(T)$;

(III)
$F(T)=\stackrel{\u02c6}{F}(T)$.
If T satisfies (I) and (II), then T is said to be relatively quasinonexpansive. It is easy to see that the class of relatively quasinonexpansive mappings contains the class of relatively nonexpansive mappings.
Let E be a real Banach space. The modulus of smoothness of E is the function ${\rho}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ defined by
E is uniformly smooth if and only if
Let $dimE\ge 2$. The modulus of convexity of E is the function ${\delta}_{E}(\u03f5):=inf\{1\parallel \frac{x+y}{2}\parallel :\parallel x\parallel =\parallel y\parallel =1;\u03f5=\parallel xy\parallel \}$. E is uniformly convex if for any $\u03f5\in (0,2]$, there exists $\delta =\delta (\u03f5)>0$ such that if $x,y\in E$ with $\parallel x\parallel \le 1$, $\parallel y\parallel \le 1$ and $\parallel xy\parallel \ge \u03f5$, then $\parallel \frac{1}{2}(x+y)\parallel \le 1\delta $. Equivalently, E is uniformly convex if and only if ${\delta}_{E}(\u03f5)>0$ for all $\u03f5\in (0,2]$. A normed space E is called strictly convex if for all $x,y\in E$, $x\ne y$, $\parallel x\parallel =\parallel y\parallel =1$, we have $\parallel \lambda x+(1\lambda )y\parallel <1$, $\mathrm{\forall}\lambda \in (0,1)$.
Let ${E}^{\ast}$ be the dual space of E. We denote by J the normalized duality mapping from E to ${2}^{{E}^{\ast}}$ defined by
The following properties of J are well known (see [5–7] for more details):

(1)
If E is uniformly smooth, then J is normtonorm uniformly continuous on each bounded subset of E.

(2)
If E is reflexive, then J is a mapping from E onto ${E}^{\ast}$.

(3)
If E is smooth, then J is single valued.
Throughout this paper, we denote by ϕ the functional on $E\times E$ defined by
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 ${\mathrm{\Pi}}_{C}$ from E onto C is defined by
The existence and uniqueness of ${\mathrm{\Pi}}_{C}$ follows from the property of the functional $\varphi (x,y)$ and strict monotonicity of the mapping J. It is obvious that
Next, we recall the notion of generalized fprojection operator and its properties. Let $G:C\times {E}^{\ast}\to R\cup \{+\mathrm{\infty}\}$ be a functional defined as follows:
where $\xi \in C$, $\phi \in {E}^{\ast}$, ρ is a positive number and $f:C\to R\cup \{+\mathrm{\infty}\}$ is proper, convex, and lower semicontinuous. From the definitions of G and f, it is easy to see the following properties:

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

(ii)
$G(\xi ,\phi )$ is convex and lower semicontinuous with respect to ξ when φ is fixed.
Definition 1.1 [9]
Let E be a real Banach space with its dual ${E}^{\ast}$. Let C be a nonempty, closed, and convex subset of E. We say that ${\mathrm{\Pi}}_{C}^{f}:{E}^{\ast}\to {2}^{C}$ is a generalized fprojection operator if
For the generalized fprojection 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 ${E}^{\ast}$. Let C be a nonempty, closed, and convex subset of E. Then the following statements hold:

(i)
${\mathrm{\Pi}}_{C}^{f}$ is a nonempty closed convex subset of C for all $\phi \in {E}^{\ast}$.

