%------------------------------------------------------------------------------ % Here please write the date of submission of paper or its revisions: %------------------------------------------------------------------------------ % \documentclass[12pt, reqno]{amsart} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref} \textheight 22.5truecm \textwidth 14.5truecm \setlength{\oddsidemargin}{0.35in}\setlength{\evensidemargin}{0.35in} \setlength{\topmargin}{-.5cm} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{summary}[theorem]{Summary} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{problem}[theorem]{Problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \input{mathrsfs.sty} \begin{document} \setcounter{page}{10} \noindent\parbox{2.95cm}{\includegraphics*[keepaspectratio=true,scale=0.125]{AFA.jpg}} \noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent \qquad Ann. Funct. Anal. 2 (2011), no. 2, 10--21\\ {\footnotesize \qquad \textsc{\textbf{$\mathscr{A}$}nnals of \textbf{$\mathscr{F}$}unctional \textbf{$\mathscr{A}$}nalysis}\\ \qquad ISSN: 2008-8752 (electronic)\\ \qquad URL: \textcolor[rgb]{0.00,0.00,0.99}{www.emis.de/journals/AFA/} }\\[.5in]} \title[Iterative algorithm for nonexpansive mappings]{A general iterative algorithm for nonexpansive mappings in Banach spaces} \author[B. Ali, G.C. Ugwunnadi, Y. Shehu]{ Bashir Ali$^{1}$, Godwin C. Ugwunnadi$^{2}$ and Yekini Shehu$^*$$^{2}$ } \address{$^{1}$ Department of Mathematical Sciences, Bayero University, Kano.} \email{\textcolor[rgb]{0.00,0.00,0.84}{bashiralik@yahoo.com}} \address{$^{2}$ Department of Mathematics, University of Nigeria, Nsukka.} \email{\textcolor[rgb]{0.00,0.00,0.84}{ugwunnadi4u@yahoo.com}} \email{\textcolor[rgb]{0.00,0.00,0.84}{deltanougt2006@yahoo.com}} \dedicatory{{\rm Communicated by H.-K. Xu}} \subjclass[2010]{Primary 47H09; Secondary 47H10, 47J20.} \keywords{$\eta-$strongly accretive maps, $\kappa-$Lipschitzian maps, nonexpansive maps, $q-$uniformly smooth Banach spaces. } \date{Received: 22 September 2010; Revised: 1 April 2011; Accepted: 30 May 2011. \newline \indent $^{*}$ Corresponding author} \begin{abstract} Let $E$ be a real $q$-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let $T:E\to E$ be a nonexpansive mapping with $F(T)\neq\emptyset.$ Let $A:E\to E$ be an $\eta$-strongly accretive map which is also $\kappa$-Lipschitzian. Let $f:E\to E$ be a contraction map with coefficient $0<\alpha<1.$ Let a sequence $\{y_{n}\}$ be defined iteratively by $y_{0}\in E,~~ y_{n+1}=\alpha_n\gamma f(y_n)+(I-\alpha_n\mu A)Ty_n,n\geq0,$ where $\{\alpha_n\},~~\gamma$ and $\mu$ satisfy some appropriate conditions. Then, we prove that $\{y_{n}\}$ converges strongly to the unique solution $x^{*} \in F(T)$ of the variational inequality $\langle(\gamma f-\mu A)x^{*},j(y-x^{*})\rangle\leq0,~\forall~y\in F(T).$ Convergence of the correspondent implicit scheme is also proved without the assumption that $E$ has weakly sequentially continuous duality map. Our results are applicable in $l_{p}$ spaces, $1< p<\infty$. \end{abstract} \maketitle \section{Introduction} Let $E$ be a real Banach space and $E^{*}$ be the dual space of $E.$ A mapping $\varphi:[0,\infty)\to [0,\infty)$ is called a gauge function if it is strictly increasing, continuous and $\varphi(0)=0.$ Let $\varphi$ be a gauge function, a generalized duality mapping with respect to $\varphi,$ $J_{\varphi}:E\to 2^{E^{*}}$ is defined by, $x\in E,$ $$J_{\varphi}x=\{x^{*}\in E^{*}:\langle x,x^{*} \rangle=\Vert x \Vert \varphi(\Vert x\Vert),\Vert x^{*}\Vert=\varphi(\Vert x\Vert)\},$$ where $\langle . ,.\rangle$ denotes the duality pairing between element of $E$ and that of $E^{*}$. If $\varphi(t)=t,$ then $J_{\varphi}$ is simply called the normalized duality mapping and is denoted by $J.