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A necessary and sufficient condition for the strong convergence of nonexpansive mappings in Banach spaces
Fixed Point Theory and Applicationsvolume 2014, Article number: 106 (2014)
Abstract
In this paper, we establish a necessary and sufficient condition for the strong convergence of nonexpansive mappings in a uniformly convex and 2uniformly smooth Banach space. It is worth pointing out that we remove some quite restrictive conditions in the corresponding results. An appropriate example, such that all conditions of this result are satisfied and that other conditions are not satisfied, is provided.
MSC:47H05, 47H09, 47H10.
1 Introduction
Let X be a real Banach space with the dual space ${X}^{\ast}$. The value of $f\in {X}^{\ast}$ at $x\in X$ is denoted by $\u3008x,f\u3009$. The normal duality mapping J from X into a family of nonempty (by the HahnBanach theorem) weakstar compact subsets of ${X}^{\ast}$ is defined by
Let $U=\{x\in X:\parallel x\parallel =1\}$. A Banach space X is said to be uniformly convex if for each $\u03f5\in (0,2]$, there exists $\delta >0$ such that for any $x,y\in U$,
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space X is said to be smooth if the limit
exists for all $x,y\in U$. It is said to be uniformly smooth if limit (1.1) is attained uniformly for $x,y\in U$. It is well known that if X is smooth, then J is singlevalued and continuous from the norm topology of X to the weakstar topology of ${X}^{\ast}$, i.e., norm to weak^{∗} continuous. It is also well known that if X is uniformly smooth, then J is uniformly continuous on bounded subsets of X from the strong topology of X to the strong topology of ${X}^{\ast}$, i.e., uniformly normtonorm continuous on any bounded subset of X; see [1, 2] for more details.
Also, we define a function $\rho :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ called the modulus of smoothness of X as follows:
It is known that X is uniformly smooth if and only if ${lim}_{\tau \to 0}\rho (\tau )/\tau =0$. Let q be a fixed real number with $1<q\le 2$. Then a Banach space X is said to be quniformly smooth if there exists a constant $c>0$ such that $\rho (\tau )\le c{\tau}^{q}$ for all $\tau >0$. One should note that no Banach space is quniformly smooth for $q>2$; see [3] for more details. So, in this paper, we focus on a 2uniformly smooth Banach space. It is well known that Hilbert spaces and Lebesgue ${L}^{p}$ ($p\ge 2$) spaces are uniformly convex and 2uniformly smooth.
Recall that a mapping $T:X\to X$ is said to be nonexpansive if
A point $x\in X$ is a fixed point of T if $Tx=x$. Let $Fix(T)$ denote the set of fixed points of T; that is, $Fix(T)=\{x\in X:Tx=x\}$.
A mapping $\overline{f}:X\to X$ is called strongly pseudocontractive if there exists a constant $\rho \in (0,1)$ and $j(xy)\in J(xy)$ satisfying
A mapping $f:X\to X$ is a contraction if there exists a constant $\alpha \in (0,1)$ such that
Since $\u3008f(x)f(y),j(xy)\u3009\le \parallel f(x)f(y)\parallel \parallel xy\parallel \le \alpha {\parallel xy\parallel}^{2}$, we have that f is a strong pseudocontraction.
Let $\eta >0$, a mapping $\overline{F}$ of X into X is said to be ηstrongly accretive if there exists $j(xy)\in J(xy)$ such that
for all $x,y\in X$. A mapping $\overline{F}$ of X into X is said to be kLipschitzian if, for $k>0$,
for all $x,y\in X$. It is well known that if X is a Hilbert space, then an ηstrongly accretive operator coincides with an ηstrongly monotone operator.