(ii)
If E is smooth, then for all $\phi \in {E}^{\ast}$, $x\in {\mathrm{\Pi}}_{C}^{f}\phi $ if and only if
$$\u3008xy,\phi Jx\u3009+\rho f(y)\rho f(x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C.$$ 
(iii)
If E is strictly convex and $f:C\to R\cup \{+\mathrm{\infty}\}$ is positive homogeneous (i.e., $f(tx)=tf(x)$ for all $t>0$ such that $tx\in C$ where $x\in C$), then ${\mathrm{\Pi}}_{C}^{f}$ is a singlevalued 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 ${E}^{\ast}$ and C a nonempty, closed, and convex subset of E. Then if E is strictly convex, then ${\mathrm{\Pi}}_{C}^{f}$ is a singlevalued mapping.
Recall that J is a singlevalued mapping when E is a smooth Banach space. There exists a unique element $\phi \in {E}^{\ast}$ such that $\phi =Jx$ for each $x\in E$. This substitution in (1.3) gives
Now, we consider the second generalized fprojection 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 ${\mathrm{\Pi}}_{C}^{f}:E\to {2}^{C}$ is a generalized fprojection operator if
Obviously, the definition of relatively quasinonexpansive mapping T is equivalent to

(1)
$F(T)\ne \mathrm{\varnothing}$;

(2)
$G(p,JTx)\le G(p,Jx)$, $\mathrm{\forall}x\in C$, $p\in F(T)$.
Lemma 1.5 [12]
Let E be a Banach space and $f:E\to R\cup \{+\mathrm{\infty}\}$ be a lower semicontinuous convex functional. Then there exist $x\in {E}^{\ast}$ and $\alpha \in R$ such that
We know that the following lemmas hold for operator ${\mathrm{\Pi}}_{C}^{f}$.
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:

(i)
${\mathrm{\Pi}}_{C}^{f}$ is a nonempty, closed, and convex subset of C for all $x\in E$;

(ii)
for all $x\in E$, $\stackrel{\u02c6}{x}\in {\mathrm{\Pi}}_{C}^{f}x$ if and only if
$$\u3008\stackrel{\u02c6}{x}y,JxJ\stackrel{\u02c6}{x}\u3009+\rho f(y)\rho f(x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C;$$ 
(iii)
if E is strictly convex, then ${\mathrm{\Pi}}_{C}^{f}x$ is a singlevalued mapping.
Lemma 1.7 [13]
Let C be a nonempty, closed, and convex subset of a smooth and reflexive Banach space E. Let $x\in E$ and $\stackrel{\u02c6}{x}\in {\mathrm{\Pi}}_{C}^{f}x$. Then
The fixed points set $F(T)$ of a relatively quasinonexpansive mapping is closed convex as given in the following lemma.
Let C be a nonempty closed convex subset of a smooth, uniformly convex Banach space E. Let T be a closed relatively quasinonexpansive 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 ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ and ${\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$ be sequences in E such that either ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ or ${\{{y}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is bounded. If ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{y}_{n})=0$, then ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0$.
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 $g:{R}^{+}\to {R}^{+}$, $g(0)=0$, such that
for all $x,y\in {B}_{r}$ and $0\le \lambda \le 1$, where ${W}_{p}(\lambda )=\lambda {(1\lambda )}^{p}+{\lambda}^{p}(1\lambda )$.
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 ${B}_{r}(0):=\{x\in E:\parallel x\parallel \le r\}$ and $\lambda \in [0,1]$. Then there exists a continuous strictly increasing convex function
such that for every $x,y\in {B}_{r}(0)$, the following inequality holds:
Let F be a bifunction of $C\times C$ into R. The equilibrium problem is to find ${x}^{\ast}\in C$ such that $F({x}^{\ast},y)\ge 0$, for all $y\in C$. 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 $F:C\times C\to R$, let us assume that F satisfies the following conditions:
(A1) $F(x,x)=0$ for all $x\in C$;
(A2) F is monotone, i.e., $F(x,y)+F(y,x)\le 0$ for all $x,y\in C$;
(A3) for each $x,y\in C$, ${lim}_{t\to 0}F(tz+(1t)x,y)\le F(x,y)$;
(A4) for each $x\in C$, $y\mapsto F(x,y)$ is convex and lower semicontinuous.
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 $C\times C$ into R satisfying (A1)(A4). Let $r>0$ and $x\in E$. Then there exists $z\in C$ such that
Lemma 1.12 [19]
Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E. Assume that $F:C\times C\to R$ satisfies (A1)(A4). For $r>0$ and $x\in E$, define a mapping ${T}_{r}^{F}:E\to C$ as follows:
for all $z\in E$. Then the following hold:

