 Research
 Open Access
 Published:
Strong convergence theorems for modifying Halpern iterations for a totally quasiϕasymptotically nonexpansive multivalued mapping in reflexive Banach spaces
Journal of Inequalities and Applications volume 2013, Article number: 126 (2013)
Abstract
In this paper, we discuss an iterative sequence for a totally quasiϕasymptotically nonexpansive multivalued mapping for modifying Halpern’s iterations and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:64896497, 2012).
MSC:47J05, 47H09, 49J25.
1 Introduction
Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. 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. Let N(D) and CB(D) denote the family of nonempty subsets and nonempty bounded closed subsets of D, respectively. The Hausdorff metric on CB(D) is defined by
for {A}_{1},{A}_{2}\in CB(D), where d(x,{A}_{2})=inf\{\parallel xy\parallel ,y\in {A}_{2}\}. The multivalued mapping T:D\to CB(D) is called nonexpansive if H(Tx,Ty)\le \parallel xy\parallel for all x,y\in D. An element p\in D is called a fixed point of T:D\to CB(D) if p\in T(p). The set of fixed points of T is represented by F(T).
In the sequel, denote S(X)=\{x\in X:\parallel x\parallel =1\}. A Banach space X is said to be strictly convex if \parallel \frac{x+y}{2}\parallel \le 1 for all x,y\in S(X) 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 S(X) and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =0. The norm of the Banach space X is said to be Gâteaux differentiable if for each x,y\in S(X), the limit
exists. In this case, X is said to be smooth. The norm of the 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.
Let X be a real Banach space with dual {X}^{\ast}. We 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.
Remark 1.1 The following basic properties for the Banach space X and for the normalized duality mapping J can be found in Cioranescu [1].

(1)
X ({X}^{\ast}, resp.) is uniformly convex if and only if {X}^{\ast} (X, resp.) is uniformly smooth.

(2)
If X is smooth, then J is singlevalued and normtoweak^{∗} continuous.

(3)
If X is reflexive, then J is onto.

(4)
If X is strictly convex, then Jx\cap Jy\ne \mathrm{\Phi} for all x,y\in X.

(5)
If X has a Fréchet differentiable norm, then J is normtonorm continuous.

(6)
If X is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of X.

(7)
Each uniformly convex Banach space X has the KadecKlee property, i.e., for any sequence \{{x}_{n}\}\subset X, if {x}_{n}\rightharpoonup x\in X and \parallel {x}_{n}\parallel \to \parallel x\parallel, then {x}_{n}\to x\in X.
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 bifunction defined by
It is obvious from the definition of the function ϕ that
and
for all \alpha \in [0,1] and x,y,z\in X.
Following Alber [2], 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.
Remark 1.2 (see [3])
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.
In 1953, Mann [4] introduced the following iterative sequence \{{x}_{n}\}:
where the initial guess {x}_{1}\in D is arbitrary and \{{\alpha}_{n}\} is a real sequence in [0,1]. It is known that under appropriate settings the sequence \{{x}_{n}\} converges weakly to a fixed point of T. However, even in a Hilbert space, the Mann iteration may fail to converge strongly [5]. Some attempts to construct an iteration method guaranteeing the strong convergence have been made. For example, Halpern [6] proposed the following socalled Halpern iteration:
where u,{x}_{1}\in D are arbitrarily given and \{{\alpha}_{n}\} is a real sequence in [0,1]. Another approach was proposed by Nakajo and Takahashi [7]. They generated a sequence as follows:
where \{{\alpha}_{n}\} is a real sequence in [0,1] and {P}_{K} denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. It should be noted here that the iteration above works only in the Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings and quasiϕnonexpansive mappings have been introduced by Aoyama et al. [8], Chang et al. [9, 10], Chidume et al. [11], Matsushita et al. [12–14], Qin et al. [15], Song et al. [16], Wang et al. [17] and others.
Inspired by the work of Matsushita and Takahashi, in this paper, we introduce modifying HalpernMann iterations sequence for finding a fixed point of a multivalued mapping T:D\to CB(D) and prove some strong convergence theorems. The results presented in the paper improve and extend the corresponding results in [9].
2 Preliminaries
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 2.1 (see [2])
Let X be a smooth, strictly convex and reflexive Banach space, and let 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.
Lemma 2.2 (see [9])
Let X be a real uniformly smooth and strictly convex Banach space with the KadecKlee property, and let D be a nonempty closed convex subset of X. Let \{{x}_{n}\} and \{{y}_{n}\} be two sequences in D such that {x}_{n}\to p and \varphi ({x}_{n},{y}_{n})\to 0, where ϕ is the function defined by (1.2), then {y}_{n}\to p.
Definition 2.1 A point p\in D is said to be an asymptotic fixed point of a multivalued mapping T:D\to CB(D) if there exists a sequence \{{x}_{n}\}\subset D such that {x}_{n}\rightharpoonup x\in X and d({x}_{n},T({x}_{n}))\to 0. Denote the set of all asymptotic fixed points of T by \stackrel{\u02c6}{F}(T).
Definition 2.2