Yamada [4] introduced the following hybrid iterative method for solving the variational inequality in a Hilbert space:
where F is a kLipschitzian and ηstrongly monotone operator with $k>0$, $\eta >0$ and $0<\mu <2\eta /{k}^{2}$. Let a sequence $\{{\lambda}_{n}\}$ of real numbers in $(0,1)$ satisfy the conditions below:
He proved that $\{{x}_{n}\}$ generated by (1.2) converges strongly to the unique solution of the variational inequality
An example of sequence $\{{\lambda}_{n}\}$ which satisfies conditions (A1)(A3) is given by ${\lambda}_{n}=1/{n}^{\sigma}$, where $0<\sigma <1$. We note that condition (A3) was first used by Lions [5]. It was observed that Lion’s conditions on the sequence $\{{\lambda}_{n}\}$ excluded the canonical choice ${\lambda}_{n}=1/n$. This was overcome in 2003 by Xu and Kim [6] in a Hilbert space. They proved that if $\{{\lambda}_{n}\}$ satisfies conditions (A1), (A2) and (A4)
then $\{{x}_{n}\}$ is strongly convergent to the unique solution ${u}^{\ast}$ of the variational inequality $\u3008F{u}^{\ast},v{u}^{\ast}\u3009\ge 0$, $\mathrm{\forall}v\in C$. It is easy to see that condition (A4) is strictly weaker than condition (A3), coupled with conditions (A1) and (A2). Moreover, (A4) includes the important and natural choice $\{1/n\}$ of $\{{\lambda}_{n}\}$.
In 2010, Tian [7] improved Yamada’s method (1.2) and established the following strong convergence theorems.
Theorem 1.1 ([[7], Theorem 3.1])
Let H be a Hilbert space. Let $T:H\to H$ be a nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing}$, and let $f:H\to H$ be a contraction mapping with $\alpha \in (0,1)$. Assume that $\{{x}_{t}\}$ is defined by
where F is a kLipschitzian and ηstrongly monotone operator on a Hilbert space H with $k>0$, $\eta >0$. Let $0<\mu <2\eta /{k}^{2}$, $0<\gamma <\mu (\eta \frac{\mu {k}^{2}}{2})/\alpha =\tau /\alpha $ and $0<t<1$. Then ${x}_{t}$ converges strongly as $t\to 0$ to a fixed point $\tilde{x}$ of T, which solves the variational inequality $\u3008(\mu F\gamma f)\tilde{x},\tilde{x}z\u3009\le 0$, $z\in Fix(T)$.
Theorem 1.2 ([[7], Theorem 3.1])
Let H be a Hilbert space. Let $T:H\to H$ be a nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing}$, let $f:H\to H$ be a contraction mapping with $\alpha \in (0,1)$, and let F be a kLipschitzian and ηstrongly monotone operator on H with $k>0$, $\eta >0$ and $0<\mu <2\eta /{k}^{2}$. For an arbitrary ${x}_{0}\in H$, let $\{{x}_{n}\}$ be generated by
where $0<\gamma <\mu (\eta \frac{\mu {k}^{2}}{2})/\alpha =\tau /\alpha $ and $\{{\alpha}_{n}\}\subset (0,1)$ satisfies
(C1) ${\alpha}_{n}\to 0$,
(C2) ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$,
(C3) either ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty}$ or ${lim}_{n\to \mathrm{\infty}}\frac{{\alpha}_{n+1}}{{\alpha}_{n}}=1$.
Then $\{{x}_{n}\}$ converges strongly to $\tilde{x}$ that is obtained in Theorem 1.1.
We remind the reader of the following facts: (i) The results are obtained when the underlying space is a Hilbert space in Yamada [4], Xu [6] and Tian [7]. (ii) In order to guarantee the strong convergence of the iterative sequence $\{{x}_{n}\}$, there is at least one parameter sequence converging to zero (i.e., ${\alpha}_{n}\to \mathrm{\infty}$ or ${\lambda}_{n}\to 0$) in Yamada [4], Xu [6] and Tian [7]. (iii) The parameter sequence satisfies the condition ${lim}_{n\to \mathrm{\infty}}{\lambda}_{n}/{\lambda}_{n+1}=1$ (or ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n+1}/{\alpha}_{n}=1$).
In this paper, we establish a necessary and sufficient condition for the strong convergence of $\{{x}_{n}\}$ generated by (3.7) (defined below) in a uniformly convex and 2uniformly smooth Banach space. In the meantime, we remove the control condition (C1) and replace condition (C3) with (C3′) (defined below) in the result of Tian [7]. It is worth pointing out that we use a new method to prove our main results. The results presented in this paper can be viewed as an improvement, supplement and extension of the results obtained in [4, 6, 7].
2 Preliminaries
For the sequence $\{{x}_{n}\}$ in X, we write ${x}_{n}\rightharpoonup x$ to indicate that the sequence $\{{x}_{n}\}$ converges weakly to x. ${x}_{n}\to x$ means that $\{{x}_{n}\}$ converges strongly to x. In order to prove our main results, we need the following lemmas.