(1)
${T}_{r}^{F}$ is single valued;

(2)
${T}_{r}^{F}$ is a firmly nonexpansivetype mapping, i.e., for any $x,y\in E$,
$$\u3008{T}_{r}^{F}x{T}_{r}^{F}y,J{T}_{r}^{F}xJ{T}_{r}^{F}y\u3009\le \u3008{T}_{r}^{F}x{T}_{r}^{F}y,JxJy\u3009;$$ 
(3)
$F({T}_{r}^{F})=EP(F)$;

(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 $F:C\times C\to R$ satisfies (A1)(A4) and let $r>0$. Then for each $x\in E$ and $q\in F({T}_{r}^{F})$,
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

(i)
$(\{{T}_{n}\},B)$ satisfies condition AKTT (see [15]) if
$$\sum _{n=1}^{\mathrm{\infty}}sup\{\parallel {T}_{n+1}x{T}_{n}x\parallel :x\in B\}<\mathrm{\infty};$$ 
(ii)
$(\{{T}_{n}\},B)$ satisfies condition ^{∗}AKTT (see [15]) if
$$\sum _{n=1}^{\mathrm{\infty}}sup\{\parallel J{T}_{n+1}xJ{T}_{n}x\parallel :x\in B\}<\mathrm{\infty}.$$
Recently, Shehu [11] proved strong convergence theorems for approximation of common element of set of common fixed points of countably infinite family of relatively quasinonexpansive 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 fprojection 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 $k=1,2,\dots ,m$, let ${F}_{k}$ be a bifunction from $C\times C$ satisfying (A1)(A4) and let ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ be an infinite family of relatively quasinonexpansive mappings of C into itself such that $F:=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\cap ({\bigcap}_{k=1}^{m}EP({F}_{k}))\ne \mathrm{\varnothing}$. Let $f:E\to R$ be a convex and lower semicontinuous mapping with $C\subset int(D(f))$ and suppose ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is iteratively generated by ${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,
where J is the duality mapping on E. Suppose ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a sequence in $(0,1)$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0$ ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) satisfying ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). 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 $Tx={lim}_{n\to \mathrm{\infty}}{T}_{n}x$ for all $x\in C$ and suppose that T is closed and $F(T)={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$. Then ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ converges strongly to ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$.
In this paper we will construct a new iterative scheme and will get strong convergence theorem for a countable family of relatively quasinonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized fprojection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasinonexpansive 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 quasinonexpansive 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 ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}:C\to E$ be a sequence of mappings of C into E such that ${\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$ is nonempty. ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is said to be uniformly closed, if $p\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$, whenever $\{{x}_{n}\}\subset C$ converges strongly to p and $\parallel {x}_{n}{T}_{n}{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$.
Example 1 Let $E={l}^{2}$, where
It is well known that ${l}^{2}$ is a Hilbert space, so that ${({l}^{2})}^{\ast}={l}^{2}$. Let $\{{x}_{n}\}\subset E$ be a sequence defined by
where
for all $n\ge 1$.
Define a countable family of mappings ${T}_{n}:E\to E$ as follows:
for all $n\ge 0$.
Conclusion 2.2 ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ has a unique fixed point 0, that is, $F({T}_{n})=\{0\}\ne \mathrm{\varnothing}$, $\mathrm{\forall}n\ge 0$.
Proof The conclusion is obvious. □
Let ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ be a countable family of quasirelatively quasinonexpansive mappings, if
the ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is said to be a countable family of relatively nonexpansive mappings in the sense of functional G, where
is said to be the asymptotic fixed point set of ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$.
Conclusion 2.3 ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of relatively quasinonexpansive 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 $G(0,J{T}_{n}x)\le G(0,Jx)$, $\mathrm{\forall}x\in E$. Note that $E={l}^{2}$ is a Hilbert space, for any $n\ge 0$ we can derive
It is obvious that $\{{x}_{n}\}$ converges weakly to ${x}_{0}=(1,0,0,\dots )$, and
as $n\to \mathrm{\infty}$, so ${x}_{0}$ is an asymptotic fixed point of ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$. Joining with Conclusion 2.2, we can obtain ${\bigcap}_{n=0}^{\mathrm{\infty}}F({T}_{n})\ne \stackrel{\u02c6}{F}({\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}})$.
Thus, ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of relatively quasinonexpansive mappings but not a countable family of relatively nonexpansive mappings in the sense of G. □
Conclusion 2.4 ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of uniformly closed relatively quasinonexpansive mappings in the sense of functional G.
Proof In fact, for any strong convergent sequence $\{{z}_{n}\}\subset E$ such that ${z}_{n}\to {z}_{0}$ and $\parallel {z}_{n}{T}_{n}{z}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, there exists a sufficiently large nature number N, such that ${z}_{n}\ne {x}_{m}$ 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 $\parallel {z}_{n}{T}_{n}{z}_{n}\parallel \to 0$ that $2{z}_{n}\to 0$ and hence ${z}_{n}\to {z}_{0}=0$.
Therefore, ${\{{T}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a countable family of uniformly closed relatively quasinonexpansive 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 quasinonexpansive mappings but not satisfied condition AKTT and ^{∗}AKTT.
Example 2 Let $X={\mathrm{\Re}}^{2}$. For any complex number $x=r{e}^{i\theta}\in X$, define a countable family of quasinonexpansive mappings as follows:
Proof It is easy to see that ${\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})=\{0\}$. We first prove that $\{{T}_{n}\}$ is uniformly closed. In fact, for any strong convergent sequence $\{{x}_{n}\}\subset X$ such that ${x}_{n}\to {x}_{0}$ and $\parallel {x}_{n}{T}_{n}{x}_{n}\parallel \to 0$ as $n\to \mathrm{\infty}$, there must be ${x}_{0}=0\in {\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$. Otherwise, if ${x}_{n}\to {x}_{0}\ne 0$, and
since ${T}_{1}$ is continuous, we have
This is a contradiction. Therefore, $\{{T}_{n}\}$ is uniformly closed.
Besides, take any $x=r{e}^{i\theta}\ne 0$. For any n by the definition of ${T}_{n}$, we have
and
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 ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ be a countable family of uniformly closed relatively quasinonexpansive 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 $Tx={lim}_{n\to \mathrm{\infty}}{T}_{n}x$ for all $x\in C$ and suppose that T is closed and $F(T)={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$ ’. Then the sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ generated by (1.5) converges strongly to ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$.
Proof We first show that ${C}_{n}$, $\mathrm{\forall}n\ge 1$, 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 $z\in {C}_{n+1}$ implies $G(z,J{u}_{n})\le G(z,J{x}_{n})$. This is equivalent to
This implies that ${C}_{n+1}$ is closed convex for the same $n>1$. Hence, ${C}_{n}$ is closed and convex for all $n\ge 1$. This shows that ${\mathrm{\Pi}}_{{C}_{n+1}}^{f}{x}_{0}$ is well defined for all $n\ge 0$.
By taking ${\theta}_{n}^{k}={T}_{{r}_{k,n}}^{{F}_{k}}{T}_{{r}_{k1,n}}^{{F}_{k1}}\cdots {T}_{{r}_{2,n}}^{{F}_{2}}{T}_{{r}_{1,n}}^{{F}_{1}}$, $k=1,2,\dots ,m$ and ${\theta}_{n}^{0}=I$ for all $n\ge 1$, we obtain ${u}_{n}={\theta}_{n}^{m}{y}_{n}$.