(1)
A multivalued mapping T:D\to CB(D) is said to be relatively nonexpansive if F(T)\ne \mathrm{\Phi}, \stackrel{\u02c6}{F}(T)=F(T) and \varphi (p,z)\le \varphi (p,x), \mathrm{\forall}x\in D, p\in F(T), z\in T(x).

(2)
A multivalued mapping T:D\to CB(D) is said to be closed if for any sequence \{{x}_{n}\}\subset D with {x}_{n}\to x\in X and d(y,T({x}_{n}))\to 0, then d(y,T(x))=0.
Remark 2.1 If H is a real Hilbert space, then \varphi (x,y)={\parallel xy\parallel}^{2} and {\mathrm{\Pi}}_{D} is the metric projection {P}_{D} of H onto D.
Next, we present an example of a relatively nonexpansive multivalued mapping.
Example 2.1 (see [18])
Let X be a smooth, strictly convex and reflexive Banach space, let D be a nonempty closed and convex subset of X, and let 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 D and define a multivalued mapping {T}_{r}:D\to N(D) as follows:
then (1) {T}_{r} is singlevalued, and so \{z\}={T}_{r}(x); (2) {T}_{r} is a relatively nonexpansive mapping, therefore, it is a closed quasiϕnonexpansive mapping; (3) F({T}_{r})=EP(f).
Definition 2.3

(1)
A multivalued mapping T:D\to CB(D) is said to be quasiϕnonexpansive if F(T)\ne \mathrm{\Phi} and \varphi (p,z)\le \varphi (p,x), \mathrm{\forall}x\in D, p\in F(T), z\in Tx.

