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Strong convergence theorems for a countable family of totally quasiϕasymptotically nonexpansive nonself mappings in Banach spaces with applications
Journal of Inequalities and Applications volume 2012, Article number: 268 (2012)
Abstract
In this paper, we introduce a class of totally quasiϕasymptotically nonexpansive nonself mappings and study the strong convergence under a limit condition only in the framework of Banach spaces. Meanwhile, our results are applied to study the approximation problem of a solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:78647870, 2012).
MSC:47J05, 47H09, 49J25.
1 Introduction
Assume that X is a real Banach space with the dual {X}^{\ast}, D is a nonempty closed convex subset of X. We also denote by J the normalized duality mapping from X to {2}^{{X}^{\ast}} which is defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing.
Let D be a nonempty closed subset of a real Banach space X. A mapping T:D\to D is said to be nonexpansive if \parallel TxTy\parallel \le \parallel xy\parallel for all x,y\in D. An element p\in D is called a fixed point of T:D\to D if p=T(p). The set of fixed points of T is represented by F(T).
A Banach space X is said to be strictly convex if \parallel \frac{x+y}{2}\parallel \le 1 for all x,y\in X with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. A Banach space is said to be uniformly convex if {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0 for any two sequences \{{x}_{n}\},\{{y}_{n}\}\subset X with \parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1 and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =0.
The norm of a Banach space X is said to be Gâteaux differentiable if for each x,y\in S(x), the limit
exists, where S(x)=\{x:\parallel x\parallel =1,x\in X\}. In this case, X is said to be smooth. The norm of a Banach space X is said to be Fréchet differentiable if for each x\in S(x), the limit (1.1) is attained uniformly for y\in S(x) and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for x,y\in S(x). In this case, X is said to be uniformly smooth.
A subset D of X is said to be a retract of X if there exists a continuous mapping P:X\to D such that Px=x for all x\in X. It is well known that every nonempty closed convex subset of a uniformly convex Banach space X is a retract of X. A mapping P:X\to D is said to be a retraction if {P}^{2}=P. It follows that if a mapping P is a retraction, then Py=y for all y in the range of P. A mapping P:X\to D is said to be a nonexpansive retraction if it is nonexpansive and it is a retraction from X to D.
Next, we assume that X is a smooth, strictly convex and reflexive Banach space and D is a nonempty closed convex subset of X. In the sequel, we always use \varphi :X\times X\to {R}^{+} to denote the Lyapunov functional defined by
It is obvious from the definition of the function ϕ that
and
for all \lambda \in [0,1] and x,y,z\in X.
Following Alber [1], the generalized projection {\mathrm{\Pi}}_{D}:X\to D is defined by
Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.
In the sequel, we denote the strong convergence and weak convergence of the sequence \{{x}_{n}\} by {x}_{n}\to x and {x}_{n}\rightharpoonup x, respectively.
Lemma 1.1 (see [1])
Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty closed convex subset of X. Then the following conclusions hold:

(a)
\varphi (x,y)=0 if and only if x=y;

(b)
\varphi (x,{\mathrm{\Pi}}_{D}y)+\varphi ({\mathrm{\Pi}}_{D}y,y)\le \varphi (x,y), \mathrm{\forall}x,y\in D;

(c)
if x\in X and z\in D, then z={\mathrm{\Pi}}_{D}x if and only if \u3008zy,JxJz\u3009\ge 0, \mathrm{\forall}y\in D.
Remark 1.1 (see [2])
Let {\mathrm{\Pi}}_{D} be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty closed convex subset D of X. Then {\mathrm{\Pi}}_{D} is a closed and quasiϕnonexpansive from X onto D.
Remark 1.2 (see [2])
If H is a real Hilbert space, then \varphi (x,y)={\parallel xy\parallel}^{2}, and {\mathrm{\Pi}}_{D} is the metric projection of H onto D.
Definition 1.1 Let P:X\to D be a nonexpansive retraction.