We next show that $F\subset {C}_{n}$, $\mathrm{\forall}n\ge 1$. From Lemma 1.12, one sees that ${T}_{{r}_{k,n}}^{{F}_{k}}$, $k=1,2,\dots ,m$, is relatively nonexpansive mapping. For $n=1$, we have $F\subset C={C}_{1}$. Now, assume that $F\subset {C}_{n}$ for some $n\ge 2$. Then for each ${x}^{\ast}\in F$, we obtain
So, ${x}^{\ast}\in {C}_{n}$. This implies that $F\subset {C}_{n}$, $\mathrm{\forall}n\ge 1$ and the sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ generated by (1.5) is well defined.
We now show that ${lim}_{n\to \mathrm{\infty}}G({x}_{n},J{x}_{0})$ exists. Since $f:E\to R$ is a convex and lower semicontinuous, applying Lemma 1.5, we see that there exist ${u}^{\ast}\in {E}^{\ast}$ and $\alpha \in R$ such that
It follows that
Since ${x}_{n}={\mathrm{\Pi}}_{{C}_{n}}^{f}{x}_{0}$, it follows from (2.2) that
for each ${x}^{\ast}\in F(T)$. This implies that ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is bounded and so is ${\{G({x}_{n},J{x}_{0})\}}_{n=0}^{\mathrm{\infty}}$. By the construction of ${C}_{n}$, we have ${C}_{m}\subset {C}_{n}$ and ${x}_{m}={\mathrm{\Pi}}_{{C}_{m}}^{f}{x}_{0}\in {C}_{n}$ for any positive integer $m\ge n$. It then follows from Lemma 1.7 that
It is obvious that
In particular,
and
and so ${\{G({x}_{n},J{x}_{0})\}}_{n=0}^{\mathrm{\infty}}$ is nondecreasing. It follows that the limit of ${\{G({x}_{n},J{x}_{0})\}}_{n=0}^{\mathrm{\infty}}$ exists.
By the fact that ${C}_{m}\subset {C}_{n}$ and ${x}_{m}={\mathrm{\Pi}}_{{C}_{m}}^{f}{x}_{0}\in {C}_{n}$ for any positive integer $m\ge n$, we obtain
Now, (2.3) implies that
Taking the limit as $m,n\to \mathrm{\infty}$ in (2.4), we obtain
It then follows from Lemma 1.9 that $\parallel {x}_{m}{x}_{n}\parallel \to 0$ as $m,n\to \mathrm{\infty}$. Hence, ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is a Cauchy sequence. Since E is a Banach space and C is closed and convex, there exists $p\in C$ such that ${x}_{n}\to p$ as $n\to \mathrm{\infty}$.
Now since $\varphi ({x}_{m},{x}_{n})\to 0$ as $m,n\to \mathrm{\infty}$ we have in particular that $\varphi ({x}_{n+1},{x}_{n})\to 0$ as $n\to \mathrm{\infty}$ and this further implies that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}{x}_{n}\parallel =0$. Since ${x}_{n+1}={\mathrm{\Pi}}_{{C}_{n=1}}^{f}{x}_{0}\in {C}_{n+1}$ we have
Then we obtain
Since E is uniformly convex and smooth, we have from Lemma 1.9
So,
Hence,
Since J is uniformly normtonorm continuous on bounded sets and ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{u}_{n}\parallel =0$, we obtain
Let $r={sup}_{n\ge 1}\{\parallel {x}_{n}\parallel ,\parallel {T}_{n}{x}_{n}\parallel \}$. Since E is uniformly smooth, we know that ${E}^{\ast}$ is uniformly convex. Then from Lemma 1.10, we have
It then follows that
But
From (2.5) and (2.6), we obtain
Using the condition ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0$, we have
By the properties of g, we have ${lim}_{n\to \mathrm{\infty}}\parallel J{x}_{n}J{T}_{n}{x}_{n}\parallel =0$. Since ${J}^{1}$ is also uniformly normtonorm continuous on bounded sets, we have
Since ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ are uniformly closed, and ${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a Cauchy sequence. Then $p\in F(T)={\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n})$.
Next, we show that $p\in {\bigcap}_{k=1}^{m}EP({F}_{k})$. From (2.1), we obtain
Since ${x}^{\ast}\in EP({F}_{m})=F({T}_{{r}_{m,n}}^{{F}_{m}})$ for all $n\ge 1$, it follows from (2.7) and Lemma 1.13 that
From (2.5) and (2.6), we obtain ${lim}_{n\to \mathrm{\infty}}\varphi ({\theta}_{n}^{m}{y}_{n},{\theta}_{n}^{m1}{y}_{n})={lim}_{n\to \mathrm{\infty}}\varphi ({u}_{n},{\theta}_{n}^{m1}{y}_{n})=0$. From Lemma 1.9, we have
Hence, we have from (2.8) that
Again, since ${x}^{\ast}\in EP({F}_{m1})=F({T}_{{r}_{m1,n}}^{{F}_{m1}})$ for all $n\ge 1$, it follows from (2.7) and Lemma 1.13 that