Lemma 2.1 ([8])
Let q be a given real number with $1<q\le 2$, and let X be a quniformly smooth Banach space. Then
for all $x,y\in X$, where K is a quniformly smooth constant of X and ${J}_{q}$ is the generalized duality mapping from X into ${2}^{{X}^{\ast}}$ defined by
for all $x\in X$.
Lemma 2.2 ([2])
Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X, and let T be a nonexpansive mapping of C into itself. If $\{{x}_{n}\}$ is a sequence of C such that ${x}_{n}\rightharpoonup x$ and ${x}_{n}T{x}_{n}\to 0$, then x is a fixed point of T.
Let $\{{s}_{n}\}$ be a sequence of nonnegative real numbers satisfying
where $\{{\lambda}_{n}\}$, $\{{\delta}_{n}\}$ and $\{{\gamma}_{n}\}$ satisfy the following conditions: (i) $\{{\lambda}_{n}\}\subset [0,1]$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n}=\mathrm{\infty}$, (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0$ or ${\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n}{\delta}_{n}<\mathrm{\infty}$, (iii) ${\gamma}_{n}\ge 0$ ($n\ge 0$), ${\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}<\mathrm{\infty}$. Then ${lim}_{n\to \mathrm{\infty}}{s}_{n}=0$.
The following lemma is easy to prove.
Lemma 2.4 Let X be a Banach space, let $\overline{f}:X\to X$ be a strongly pseudocontractive operator with $0<\rho <1$, and let $\overline{F}:X\to X$ be a kLipschitzian and ηstrongly accretive operator with $k>0$, $\eta >0$. Then, for $\rho <\mu \eta $,
that is, $\mu \overline{F}\overline{f}$ is a strongly accretive operator with coefficient $\mu \eta \rho $.
Lemma 2.5 Let X be a real 2uniformly smooth Banach space. Let t be a number in $(0,1)$, and let $\mu >0$. Let $\overline{F}:X\to X$ be an operator such that, for some constant $0<\eta \le \sqrt{2}kK$, $\overline{F}$ is kLipschitzian and ηstrongly accretive. Then $S=(It\mu \overline{F}):X\to X$ is a contraction provided $\mu \le \eta /(2{k}^{2}{K}^{2})$, that is,
where $\tau =1\sqrt{12\mu \eta +2{\mu}^{2}{k}^{2}{K}^{2}}\in (0,1]$.
Proof Using Lemma 2.1, we have
It follows from $0<\eta \le \sqrt{2}kK$ and $0<\mu \le \eta /(2{k}^{2}{K}^{2})$ that
for all $x,y\in X$. Therefore, we have
where $\tau =1\sqrt{12\mu \eta +2{\mu}^{2}{k}^{2}{K}^{2}}\in (0,1]$. □
3 Main results
Throughout this section, let X be a uniformly convex and 2uniformly smooth Banach space. Let $T:X\to X$ be a nonexpansive mapping with $Fix(T)\ne \mathrm{\varnothing}$. Let $\overline{F}:X\to X$ be a kLipschitzian and ηstrongly accretive operator with $0<\eta \le \sqrt{2}kK$. Let $\mu \in (0,\eta /(2{k}^{2}{K}^{2})]$ and $\tau =1\sqrt{12\mu \eta +2{\mu}^{2}{k}^{2}{K}^{2}}$. Let $\overline{f}:X\to X$ be a Lipschitzian and strongly pseudocontractive operator with $0<\rho <\tau $. Let t be a number in $(0,1)$. Consider a mapping ${S}_{t}$ on X defined by
It is easy to see that the mapping ${S}_{t}$ is strongly pseudocontractive. Indeed, from Lemma 2.5, we have
for all $x,y\in X$. Since $\overline{f}$ is Lipschitzian and $It\mu \overline{F}$ is contractive, hence ${S}_{t}$ is continuous. So, by Deimling [11], we can obtain that ${S}_{t}$ has a unique fixed point which we denoted by ${x}_{t}$, that is,
Our first main result below shows that $\{{x}_{t}\}$ converges strongly as $t\to {0}^{+}$ to a fixed point of T which solves some variational inequality.