$ For any $x\in E,$ an element of $J_{\varphi}x$ is denoted by $j_{\varphi}(x)$.\\ If however $\varphi(t)=t^{q-1},$ for some $q>1,$ then $J_{\varphi}$ is still called the generalized duality mapping and is denoted by $J_{q}$ (see, for example \cite{yongfusu, afa}). \\[2mm] The space $E$ is said to have weakly (sequentially) continuous duality map if there exists a gauge function $\varphi$ such that $J_{\varphi}$ is singled valued and (sequentially) continous from $E$ with weak topology to $E^{*}$ with weak$^{*}$ topology. It is well known that all $l_{p}$ spaces, $(10,$ $\Vert A(x)-A(y)\Vert \le \kappa \Vert x-y \Vert$ $\forall~~x,y \in D(A)$. A mapping $T:E\rightarrow E$ is called {\it nonexpansive} if \begin{eqnarray*} ||T x-T y|| \leq ||x-y||,~~\forall x, y \in E. \end{eqnarray*} A point $x \in E$ is called {\it a fixed point} of $T$ if $Tx=x$. The set of fixed points of $T$ is denoted by $F(T):=\{x \in E:Tx=x\}$. In Hilbert spaces, accretive operators are called {\it monotone} where inequalities \eqref{31} and \eqref{accretive} hold with $j_{q}$ replaced by the identity map on $H.$ \\ \noindent Moudafi \cite{moudafi} introduced the viscosity approximation method for nonexpansive mappings. Let $f$ be a contraction on $H$, starting with an arbitrary $x_{0}\in H$, define a sequence $\{x_{n}\}$ recursively by \begin{eqnarray}\label{mouda} x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})Tx_{n},~~n\geq0, \end{eqnarray} where $\{\alpha_{n}\}$ is a sequence in (0,1). Xu \cite{xuk} proved that under certain appropriate conditions on $\{\alpha_{n}\}$, the sequence $\{x_{n}\}$ generated by \eqref{mouda} strongly converges to the unique solution $x^{*}$ in $F(T)$ of the variational inequality \begin{eqnarray*} \langle(I-f)x^{*},x-x^{*}\rangle\geq0,~~{\rm for}~x\in F(T). \end{eqnarray*} In \cite{xu4}, it is proved, under some conditions on the real sequence $\{\alpha_{n}\},$ that the sequence $\{x_{n}\}$ defined by $x_{0}\in H$ chosen arbitrary, \begin{eqnarray}\label{05} x_{n+1}=\alpha_{n}b+(I-\alpha_{n}A)Tx_{n},~~n\geq0, \end{eqnarray} converges strongly to $x^{*} \in F(T)$ which is the unique solution of the minimization problem \begin{eqnarray*} \underset{x\in F(T)}{\min}\frac{1}{2}\langle Ax,x\rangle-\langle x,b\rangle, \end{eqnarray*} where $A$ is a strongly positive bounded linear operator. That is, there is a constant $\bar{\gamma}>0$ with the property \begin{eqnarray*} \langle Ax,x\rangle\geq\bar{\gamma}||x||^{2},~~\forall x\in H. \end{eqnarray*} Combining the iterative method \eqref{mouda} and \eqref{05}, Marino and Xu \cite{xu6} consider the following general iterative method: \begin{eqnarray}\label{06} x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}A)Tx_{n},~~~n\geq0. \end{eqnarray} They proved that if the sequence $\{\alpha_{n}\}$ of parameters satisfies appropriate conditions, then the sequence $\{x_{n}\}$ generated by \eqref{06} converges strongly to $x^{*} \in F(T)$ which solves the variational inequality \begin{eqnarray*} \langle(\gamma f-A)x^{*},x-x^{*}\rangle\leq0,~~x\in F(T), \end{eqnarray*} which is the optimality condition for the minimization problem \begin{eqnarray*} \underset{x\in F(T)}{\min}\frac{1}{2}\langle Ax,x\rangle-h(x), \end{eqnarray*} where $h$ is a potential function for $\gamma f$ (i.e. $h'(x)=\gamma f(x)$ for $x\in H$).