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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. □
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The authors are very grateful to both reviewers for carefully reading this paper and for their comments.
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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
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Keywords
 multivalued mapping
 quasiϕasymptotically nonexpansive
 total quasiϕasymptotically nonexpansive
 Halpern iterative sequence