Again, from (2.5) and (2.6), we obtain ${lim}_{n\to \mathrm{\infty}}\varphi ({\theta}_{n}^{m1}{y}_{n},{\theta}_{n}^{m2}{y}_{n})=0$. From Lemma 1.9, we have
and hence,
In a similar way, we can verify that
From (2.8), (2.10), and (2.12), we can conclude that
Since ${x}_{n}\to p$, $n\to \mathrm{\infty}$, we obtain from (2.5) that ${u}_{n}\to p$, $n\to \mathrm{\infty}$. Again, from (2.8), (2.10), (2.12), and ${u}_{n}\to p$, $n\to \mathrm{\infty}$, we have that ${\theta}_{n}^{k}{y}_{n}\to p$, $n\to \mathrm{\infty}$ for each $k=1,2,\dots ,m$. Also, using (2.13), we obtain
Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$, $k=1,2,\dots ,m$,
By Lemma 1.12, we have for each $k=1,2,\dots ,m$
Furthermore, using (A2) we obtain
By (A4), (2.14), and ${\theta}_{n}^{k}{y}_{n}\to p$, we have for each $k=1,2,\dots ,m$
For fixed $y\in C$, let ${z}_{t}=ty+(1t)p$ for all $t\in (0,1]$. This implies that ${z}_{t}\in C$. This yields ${F}_{k}({z}_{t},p)\le 0$. It follows from (A1) and (A4) that
and hence
From condition (A3), we obtain
This implies that $p\in EP({F}_{k})$, $k=1,2,\dots ,m$. Thus, $p\in {\bigcap}_{k=1}^{m}EP({F}_{k})$. Hence, we have $p\in F={\bigcap}_{k=1}^{m}EP({F}_{k})\cap ({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))$.
Finally, we show that $p={\mathrm{\Pi}}_{F}^{f}{x}_{0}$. Since $F={\bigcap}_{k=1}^{m}EP({F}_{k})\cap ({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))$ is a closed and convex set, from Lemma 1.6, we know that ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$ is single valued and denote $w={\mathrm{\Pi}}_{F}^{f}{x}_{0}$. Since ${x}_{n}={\mathrm{\Pi}}_{{c}_{n}}^{f}{x}_{0}$ and $w\in F\subset {C}_{n}$, we have
We know that $G(\xi ,J\phi )$ is convex and lower semicontinuous with respect to ξ when φ is fixed. This implies that
From the definition of ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$ and $p\in F$, 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 $k=1,2,\dots ,m$, let ${F}_{k}$ be a bifunction from $C\times C$ satisfying (A1)(A4) and let ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ be a countable family of uniformly closed relatively quasinonexpansive mappings of C into itself such that $F:=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\cap ({\bigcap}_{k=1}^{m}EP({F}_{k}))\ne \mathrm{\varnothing}$. Suppose ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is iteratively generated by ${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,
where J is the duality mapping on E. Suppose ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a sequence in $(0,1)$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0$, and ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) satisfying ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). Then ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ converges strongly to ${\mathrm{\Pi}}_{F}{x}_{0}$.
Proof Take $f(x)=0$ for all $x\in E$ in Theorem 2.5, then $G(\xi ,Jx)=\varphi (\xi ,x)$ and ${\mathrm{\Pi}}_{C}^{f}{x}_{0}={\mathrm{\Pi}}_{C}{x}_{0}$. Then Corollary 2.6 holds. □
Take ${F}_{k}\equiv 0$ ($k=1,2,\dots ,m$), 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 ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ be a countable family of uniformly closed relatively quasinonexpansive mappings of C into itself such that $F=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\ne \mathrm{\varnothing}$. Let $f:E\to R$ be a convex and lower semicontinuous mapping with $C\subset int(D(f))$ and suppose ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is iteratively generated by ${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,
where J is the duality mapping on E. Suppose ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a sequence in $(0,1)$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0$, and ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) satisfying ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). Then ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ converges strongly to ${\mathrm{\Pi}}_{F}{x}_{0}$.