(1)
A nonself mapping T:D\to X is said to be quasiϕnonexpansive if F(T)\ne \mathrm{\Phi}, and
\varphi (p,T{(PT)}^{n1}x)\le \varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),\mathrm{\forall}n\ge 1;(1.7) 
(2)
A nonself mapping T:D\to X is said to be quasiϕasymptotically nonexpansive if F(T)\ne \mathrm{\Phi}, and there exists a real sequence {k}_{n}\subset [1,+\mathrm{\infty}), {k}_{n}\to 1 (as n\to \mathrm{\infty}), such that
\varphi (p,T{(PT)}^{n1}x)\le {k}_{n}\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),\mathrm{\forall}n\ge 1;(1.8) 
(3)
A nonself mapping T:D\to X is said to be totally quasiϕasymptotically nonexpansive if F(T)\ne \mathrm{\Phi}, and there exist nonnegative real sequences \{{v}_{n}\}, \{{\mu}_{n}\} with {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that
\varphi (p,T{(PT)}^{n1}x)\le \varphi (p,x)+{v}_{n}\zeta [\varphi (p,x)]+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,\mathrm{\forall}n\ge 1,p\in F(T).(1.9)
Remark 1.3 From the definitions, it is obvious that a quasiϕnonexpansive nonself mapping is a quasiϕasymptotically nonexpansive nonself mapping, and a quasiϕasymptotically nonexpansive nonself mapping is a totally quasiϕasymptotically nonexpansive nonself mapping, but the converse is not true.
Next, we present an example of a quasiϕnonexpansive nonself mapping.
Example 1.1 (see [2])
Let H be a real Hilbert space, D be a nonempty closed and convex subset of H and f:D\times D\to R be a bifunction satisfying the conditions: (A1) f(x,x)=0, \mathrm{\forall}x\in D; (A2) f(x,y)+f(y,x)\le 0, \mathrm{\forall}x,y\in D; (A3) for each x,y,z\in D, {lim}_{t\to 0}f(tz+(1t)x,y)\le f(x,y); (A4) for each given x\in D, the function y\mapsto f(x,y) is convex and lower semicontinuous. The socalled equilibrium problem for f is to find an {x}^{\ast}\in D such that f({x}^{\ast},y)\ge 0, \mathrm{\forall}y\in D. The set of its solutions is denoted by EP(f).
Let r>0, x\in H and define a mapping {T}_{r}:D\to D\subset H as follows:
then (1) {T}_{r} is singlevalued, and so z={T}_{r}(x); (2) {T}_{r} is a relatively nonexpansive nonself mapping, therefore it is a closed quasiϕnonexpansive nonself mapping; (3) F({T}_{r})=EP(f) and F({T}_{r}) is a nonempty and closed convex subset of D; (4) {T}_{r}:D\to D is nonexpansive. Since F({T}_{r}) is nonempty, and so it is a quasiϕnonexpansive nonself mapping from D to H, where \varphi (x,y)={\parallel xy\parallel}^{2}, x,y\in H.
Now, we give an example of a totally quasiϕasymptotically nonexpansive nonself mapping.
Example 1.2 (see [2])
Let D be a unit ball in a real Hilbert space {l}^{2}, and let T:D\to {l}^{2} be a nonself mapping defined by
where \{{a}_{i}\} is a sequence in (0,1) such that {\prod}_{i=2}^{\mathrm{\infty}}{a}_{i}=\frac{1}{2}.
It is proved in Goebal and Kirk [3] that

(i)
\parallel TxTy\parallel \le 2\parallel xy\parallel, \mathrm{\forall}x,y\in D;