\\ \noindent Let $K$ be a nonempty, closed and convex subset of a real Hilbert space $H$. The variational inequality problem: Find a point $x^{*}\in K$ such that \begin{eqnarray*} \langle Ax^{*},y-x^{*}\rangle \ge 0, ~~\forall y\in K \end{eqnarray*} is equivalent to the following fixed point equation \begin{equation}\label{fixpoint} x^{*}=P_{K}(x^{*}-\delta Ax^{*}), \end{equation} where $\delta>0$ is an arbitrary fixed constant, $A$ is a nonlinear operator on $K$ and $P_{K}$ is the {\it nearest point projection map} from $H$ onto $K,$ i.e., $P_{K}x=y$ where $\Vert x-y \Vert = \underset{u\in K}{inf}\Vert x-u \Vert$ for $x\in H$. Consequently, under appropriate conditions on $A$ and $\delta,$ fixed point methods can be used to find or approximate a solution of the variational inequality. Considerable efforts have been devoted to this problem (see, for example, \cite{xukim, yamada} and the references contained therein). For instance, if $A$ is strongly monotone and Lipschitz then, a mapping $B:H\to H$ defined by $Bx= P_{K}(x-\delta Ax),$ $x\in H$ with $\delta>0$ sufficiently small is a strict contraction. Hence, the {\it Picard iteration,} $x_{0}\in H,$ $x_{n+1}=Bx_{n},~~n\ge 0$ of the classical Banach contraction mapping principle converges to the unique solution of the variational inequality. It has been observed that the projection operator $P_{K}$ in the fixed point formulation \eqref{fixpoint} may make the computation of the iterates difficult due to possible complexity of the convex set $K.$ In order to reduce the possible difficulty with the use of $P_{K},$ Yamada \cite{yamada} introduced the following hybrid descent method for solving the variational inequality: \begin{equation}\label{recur-1} x_{n+1}=Tx_{n}-\lambda_n\mu A(Tx_{n}), ~~~n\geq 0, \end{equation} where $T$ is a nonexpansive mapping, $A$ is an $\eta-$strongly monotone and $\kappa-$Lipschitz operator with $\eta > 0,~~\kappa> 0,~~0<\mu <\frac{2\eta}{\kappa^2}$. He proved that if $\{\lambda_n\}$ satisfies appropriate conditions then, $\{x_n\}$ converges strongly to the unique solution $x^{*}$ of the variational inequality \begin{eqnarray*} \langle Ax^{*},x-x^{*}\rangle\geq0,~~x\in F(T). \end{eqnarray*} \noindent Very recently, Tian \cite{tian} combined the Yamada's method \eqref{recur-1} with the iterative method \eqref{06} and introduced the following general iterative method in Hilbert spaces: \begin{eqnarray}\label{09} x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}\mu A)Tx_{n},~~n\geq0. \end{eqnarray} Then, he proved that the sequence $\{x_{n}\}$ generated by \eqref{09} converges strongly to the unique solution $x^{*}\in F(T)$ of the variational inequality \begin{eqnarray*} \langle(\gamma f-\mu A)x^{*},x-x^{*}\rangle\leq0,~~x\in F(T). \end{eqnarray*} \noindent We remark immediately here that the results of Tian \cite{tian} improved the results of Yamada \cite{yamada}, Moudafi \cite{moudafi}, Xu \cite{xuk} and Marino and Xu \cite{xu4} in Hilbert spaces.\\ \noindent In this paper, motivated and inspired by the above research results, our purpose is to extend the result of Tian \cite{tian} to $q$-uniformly smooth Banach space whose duality mapping is weakly sequentially continuous. Thus, our results are applicable in $l_{p}$ spaces, $1< p<\infty$. Furthermore, our results extend the results of Moudafi \cite{moudafi}, Xu \cite{xuk} and Marino and Xu \cite{xu4} to Banach spaces much more general than Hilbert. \section{Preliminaries} \noindent Let $E$ be a real Banach space. \noindent Let $K$ be a nonempty closed convex and bounded subset of a Banach space $E$ and let the diameter of $K$ be defined by $d(K):=sup\{\|x-y\|: x, y\in K\}$. For each $x\in K,$ let $r(x, K):=sup\{\|x-y\|: y\in K\}$ and let $r(K):=inf\{r(x, K): x \in K\}$ denote the Chebyshev radius of $K$ relative to itself. The \emph{normal structure coefficient} $N(E)$ of $E$ (see, for example, \cite{bynum}) is defined by $N(E):=inf\Big \{\frac{d(K)}{r(K)}: K $ is a closed convex and bounded subset of E with $ d(K)>0\Big \}.$ A space $E$ such that $N(E)>1$ is said to have uniform normal structure. It is known that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see, for example, \cite{chiliudo, limxu}).