Theorem 3.1 $\{{x}_{t}\}$ generated by the implicit method (3.1) converges in norm as $t\to {0}^{+}$ to the unique solution ${x}^{\ast}\in Fix(T)$ of the variational inequality
Proof It is easy to see the uniqueness of a solution of variational inequality (3.2). By Lemma 2.4, $\mu \overline{F}\overline{f}$ is strongly accretive, so variational inequality (3.2) has only one solution. Below we use ${x}^{\ast}\in Fix(T)$ to denote the unique solution of (3.2).
Next, we prove that $\{{x}_{t}\}$ is bounded. Take $u\in Fix(T)$, from (3.1) and using Lemma 2.5, we have
It follows that
Therefore, $\{{x}_{t}\}$ is bounded, and so are the nets $\{\overline{f}({x}_{t})\}$ and $\{\overline{F}T{x}_{t}\}$.
On the other hand, from (3.1) we obtain
Next, we show that $\{{x}_{t}\}$ is relatively normcompact as $t\to {0}^{+}$. Assume that $\{{t}_{n}\}\in (0,1)$ such that ${t}_{n}\to {0}^{+}$ as $n\to \mathrm{\infty}$. Put ${x}_{n}:={x}_{{t}_{n}}$. It follows from (3.3) that
For a given $u\in Fix(T)$, by (3.1) and using Lemma 2.5, we have
that is,
In particular,
Since $\{{x}_{t}\}$ is bounded, without loss of generality, we may assume that $\{{x}_{n}\}$ converges weakly to a point $\tilde{x}$. By (3.4) and using Lemma 2.2, we have $\tilde{x}\in Fix(T)$. Then by (3.5), ${x}_{n}\to \tilde{x}$. This has proved the relative norm compactness of the net $\{{x}_{t}\}$ as $t\to {0}^{+}$.
We next show that $\tilde{x}$ solves variational inequality (3.2). Observe that
Since $IT$ is accretive (this is due to the nonexpansiveness of T), for any $u\in Fix(T)$, we can deduce immediately that
Therefore, for any $u\in Fix(T)$,
Now, replace t in (3.6) with ${t}_{n}$. Noting that $\overline{F}{x}_{{t}_{n}}\overline{F}T{x}_{{t}_{n}}\to \overline{F}\tilde{x}\overline{F}\tilde{x}=0$ for $\tilde{x}\in Fix(T)$ as $n\to \mathrm{\infty}$, we have
That is $\tilde{x}\in Fix(T)$ is a solution of (3.2), hence $\tilde{x}={x}^{\ast}$ by uniqueness. In summary, we have shown that each cluster point of $\{{x}_{t}\}$ (as $t\to {0}^{+}$) equals ${x}^{\ast}$. Therefore, ${x}_{t}\to {x}^{\ast}$ as $t\to {0}^{+}$. □
Remark 3.2 Compared with Theorem 3.1 of Tian [7], our Theorem 3.1 improves and extends Theorem 3.1 of Tian [7] in the following aspects:

(i)
The framework of a Hilbert space is extended to a uniformly convex and 2uniformly smooth Banach space.

(ii)
The ηstrongly monotone operator F is extended to the case of an ηstrongly accretive operator $\overline{F}$. The contraction $f:H\to H$ is extended to the case of a Lipschitzian and strongly pseudocontractive operator $\overline{f}:X\to X$.

(iii)
If we put $X=H$, $\overline{F}=F$ and $\overline{f}=\gamma f$, then our Theorem 3.1 reduces to Theorem 3.1 of Tian [7]. Thus, our Theorem 3.1 covers Theorem 3.1 of Tian [7] as a special case.
Next we consider the following iteration process: the initial guess ${x}_{0}$ is selected in X arbitrarily and the $(n+1)$th iterate ${x}_{n+1}$ is defined by
where $f:X\to X$ is a contractive mapping with $0<\gamma \alpha <\tau $, $\{{\alpha}_{n}\}$ is a sequence in $(0,1)$ satisfying conditions (C2) and
(C3′) ${\alpha}_{n+1}{\alpha}_{n}\le o({\alpha}_{n+1})+{\sigma}_{n}$ with ${\sigma}_{n}\ge 0$ and ${\sum}_{n=0}^{\mathrm{\infty}}{\sigma}_{n}<\mathrm{\infty}$.
Besides the basic condition (C2) on the sequence ${\alpha}_{n}$, we have the control condition (C3′). It can obviously be replaced by one of the following:
(C31) ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n+1}{\alpha}_{n}<\mathrm{\infty}$;
(C32) ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n+1}/{\alpha}_{n}=1$.