(ii)
\parallel {T}^{n}x{T}^{n}y\parallel \le 2{\prod}_{j=2}^{n}{a}_{j}, \mathrm{\forall}x,y\in D, n\ge 2.
Let \sqrt{{k}_{1}}=2, \sqrt{{k}_{n}}=2{\prod}_{j=2}^{n}{a}_{j}, n\ge 2, then {lim}_{n\to \mathrm{\infty}}{k}_{n}=1. Letting {\nu}_{n}={k}_{n}1 (n\ge 2), \zeta (t)=t (t\ge 0) and \{{\mu}_{n}\} be a nonnegative real sequence with {\mu}_{n}\to 0, then from (i) and (ii) we have
Since D is a unit ball in a real Hilbert space {l}^{2}, it follows from Remark 1.2 that \varphi (x,y)={\parallel xy\parallel}^{2}, \mathrm{\forall}x,y\in D. The above inequality can be written as
Again, since 0\in D and 0\in F(T), this implies that F(T)\ne \mathrm{\Phi}. From above inequality, we get that
where P is the nonexpansive retraction. This shows that the mapping T defined as above is a totally quasiϕasymptotically nonexpansive nonself mapping.
Lemma 1.2 (see [4])
Let X be a uniformly convex and smooth Banach space, and let \{{x}_{n}\} and \{{y}_{n}\} be two sequences of X such that \{{x}_{n}\} and \{{y}_{n}\} are bounded; if \varphi ({x}_{n},{y}_{n})\to 0, then \parallel {x}_{n}{y}_{n}\parallel \to 0.
Lemma 1.3 Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty closed convex subset of X. Let T:D\to X be a totally quasiϕasymptotically nonexpansive nonself mapping with {\mu}_{1}=0, then F(T) is a closed and convex subset of D.
Proof Let \{{x}_{n}\} be a sequence in F(T) such that {x}_{n}\to p. Since T is a totally quasiϕasymptotically nonexpansive nonself mapping, we have
for all n\in N. Therefore,
By Lemma 1.2, we obtain Tp=p. So, we have p\in F(T). This implies F(T) is closed.
Let p,q\in F(T) and t\in (0,1), and put w=tp+(1t)q. We prove that w\in F(T). Indeed, in view of the definition of ϕ, let \{{u}_{n}\} be a sequence generated by {u}_{1}=Tw, {u}_{2}=T(PT)w, {u}_{3}=T{(PT)}^{2}w,\dots ,{u}_{n}=T{(PT)}^{n1}w=TP{u}_{n1}, we have
Since
Substituting (1.10) into (1.11) and simplifying it, we have
Hence, we have {u}_{n}\to w. This implies that {u}_{n+1}\to w. Since TP is closed and {u}_{n+1}=T{(PT)}^{n}w=TP{u}_{n}, we have TPw=w. Since w\in C, and so Tw=w, i.e., w\in F(T). This implies F(T) is convex. This completes the proof of Lemma 1.3. □
Definition 1.2 (1) (see [5]) A countable family of nonself mappings \{{T}_{i}\}:D\to X is said to be uniformly quasiϕasymptotically nonexpansive if {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\Phi}, and there exist nonnegative real sequences {k}_{n}\subset [1,+\mathrm{\infty}), {k}_{n}\to 1, such that for each i\ge 1,

(2)
A countable family of nonself mappings \{{T}_{i}\}:D\to X is said to be uniformly totally quasiϕasymptotically nonexpansive if {\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i})\ne \mathrm{\Phi}, and there exist nonnegative real sequences \{{v}_{n}\}, \{{\mu}_{n}\} with {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that for each i\ge 1,
\varphi (p,{T}_{i}{(P{T}_{i})}^{n1}x)\le \varphi (p,x)+{v}_{n}\zeta [\varphi (p,x)]+{\mu}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,\mathrm{\forall}n\ge 1,p\in F(T).(1.14)
(3) (see [5]) A nonself mapping T:D\to X is said to be uniformly LLipschitz continuous if there exists a constant L>0 such that
Considering the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasiϕnonexpansive and quasiϕasymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [4–19]).
The purpose of this paper is to modify the Halpern and Manntype iteration algorithm for a family of totally quasiϕasymptotically nonexpansive nonself mappings to have the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. [5, 6, 20, 21], Su et al. [16], Kiziltunc et al. [10], Yildirim et al. [11], Yang et al. [22], Wang [18, 19], Pathak et al. [14], Thianwan [17], Qin et al. [15], Hao et al. [9], Guo et al. [7], Nilsrakoo et al. [13] and others.
2 Main results
Theorem 2.1 Let X be a real uniformly smooth and uniformly convex Banach space, D be a nonempty closed convex subset of X. Let \{{T}_{i}\}:D\to X be a family of uniformly totally quasiϕasymptotically nonexpansive nonself mappings with sequences \{{v}_{n}\}, \{{\mu}_{n}\}, with {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}), and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0 such that for each i\ge 1, \{{T}_{i}\}:D\to X is uniformly {L}_{i}Lipschitz continuous. Let \{{\alpha}_{n}\} be a sequence in [0,1] and \{{\beta}_{n}\} be a sequence in (0,1) satisfying the following conditions:

