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A necessary and sufficient condition for the strong convergence of nonexpansive mappings in Banach spaces
- Shuang Wang^{1, 2} and
- Dingbian Qian^{1}Email author
https://doi.org/10.1186/1687-1812-2014-106
© Wang and Qian; licensee Springer. 2014
- Received: 24 January 2014
- Accepted: 8 April 2014
- Published: 6 May 2014
Abstract
In this paper, we establish a necessary and sufficient condition for the strong convergence of nonexpansive mappings in a uniformly convex and 2-uniformly 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.
Keywords
- strong convergence
- fixed points
- nonexpansive mapping
- η-strongly accretive
- 2-uniformly smooth Banach spaces
1 Introduction
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 single-valued and continuous from the norm topology of X to the weak-star 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 norm-to-norm continuous on any bounded subset of X; see [1, 2] for more details.
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 q-uniformly 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 q-uniformly smooth for $q>2$; see [3] for more details. So, in this paper, we focus on a 2-uniformly smooth Banach space. It is well known that Hilbert spaces and Lebesgue ${L}^{p}$ ($p\ge 2$) spaces are uniformly convex and 2-uniformly smooth.
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, $Fix(T)=\{x\in X:Tx=x\}$.
Since $\u3008f(x)-f(y),j(x-y)\u3009\le \parallel f(x)-f(y)\parallel \parallel x-y\parallel \le \alpha {\parallel x-y\parallel}^{2}$, we have that f is a strong pseudo-contraction.
for all $x,y\in X$. It is well known that if X is a Hilbert space, then an η-strongly accretive operator coincides with an η-strongly monotone operator.
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])
where F is a k-Lipschitzian and η-strongly 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])
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 2-uniformly 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])
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.
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.
that is, $\mu \overline{F}-\overline{f}$ is a strongly accretive operator with coefficient $\mu \eta -\rho $.
where $\tau =1-\sqrt{1-2\mu \eta +2{\mu}^{2}{k}^{2}{K}^{2}}\in (0,1]$.
where $\tau =1-\sqrt{1-2\mu \eta +2{\mu}^{2}{k}^{2}{K}^{2}}\in (0,1]$. □
3 Main results
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.
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).
Therefore, $\{{x}_{t}\}$ is bounded, and so are the nets $\{\overline{f}({x}_{t})\}$ and $\{\overline{F}T{x}_{t}\}$.
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 $t\to {0}^{+}$.
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 $t\to {0}^{+}$. □
- (i)
The framework of a Hilbert space is extended to a uniformly convex and 2-uniformly smooth Banach space.
- (ii)
The η-strongly monotone operator F is extended to the case of an η-strongly accretive operator $\overline{F}$. The contraction $f:H\to H$ is extended to the case of a Lipschitzian and strongly pseudo-contractive 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.
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:
(C3-1) ${\sum}_{n=0}^{\mathrm{\infty}}|{\alpha}_{n+1}-{\alpha}_{n}|<\mathrm{\infty}$;
(C3-2) ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n+1}/{\alpha}_{n}=1$.
Indeed, (C3-1) implies (C3′) by choosing ${\sigma}_{n}=|{\alpha}_{n+1}-{\alpha}_{n}|$, and (C3-2) 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 2-uniformly smooth Banach space.
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.
Therefore, $\{{x}_{n}\}$ is bounded. We also obtain that $\{f({x}_{n})\}$ and $\{\overline{F}T{x}_{n}\}$ are bounded.
where $M=max\{\gamma f({x}_{n-1}),\mu \overline{F}T{x}_{n-1}\}$. By Lemma 2.3, we have ${lim}_{n\to \mathrm{\infty}}\parallel {x}_{n+1}-{x}_{n}\parallel =0$.
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 $\u3008(\mu \overline{F}-\gamma f){x}^{\ast},j({x}^{\ast}-u)\u3009\le 0$, $u\in Fix(T)$.
This completes the proof. □
- (i)
The framework of a Hilbert space is extended to a uniformly convex and 2-uniformly smooth Banach space.
- (ii)
The η-strongly monotone operator F is extended to the case of an η-strongly 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.
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 $\u3008(\mu \overline{F}-\gamma f)0,j(0-u)\u3009\le 0$, $u\in Fix(T)$, that is, 0 is the solution of the variational inequality $\u3008(\mu \overline{F}-\gamma f){x}^{\ast},j({x}^{\ast}-u)\u3009\le 0$, $u\in Fix(T)$.
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.
Declarations
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).
Authors’ Affiliations
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