Indeed, (C31) implies (C3′) by choosing ${\sigma}_{n}={\alpha}_{n+1}{\alpha}_{n}$, and (C32) implies (C3′) by choosing ${\sigma}_{n}=0$. In this sense (C3′) is a weaker condition than the previous condition (C3).
Our second main result below shows that we have established a necessary and sufficient condition for the strong convergence of nonexpansive mappings in a uniformly convex and 2uniformly smooth Banach space.
Theorem 3.3 Let $\{{x}_{n}\}$ be generated by algorithm (3.7) with the sequence ${\alpha}_{n}$ of parameters satisfying conditions (C2) and (C3′). Then
where ${x}^{\ast}\in Fix(T)$ solves the variational inequality $\u3008(\mu \overline{F}\gamma f){x}^{\ast},j({x}^{\ast}u)\u3009\le 0$, $u\in Fix(T)$.
Proof On the one hand, suppose that ${\alpha}_{n}(\gamma f({x}_{n})\mu \overline{F}T{x}_{n})\to 0$ ($n\to \mathrm{\infty}$). We proceed with the following steps.
Step 1. We claim that $\{{x}_{n}\}$ is bounded. In fact, taking $u\in Fix(T)$, from (3.7) and using Lemma 2.5, we have
By induction, we have
Therefore, $\{{x}_{n}\}$ is bounded. We also obtain that $\{f({x}_{n})\}$ and $\{\overline{F}T{x}_{n}\}$ are bounded.
Step 2. We claim that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}{x}_{n}\parallel =0$. Observe that
where $M=max\{\gamma f({x}_{n1}),\mu \overline{F}T{x}_{n1}\}$. By Lemma 2.3, we have ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}{x}_{n}\parallel =0$.
Step 3. We claim that ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0$. Indeed, from Step 2, we have
Step 4. We claim that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\u3008(\gamma f\mu \overline{F}){x}^{\ast},j({x}_{n+1}{x}^{\ast})\u3009\le 0$, where ${x}^{\ast}={lim}_{t\to {0}^{+}}{x}_{t}$ and ${x}_{t}$ is defined by ${x}_{t}=t\gamma f({x}_{t})+(It\mu \overline{F})T{x}_{t}$. Since $\{{x}_{n+1}\}$ is bounded, there exists a subsequence $\{{x}_{{\{n+1\}}_{k}}\}$ of $\{{x}_{n+1}\}$ which converges weakly to ω. From Step 3, we obtain $T{x}_{{\{n+1\}}_{k}}\rightharpoonup \omega $. From Lemma 2.2, we have $\omega \in Fix(T)$. Since f is a contractive mapping, we have that γf is a Lipschitzian and strongly pseudocontractive operator with $\gamma \alpha \in (0,\tau )$. Hence, using Theorem 3.1, we have ${x}^{\ast}\in Fix(T)$ and
Step 5. We claim that $\{{x}_{n}\}$ converges strongly to ${x}^{\ast}\in Fix(T)$. From (3.7) and using Lemma 2.5, we have
It follows that
where ${\mu}_{n}={\alpha}_{n}(\tau \gamma \alpha )$ and ${\delta}_{n}=\frac{2}{\tau \gamma \alpha}\u3008\gamma f({x}^{\ast})\mu \overline{F}{x}^{\ast},j({x}_{n+1}{x}^{\ast})\u3009$. It is easy to see that ${\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}=\mathrm{\infty}$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0$. Hence, by Lemma 2.3, the sequence $\{{x}_{n}\}$ converges strongly to ${x}^{\ast}\in Fix(T)$. From ${x}^{\ast}={lim}_{t\to 0}{x}_{t}$ and Theorem 3.1, we have that ${x}^{\ast}$ is the unique solution of the variational inequality $\u3008(\mu \overline{F}\gamma f){x}^{\ast},j({x}^{\ast}u)\u3009\le 0$, $u\in Fix(T)$.
On the other hand, suppose that ${x}_{n}\to {x}^{\ast}\in Fix(T)$ as $n\to \mathrm{\infty}$, where ${x}^{\ast}$ is the unique solution of the variational inequality $\u3008(\mu \overline{F}\gamma f){x}^{\ast},j({x}^{\ast}u)\u3009\le 0$, $u\in Fix(T)$. Observe that
This completes the proof. □
Remark 3.4 Compared with Theorem 3.2 of Tian [7], our Theorem 3.3 improves and extends Theorem 3.1 of Tian [7] in the following aspects:

(i)
The framework of a Hilbert space is extended to a uniformly convex and 2uniformly smooth Banach space.