(i)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0;

(ii)
0<{lim}_{n\to \mathrm{\infty}}inf{\beta}_{n}\le {lim}_{n\to \mathrm{\infty}}sup{\beta}_{n}<1.
Let {x}_{n} be a sequence generated by
where {\xi}_{n}={v}_{n}{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}, \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), {\mathrm{\Pi}}_{{D}_{n+1}}is the generalized projection of X onto {D}_{n+1}. If ℱ is nonempty, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}.
Proof (I) First, we prove that ℱ and {D}_{n} are closed and convex subsets in D.
In fact, by Lemma 1.3 for each i\ge 1, F({T}_{i}) is closed and convex in D. Therefore, ℱ is a closed and convex subset in D. By the assumption that {D}_{1}=D is closed and convex, suppose that {D}_{n} is closed and convex for some n\ge 1. In view of the definition of ϕ, we have
This shows that {D}_{n+1} is closed and convex. The conclusions are proved.

(II)
Next, we prove that \mathcal{F}\subset {D}_{n} for all n\ge 1.
In fact, it is obvious that \mathcal{F}\subset {D}_{1}. Suppose that \mathcal{F}\subset {D}_{n}.
Let {w}_{n,i}={J}^{1}({\beta}_{n}J{x}_{n}+(1{\beta}_{n})J{T}_{i}{(P{T}_{i})}^{n1}{x}_{n}). Hence for any u\in \mathcal{F}\subset {D}_{n}, by (1.5), we have
and
Therefore, we have
where {\xi}_{n}={v}_{n}{sup}_{p\in \mathcal{F}}\zeta (\varphi (p,{x}_{n}))+{\mu}_{n}. This shows that u\in \mathcal{F}\subset {D}_{n+1} and so \mathcal{F}\subset {D}_{n}. The conclusion is proved.

(III)
Now, we prove that \{{x}_{n}\} converges strongly to some point {p}^{\ast}.
Since {x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1}, from Lemma 1.1(c), we have
Again since \mathcal{F}\subset {D}_{n}, we have
It follows from Lemma 1.1(b) that for each u\in \mathcal{F} and for each n\ge 1,
Therefore, \{\varphi ({x}_{n},{x}_{1})\} is bounded, and so is \{{x}_{n}\}. Since {x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1} and {x}_{n+1}={\mathrm{\Pi}}_{{D}_{n+1}}{x}_{1}\in {D}_{n+1}\subset {D}_{n}, we have \varphi ({x}_{n},{x}_{1})\le \varphi ({x}_{n+1},{x}_{1}). This implies that \{\varphi ({x}_{n},{x}_{1})\} is nondecreasing. Hence, {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1}) exists.
By the construction of \{{D}_{n}\}, for any m\ge n, we have {D}_{m}\subset {D}_{n} and {x}_{m}={\mathrm{\Pi}}_{{D}_{m}}{x}_{1}\in {D}_{n}. This shows that
It follows from Lemma 1.2 that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{m}{x}_{n}\parallel =0. Hence, \{{x}_{n}\} is a Cauchy sequence in D. Since D is complete, without loss of generality, we can assume that {lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast} (some point in D).
By the assumption, it is easy to see that