(2)
A multivalued mapping T:D\to CB(D) 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, such that
\varphi (p,{z}_{n})\le {k}_{n}\varphi (p,x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in D,p\in F(T),{z}_{n}\in {T}^{n}x.(2.2) 
(3)
A multivalued mapping T:D\to CB(D) 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
(2.3)
Remark 2.2 From the definitions, it is obvious that a relatively nonexpansive multivalued mapping is a quasiϕnonexpansive multivalued mapping, and a quasiϕnonexpansive multivalued mapping is a quasiϕasymptotically nonexpansive multivalued mapping, and a quasiϕasymptotically nonexpansive multivalued mapping is a total quasiϕasymptotically nonexpansive multivalued mapping, but the converse is not true.
Lemma 2.3 Let X and D be as in Lemma 2.2. Let T:D\to CB(D) be a closed and totally quasiϕasymptotically nonexpansive multivalued mapping with nonnegative real sequences \{{v}_{n}\}, \{{\mu}_{n}\} and a strictly increasing continuous function \zeta :{R}^{+}\to {R}^{+} with \zeta (0)=0. If {v}_{n},{\mu}_{n}\to 0 (as n\to \mathrm{\infty}) and {\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 {x}^{\ast}. Since T is a totally quasiϕasymptotically nonexpansive multivalued mapping, we have
for all z\in T{x}^{\ast} and for all n\in N. Therefore,
By Lemma 2.1(a), we obtain z={x}^{\ast}. Hence, T{x}^{\ast}=\{{x}^{\ast}\}. So, we have {x}^{\ast}\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. Next we prove that w\in F(T). Indeed, in view of the definition of ϕ, letting {z}_{n}\in {T}^{n}w, we have
Since
Substituting (2.4) into (2.5) and simplifying it, we have
By Lemma 2.2, we have {z}_{n}\to w. This implies that {z}_{n+1}\phantom{\rule{0.25em}{0ex}}(\in T{T}^{n}w)\to w. Since T is closed, we have Tw=\{w\}, i.e., w\in F(T). This completes the proof of Lemma 2.3. □
Definition 2.4 A mapping T:D\to CB(D) is said to be uniformly LLipschitz continuous if there exists a constant L>0 such that \parallel {x}_{n}{y}_{n}\parallel \le L\parallel xy\parallel, where x,y\in D, {x}_{n}\in {T}^{n}x, {y}_{n}\in {T}^{n}y.
3 Main results
Theorem 3.1 Let X be a real uniformly smooth and strictly convex Banach space with the KadecKlee property, let D be a nonempty closed convex subset of X, and let T:D\to CB(D) be a closed and uniformly LLipschitz continuous totally quasiϕasymptotically nonexpansive multivalued mapping with nonnegative real sequences \{{v}_{n}\}, \{{\mu}_{n}\}, {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 satisfying condition (2.3). Let \{{\alpha}_{n}\} be a sequence in [0,1] such that {\alpha}_{n}\to 0. If \{{x}_{n}\} is the sequence generated by
where {\xi}_{n}={v}_{n}{sup}_{p\in F(T)}\zeta [\varphi (p,{x}_{n})]+{\mu}_{n}, F(T) is the fixed point set of T, and {\mathrm{\Pi}}_{{D}_{n+1}} is the generalized projection of X onto {D}_{n+1}. If F(T) is nonempty and {\mu}_{1}=0, then {lim}_{n\to \mathrm{\infty}}{x}_{n}={\mathrm{\Pi}}_{F(T)}{x}_{1}.
Proof (I) First, we prove that {D}_{n} is a closed and convex subset in D.
By the assumption, {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 F(T)\subset {D}_{n} for all n\ge 1.
In fact, it is obvious that F(T)\subset {D}_{1}. Suppose that F(T)\subset {D}_{n}. Hence, for any u\in F(T)\subset {D}_{n}, by (1.5), we have
This shows that u\in F(T)\subset {D}_{n+1}, and so F(T)\subset {D}_{n}.

(III)
Now we prove that \{{x}_{n}\} converges strongly to some point {p}^{\ast}.
In fact, since {x}_{n}={\mathrm{\Pi}}_{{D}_{n}}{x}_{1}, from Lemma 2.1(c), we have
Again since F(T)\subset {D}_{n}, we have
It follows from Lemma 2.1(b) that for each u\in F(T) 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. Since X is reflexive, there exists a subsequence \{{x}_{{n}_{i}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{i}}\rightharpoonup {p}^{\ast} (some point in D={D}_{1}). Since {D}_{n} is closed and convex and {D}_{n+1}\subset {D}_{n}. This implies that {D}_{n} is weakly closed and {p}^{\ast}\in {D}_{n} for each n\ge 1. In view of {x}_{{n}_{i}}={\mathrm{\Pi}}_{{D}_{{n}_{i}}}{x}_{1}, we have
Since the norm \parallel \cdot \parallel is weakly lower semicontinuous, we have
and so
This shows that {lim}_{{n}_{i}\to \mathrm{\infty}}\varphi ({x}_{{n}_{i}},{x}_{1})=\varphi ({p}^{\ast},{x}_{1}), and we have \parallel {x}_{{n}_{i}}\parallel \to \parallel {p}^{\ast}\parallel. Since {x}_{{n}_{i}}\rightharpoonup {p}^{\ast}, by virtue of the KadecKlee property of X, we obtain that {x}_{{n}_{i}}\to {p}^{\ast}. Since \{\varphi ({x}_{n},{x}_{1})\} is convergent, this together with {lim}_{{n}_{i}\to \mathrm{\infty}}\varphi ({x}_{{n}_{i}},{x}_{1})=\varphi ({p}^{\ast},{x}_{1}) shows that {lim}_{{n}_{i}\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})=\varphi ({p}^{\ast},{x}_{1}). If there exists some subsequence \{{x}_{{n}_{j}}\}\subset \{{x}_{n}\} such that {x}_{{n}_{j}}\to q, then from Lemma 2.1 we have
i.e., {p}^{\ast}=q, and hence
By the way, from (3.4), it is easy to see that