(ii)
The ηstrongly monotone operator F is extended to the case of an ηstrongly accretive operator $\overline{F}$.

(iii)
We establish a necessary and sufficient condition for the strong convergence of nonexpansive mappings. It follows from (C1) that ${\alpha}_{n}(\gamma f({x}_{n})\mu \overline{F}T{x}_{n})\to 0$ ($n\to \mathrm{\infty}$). Hence, we can obtain Theorem 3.2 of Tian [7] immediately. Thus, our Theorem 3.3 covers Theorem 3.1 of Tian [7] as a special case.
The following example shows that all the conditions of Theorem 3.3 are satisfied. However, condition (C1) is not satisfied.
Example 3.5 Let $X=R$ be the set of real numbers. Define the mappings $T:X\to X$, $f:X\to X$ and $\overline{F}:X\to X$ as follows:
It is easy to see that $K=\frac{\sqrt{2}}{2}$, $\alpha =\frac{1}{2}$ and $Fix(T)=\{0\}$. By $\overline{F}x=x$, we have $\eta =k=1$ and hence $0<\mu \le \eta /(2{k}^{2}{K}^{2})=1$. Also, put $\mu =1$. It is easy to see that $\tau =1\sqrt{12\mu \eta +2{\mu}^{2}{k}^{2}{K}^{2}}=1$. From $0<\gamma \alpha <\tau $, we have $\gamma \in (0,2)$. Without loss of generality, we put $\gamma =1$. Given sequences $\{{\alpha}_{n}\}$ and $\{{\sigma}_{n}\}$, ${\alpha}_{n}=1/2$, $o({\alpha}_{n+1})=1/{n}^{2}$ and ${\sigma}_{n}=0$ for all $n\ge 0$. For an arbitrary ${x}_{0}\in X$, let $\{{x}_{n}\}$ be defined as
that is,
Observe that for all $n\ge 0$,
Hence we have $\parallel {x}_{n+1}0\parallel ={(\frac{1}{4})}^{n+1}\parallel {x}_{0}0\parallel $ for all $n\ge 0$. This implies that $\{{x}_{n}\}$ converges strongly to $0\in Fix(T)$.
Observe that $\u3008(\mu \overline{F}\gamma f)0,j(0u)\u3009\le 0$, $u\in Fix(T)$, that is, 0 is the solution of the variational inequality $\u3008(\mu \overline{F}\gamma f){x}^{\ast},j({x}^{\ast}u)\u3009\le 0$, $u\in Fix(T)$.
Finally, we have
Hence there is no doubt that all the conditions of Theorem 3.3 are satisfied. Since ${\alpha}_{n}=1/2$, condition (C1): ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$ of Tian [7] is not satisfied.
References
 1.
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.
 2.
Goebel K, Kirk WA: Topics on Metric FixedPoint Theory. Cambridge University Press, Cambridge; 1990.
 3.
Takahashi Y, Hashimoto K, Kato M: On sharp uniform convexity, smoothness, and strong type, cotype inequalities. J. Nonlinear Convex Anal. 2002, 3: 267–281.
 4.
Yamada I: The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mapping. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Application. Edited by: Butnariu D, Censor Y, Reich S. Elsevier, New York; 2001:473–504.
 5.
Lions PL: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sér. AB 1977, 284: 1357–1359.
 6.
Xu HK, Kim TH: Convergence of hybrid steepestdescent methods for variational inequalities. J. Optim. Theory Appl. 2003, 119: 185–201.
 7.
Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 689–694. 10.1016/j.na.2010.03.058
 8.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362546X(91)90200K
 9.
Liu LS: Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289
 10.
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
 11.
Deimling K: Zero of accretive operators. Manuscr. Math. 1974, 13: 365–374. 10.1007/BF01171148
Acknowledgements
The authors would like to thank the referees for giving useful suggestions and comments for the improvement of this paper. This study was supported by the Natural Science Foundation of Jiangsu Province under Grant (13KJB110028), and the National Science Foundation of China (11271277).
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Keywords
 strong convergence
 fixed points
 nonexpansive mapping
 ηstrongly accretive
 2uniformly smooth Banach spaces