3 Applications
Let $\phi :C\to R$ be a realvalued function. The convex minimization problem is to find ${x}^{\ast}\in C$ such that
$\mathrm{\forall}y\in C$. The set of solutions of (3.1) is denoted by $CMP(\phi )$. For each $r>0$ and $x\in E$, define the mapping
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 $k=1,2,\dots ,m$, let ${\phi}_{k}$ be a bifunction from $C\times C$ satisfying (A1)(A4) and let ${\{{T}_{n}\}}_{n=1}^{\mathrm{\infty}}$ be a countable family of uniformly closed relatively quasinonexpansive mappings of C into itself such that $F:=({\bigcap}_{n=1}^{\mathrm{\infty}}F({T}_{n}))\cap ({\bigcap}_{k=1}^{m}CMP({\phi}_{k}))\ne \mathrm{\varnothing}$. Let $f:E\to R$ be a convex and lower semicontinuous mapping with $C\subset int(D(f))$ and suppose ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ is iteratively generated by ${x}_{0}\in C$, ${C}_{1}=C$, ${x}_{1}={\mathrm{\Pi}}_{{C}_{1}}^{f}{x}_{0}$,
where J is the duality mapping on E. Suppose ${\{{\alpha}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a sequence in $(0,1)$ such that ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0$ and ${\{{r}_{k,n}\}}_{n=1}^{\mathrm{\infty}}\subset (0,\mathrm{\infty})$ ($k=1,2,\dots ,m$) satisfying ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{r}_{k,n}>0$ ($k=1,2,\dots ,m$). Then ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ converges strongly to ${\mathrm{\Pi}}_{F}^{f}{x}_{0}$.
Proof Define ${F}_{k}(x,y)={\phi}_{k}(y){\phi}_{k}(x)$, $x,y\in C$ and $k=1,2,\dots ,m$. Then $F({T}_{{r}_{k}}^{{F}_{k}})=EP({F}_{k})=CMP({\phi}_{k})=F({T}_{{r}_{k}}^{{\phi}_{k}})$ for each $k=1,2,\dots ,m$, 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.
Butnariu D, Reich S, Zaslavski AJ: Asymptotic behaviour of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 2001, 7: 151–174.
 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/NFA120023869
 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.
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.
Chidume CE Lecture Notes in Mathematics 1965. In Geometric Properties of Banach Spaces and Nonlinear Iterations. Springer, Berlin; 2009. xviii+326 pp.
 6.
Takahashi W: Nonlinear Functional AnalysisFixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2000. (in Japanese)
 7.
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.
 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.
Wu KQ, Huang NJ: The generalized f projection operator with application. Bull. Aust. Math. Soc. 2006, 73: 307–317. 10.1017/S0004972700038892
 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.
Shehu Y: Strong convergence theorems for infinite family of relatively quasinonexpansive mappings and systems of equilibrium problems. Appl. Math. Comput. 2012, 218: 5146–5156. 10.1016/j.amc.2011.11.001
 12.
Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.
 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.
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.
Nilsrakoo W, Saejung S: Strong convergence to common fixed points of countable relatively quasinonexpansive mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 312454
 16.
Kamimura S, Takahashi W: Strong convergence of a proximaltype algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611X
 17.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16(2):1127–1138.
 18.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
 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
Additional information
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.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 relatively quasinonexpansive mapping
 equilibrium problems
 generalized fprojection operator
 hybrid algorithm
 uniformly closed mappings