(IV)
Now we prove that {p}^{\ast}\in F(T).
In fact, since {x}_{n+1}\in {D}_{n+1}, from (3.1), (3.4) and (3.5), we have
Since {x}_{n}\to {p}^{\ast}, it follows from (3.6) and Lemma 2.2 that
Since \{{x}_{n}\} is bounded and T is a totally quasiϕasymptotically nonexpansive multivalued mapping, {T}^{n}{x}_{n} is bounded. In view of {\alpha}_{n}\to 0, from (3.1), we have
Since J{y}_{n}\to J{p}^{\ast}, this implies J{z}_{n}\to J{p}^{\ast}. From Remark 1.1, it yields that
Again, since
this together with (3.9) and the KadecKleeproperty of X shows that
On the other hand, by the assumption that T is LLipschitz continuous, we have
From (3.11) and {x}_{n}\to {p}^{\ast}, we have that d(T{z}_{n},{z}_{n})\to 0. In view of the closedness of T, it yields that T({p}^{\ast})=\{{p}^{\ast}\}, which implies that {p}^{\ast}\in F(T).

(V)
Finally, we prove that {p}^{\ast}={\mathrm{\Pi}}_{F(T)}{x}_{1} and so {x}_{n}\to {\mathrm{\Pi}}_{F(T)}{x}_{1}.
Let w={\mathrm{\Pi}}_{F(T)}{x}_{1}. Since w\in F(T)\subset {D}_{n}, we have \varphi ({p}^{\ast},{x}_{1})\le \varphi (w,{x}_{1}). This implies that
which yields that {p}^{\ast}=w={\mathrm{\Pi}}_{F(T)}{x}_{1}. Therefore, {x}_{n}\to {\mathrm{\Pi}}_{F(T)}{x}_{1}. The proof of Theorem 3.1 is completed. □
By Remark 2.2, the following corollaries are obtained.
Corollary 3.1 Let X and D be as in Theorem 3.1, and let T:D\to CB(D) be a closed and uniformly LLipschitz continuous relatively nonexpansive multivalued mapping. Let \{{\alpha}_{n}\} in (0,1) with {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0. Let \{{x}_{n}\} be the sequence generated by
where F(T) is the set of fixed points of T, and {\mathrm{\Pi}}_{{D}_{n+1}} is the generalized projection of X onto {D}_{n+1}, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}.
Corollary 3.2 Let X and D be as in Theorem 3.1, and T:D\to CB(D) be a closed and uniformly LLipschitz continuous quasiϕnonexpansive multivalued mapping. Let \{{\alpha}_{n}\} be a sequence of real numbers such that {\alpha}_{n}\in (0,1) for all n\in N and satisfy {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0. Let \{{x}_{n}\} be the sequence generated by (3.14). Then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}.
Corollary 3.3 Let X be a real uniformly smooth and strictly convex Banach space with the KadecKlee property, let D be a nonempty closed convex subset of X, and let T:D\to CB(D) be a closed and uniformly LLipschitz continuous quasiϕasymptotically nonexpansive multivalued mapping with nonnegative real sequences \{{k}_{n}\}\subset [1,+\mathrm{\infty}) and {k}_{n}\to 1 satisfying condition (2.2). Let \{{\alpha}_{n}\} be a sequence in (0,1) and satisfy {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0. If \{{x}_{n}\} is the sequence generated by
where {\xi}_{n}=({k}_{n}1){sup}_{p\in F(T)}\varphi (p,{x}_{n}), F(T) is the fixed point set of T, and {\mathrm{\Pi}}_{{D}_{n+1}} is the generalized projection of X onto {D}_{n+1}, if F(T) is nonempty, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{1}.
4 Application
We utilize Corollary 3.2 to study a modified Halpern iterative algorithm for a system of equilibrium problems.
Theorem 4.1 Let D, X and \{{\alpha}_{n}\} be the same as in Theorem 3.1. Let f:D\times D\to R be a bifunction satisfying conditions (A1)(A4) as given in Example 2.1. Let {T}_{r}:X\to D be a mapping defined by (2.1), i.e.,