(IV)
Now, we prove that {p}^{\ast}\in \mathcal{F}.
Since {x}_{n+1}\in {D}_{n+1}, from (2.1) and (2.6), we have
Since {x}_{n}\to {p}^{\ast}, it follows from (2.7) and Lemma 1.2 that
Since \{{x}_{n}\} is bounded and \{{T}_{i}\} is a family of uniformly total quasiϕasymptotically nonexpansive nonself mappings, we have
This implies that \{{T}_{i}{(P{T}_{i})}^{n1}{x}_{n}\} is uniformly bounded.
Since
this implies that \{{w}_{n,i}\} is also uniformly bounded.
In view of {\alpha}_{n}\to 0, from (2.1), we have that
for each i\ge 1.
Since {J}^{1} is uniformly continuous on each bounded subset of {X}^{\ast}, it follows from (2.8) and (2.9) that
for each i\ge 1. Since J is uniformly continuous on each bounded subset of X, we have
By condition (ii), we have that
Since J is uniformly continuous, this shows that
for each i\ge 1. Again, by the assumption that \{{T}_{i}\}:D\to X is uniformly {L}_{i}Lipschitz continuous for each i\ge 1, thus we have
for each i\ge 1.
We get {lim}_{n\to \mathrm{\infty}}\parallel {T}_{i}{(P{T}_{i})}^{n}{x}_{n}{T}_{i}{(P{T}_{i})}^{n1}{x}_{n}\parallel =0. Since {lim}_{n\to \mathrm{\infty}}{T}_{i}{(P{T}_{i})}^{n1}{x}_{n}={P}^{\ast} and {lim}_{n\to \mathrm{\infty}}{x}_{n}={p}^{\ast}, we have {lim}_{n\to \mathrm{\infty}}{T}_{i}P{T}_{i}{(P{T}_{i})}^{n1}{x}_{n}={p}^{\ast}.
In view of the continuity of {T}_{i}P, it yields that {T}_{i}P{p}^{\ast}={p}^{\ast}. Since {p}^{\ast}\in C, it implies that {T}_{i}{p}^{\ast}={p}^{\ast}. By the arbitrariness of i\ge 1, we have {p}^{\ast}\in \mathcal{F}.

(V)
Finally, we prove that {p}^{\ast}={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1} and so {x}_{n}\to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}={p}^{\ast}.
Let w={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}. Since w\in \mathcal{F}\subset {D}_{n} and {x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1}, we have \varphi ({x}_{n},{x}_{1})\le \varphi (w,{x}_{1}). This implies that
which yields that {p}^{\ast}=w={\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}. Therefore, {x}_{n}\to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}. The proof of Theorem 3.1 is completed. □
By Remark 1.3, the following corollary is obtained.
Corollary 2.1 Let X, D, \{{\alpha}_{n}\}, \{{\beta}_{n}\} be the same as in Theorem 2.1. Let \{{T}_{i}\}:D\to X be a family of uniformly quasiϕasymptotically nonexpansive nonself mappings with the sequence {k}_{n}\subset [1,+\mathrm{\infty}), {k}_{n}\to 1, such that for each i\ge 1, \{{T}_{i}\}:D\to X is uniformly {L}_{i}Lipschitz continuous.
Let {x}_{n} be a sequence generated by
where {\xi}_{n}=({k}_{n}1){sup}_{p\in \mathcal{F}}\varphi (p,{x}_{n}), \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{i}), {\mathrm{\Pi}}_{{D}_{n+1}} is the generalized projection of X onto {D}_{n+1}. If ℱ is nonempty, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}.
3 Application
In this section we utilize Corollary 2.1 to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result.
Theorem 3.1 Let H be a real Hilbert space, D be a nonempty closed and convex subset of H. \{{\alpha}_{n}\}, ({\beta}_{n}) be the same as in Theorem 2.1. Let \{{f}_{i}\}:D\times D\to R be a countable family of bifunctions satisfying conditions (A1)(A4) as given in Example 1.1. Let \{{T}_{r,i}:D\to D\subset H\} be the family of mappings defined by (1.9), i.e.,
Let \{{x}_{n}\} be the sequence generated by
If \mathcal{F}={\bigcap}_{i=1}^{\mathrm{\infty}}F({T}_{r,i})\ne \mathrm{\Phi}, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}{x}_{1}, which is a common solution of the system of equilibrium problems for f.
Proof In Example 1.1, we have pointed out that {u}_{n,i}={T}_{r,i}({x}_{n}), F({T}_{r,i})=EP({f}_{i}) is nonempty and convex for all i\ge 1, {T}_{r,i} is a countable family of quasiϕnonexpansive nonself mappings. Since F({T}_{r,i}) is nonempty, so {T}_{r,i} is a countable family of quasiϕnonexpansive mappings and for all i\ge 1, {T}_{r,i} is a uniformly 1Lipschitzian mapping. Hence, (3.1) can be rewritten as follows:
Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □
References
Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartosator AG. Dekker, New York; 1996:15–50.
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63(1/4):123–145.
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.
Kamimura S, Takahashi W: Strong convergence of a proximaltype algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611X
Chang SS, Wang L, Tang YK, Wang B, Qin LJ: Strong convergence theorems for a countable family of quasi ϕ asymptotically nonexpansive nonself mappings. Appl. Math. Comput. 2012, 218: 7864–7870. 10.1016/j.amc.2012.02.002
Chang SS, Lee HWJ, Chan CK, Yang L: Approximation theorems for total quasi ϕ asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 2011, 218: 2921–2931. 10.1016/j.amc.2011.08.036
Guo W: Weak convergence theorems for asymptotically nonexpansive nonselfmappings. Appl. Math. Lett. 2011, 24: 2181–2185. 10.1016/j.aml.2011.06.022
Halpren B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S000299041967118640
Hao Y, Cho SY, Qin X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 218573
Kiziltunc H, Temir S: Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces. Comput. Math. Appl. 2011, 61(9):2480–2489. 10.1016/j.camwa.2011.02.030
Yildirim I, Özdemir M: A new iterative process for common fixed points of finite families of nonselfasymptotically nonexpansive mappings. Nonlinear Anal., Theory Methods Appl. 2009, 71(3–4):991–999. 10.1016/j.na.2008.11.017
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279: 372–379. 10.1016/S0022247X(02)004584
Nilsrakoo W, Sajung S: Strong convergence theorems by HalpernMann iterations for relatively nonexpansive mappings in Banach spaces. Appl. Math. Comput. 2011, 217(14):6577–6586. 10.1016/j.amc.2011.01.040
Pathak HK, Cho YJ, Kang SM: Strong and weak convergence theorems for nonselfasymptotically perturbed nonexpansive mappings. Nonlinear Anal., Theory Methods Appl. 2009, 70(5):1929–1938. 10.1016/j.na.2008.02.092
Qin XL, Cho YJ, Kang SM, Zhou HY: Convergence of a modified Halperntype iterative algorithm for quasi ϕ nonexpansive mappings. Appl. Math. Lett. 2009, 22: 1051–1055. 10.1016/j.aml.2009.01.015
Su YF, Xu HK, Zhang X: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal. 2010, 73: 3890–3906. 10.1016/j.na.2010.08.021
Thianwan S: Common fixed points of new iterations for two asymptotically nonexpansive nonselfmappings in a Banach space. J. Comput. Appl. Math. 2009, 224(2):688–695. 10.1016/j.cam.2008.05.051
Wang L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2006, 323(1):550–557. 10.1016/j.jmaa.2005.10.062
Wang L: Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings. Comput. Math. Appl. 2007, 53(7):1012–1019. 10.1016/j.camwa.2007.01.001
Chang SS, Chan CK, Lee HWJ: Modified block iterative algorithm for quasi ϕ asymptotically nonexpansive mappings and equilibrium problem in Banach spaces. Appl. Math. Comput. 2011, 217: 7520–7530. 10.1016/j.amc.2011.02.060
Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications. Nonlinear Anal. TMA 2010, 73: 2260–2270. 10.1016/j.na.2010.06.006
Yang L, Xie X: Weak and strong convergence theorems of three step iteration process with errors for nonselfasymptotically nonexpansive mappings. Math. Comput. Model. 2010, 52(5–6):772–780. 10.1016/j.mcm.2010.05.006
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Yi, L. Strong convergence theorems for a countable family of totally quasiϕasymptotically nonexpansive nonself mappings in Banach spaces with applications. J Inequal Appl 2012, 268 (2012). https://doi.org/10.1186/1029242X2012268
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DOI: https://doi.org/10.1186/1029242X2012268