Let \{{x}_{n}\} be the sequence generated by
If F({T}_{r})\ne \mathrm{\Phi}, then \{{x}_{n}\} converges strongly to {\prod}_{F(T)}{x}_{1}, which is a common solution of the system of equilibrium problems for f.
Proof In Example 2.1, we have pointed out that {u}_{n}={T}_{r}({x}_{n}), F({T}_{r})=EP(f) and {T}_{r} is a closed quasiϕnonexpansive mapping. Hence (4.1) can be rewritten as follows:
Therefore the conclusion of Theorem 4.1 can be obtained from Corollary 3.2. □
References
Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.
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.
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
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S00029939195300548463
Genel A, Lindenstrauss J: An example concerning fixed points. Isr. J. Math. 1975, 22: 81–86. 10.1007/BF02757276
Halpren B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S000299041967118640
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
Aoyama K, Kimura Y: Strong convergence theorems for strongly nonexpansive sequences. Appl. Math. Comput. 2011, 217: 7537–7545. 10.1016/j.amc.2011.01.092
Chang SS, Lee HWJ, Chan CK, Zhang WB: A modified Halperntype iteration algorithm for totally quasi ϕ asymptotically nonexpansive mappings with applications. Appl. Math. Comput. 2012, 218: 6489–6497. 10.1016/j.amc.2011.12.019
Chang SS, Yang L, Liu JA: Strong convergence theorem for nonexpansive semigroups in Banach space. Appl. Math. Mech. 2007, 28: 1287–1297. 10.1007/s104830071002x
Chidume CE, Ofoedu EU: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2007, 333: 128–141. 10.1016/j.jmaa.2006.09.023
Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in a Banach space. Fixed Point Theory Appl. 2004, 2004: 37–47.
Matsushita S, Takahashi W: An iterative algorithm for relatively nonexpansive mappings by hybrid method and applications. Proceedings of the Third International Conference on Nonlinear Analysis and Convex Analysis 2004, 305–313.
Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007
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
Song Y: New strong convergence theorems for nonexpansive nonselfmappings without boundary conditions. Comput. Math. Appl. 2008, 56: 1473–1478. 10.1016/j.camwa.2008.03.004
Wang ZM, Su YF, Wang DX, Dong YC: A modified Halperntype iteration algorithm for a family of hemirelative nonexpansive mappings and systems of equilibrium problems in Banach spaces. J. Comput. Appl. Math. 2011, 235: 2364–2371. 10.1016/j.cam.2010.10.036
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63(1/4):123–145.
Acknowledgements
The authors are very grateful to both reviewers for carefully reading this paper and for their comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Liu, H.B., Li, Y. Strong convergence theorems for modifying Halpern iterations for a totally quasiϕasymptotically nonexpansive multivalued mapping in reflexive Banach spaces. J Inequal Appl 2013, 126 (2013). https://doi.org/10.1186/1029242X2013126
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2013126
Keywords
 multivalued mapping
 quasiϕasymptotically nonexpansive
 total quasiϕasymptotically nonexpansive
 Halpern iterative sequence