\\ \noindent Let $\mu$ be a linear continuous functional on $\ell^{\infty}$ and let $a=(a_1,a_2,\ldots) \in \ell^{\infty}$. We will sometimes write $\mu_n(a_n)$ in place of the value $\mu(a)$. A linear continuous functional $\mu$ such that $||\mu||=1=\mu(1)$ and $\mu_n(a_n)=\mu_n(a_{n+1})$ for every $a=(a_1,a_2,\ldots) \in \ell^{\infty}$ is called a {\it Banach limit}. It is known that if $\mu$ is a Banach limit, then $$\underset{n\rightarrow \infty}\liminf a_n \leq \mu_n(a_n) \leq \underset{n\rightarrow \infty}\limsup a_n$$ \noindent for every $a=(a_1,a_2,\ldots) \in \ell^{\infty}$ (see, for example, \cite{chiliudo, chidmono}).\\[2mm] Let $E$ be a normed space with ${\rm dimE} ~ \geq 2$. The {\it modulus of smoothness} of $E$ is the function $\rho_{E}: [0, \infty ) \rightarrow [0, \infty)$ defined by $$\rho_{E}(\tau):= \sup \left\{ \frac {\Vert x+y\Vert \ + \Vert x-y\Vert}{2} -1: \Vert x\Vert = 1; \Vert y \Vert = \tau \right \}.$$ The space $E$ is called {\it uniformly smooth} if and only if $\underset{t \rightarrow 0^{+}} {lim}\frac{\rho_{E}(t)}{t} = 0$. For some positive constant $q,$ $E$ is called {\it $q-$uniformly smooth} if there exists a constant $c>0$ such that $\rho_{E}(t)\le ct^{q},~~~~t>0.$ It is known that \begin{displaymath} L_{p} (or~~ l_{p})~~~{\rm spaces~~~are}~~~ \left\{ \begin{array}{ll} 2-~~~{\rm uniformly~~ smooth}, ~~~{\rm if,}~~~ 2\le p<\infty \\[3mm] p-~~~{\rm uniformly~~ smooth}, ~~~{\rm if,}~~~ 1< p\le 2. \end{array} \right. \end{displaymath} It is well known that if $E$ is smooth then the duality mapping is singled-valued, and if $E$ is uniformly smooth then the duality mapping is norm-to-norm uniformly continuous on bounded subset of E. \\[2mm] \noindent We shall make use of the following well known results. \begin{lemma}\label{lm21} Let $E$ be a real normed space. Then $$||x+y||^2 \leq ||x||^2+ 2\langle y, j(x+y) \rangle,$$ \noindent for all $x, y \in E$ and for all $j(x+y) \in J(x+y).$ \end{lemma} \begin{lemma} {\rm (Xu, \cite{xu3})}\label{xu} Let $E$ be a real $q$-uniformly smooth Banach space for some $q>1,$ then there exists some positive constant $d_{q}$ such that \begin{eqnarray*} \Vert x+y \Vert^{q} \le \Vert x \Vert^{q}+ q\langle y,j_{q}(x) \rangle + d_{q} \Vert y \Vert^{q}~~~~\forall~ x,y \in E ~{\rm and}~ j_{q}(x) \in J_{q}(x). \end{eqnarray*} \end{lemma} \begin{lemma}{\rm (Xu, \cite{xu2})}\label{xu2} Let $\{a_{n}\}$ be a sequence of nonnegative real numbers satisfying the following relation: $$a_{n+1} \leq (1-\alpha_{n})a_{n} + \alpha_{n} \sigma_{n} + \gamma_{n}, n\geq 0,$$ where, $(i) ~\{\alpha_{n}\} \subset [0,1], ~\sum \alpha_{n} = \infty;$ $(ii)~ \limsup ~\sigma_{n} \leq 0;$ $(iii)~\gamma_{n} \geq 0;~(n\geq 0),\\~\sum\gamma_{n} < \infty.$ Then, $a_{n}\rightarrow 0$ as $ n\rightarrow \infty.$ \end{lemma} \begin{lemma} {\rm (Lim and Xu, \cite{limxu})}\label{limxu} Suppose $E$ is a Banach space with uniform normal structure, $K$ is a nonempty bounded subset of $E,$ and $T:K\to K$ is uniformly $k-$Lipschitzian mapping with $k0,~~\eta \in (0,1)$. Then for $\gamma\in(0,\frac{\mu \eta}{\alpha})$, $$\langle(\mu A-\gamma f)x-(\mu A-\gamma f)y,j(x-y)\rangle\geq(\mu \eta-\gamma\alpha)||x-y||^{2},~\forall x,y\in E.$$ \noindent That is, $\mu A-\gamma f$ is strongly accretive with coefficient $\mu \eta-\gamma\alpha$. \end{lemma} \section{Main Results} \noindent We begin with the following lemma. \begin{lemma}\label{l1} Let $E$ be a $q$-uniformly smooth real Banach space with constant $d_{q},q >1.$ Let $f:E\to E$ be a contraction mapping with constant of contraction $\alpha \in (0,1)$. Let $T:E\to E$ be a nonexpansive mapping such that $F(T) \neq \emptyset$ and $A:E\to E$ be an $\eta$-strongly accretive mapping which is also $\kappa$-Lipschitzian. Let $\mu\in\Big(0,\min\Big\{1,(\frac{q\eta}{d_{q}\kappa^{q}})^\frac{1}{q-1}\Big\}\Big)$ and $\tau:=\mu \big(\eta-\frac{\mu^{q-1}d_{q}\kappa^{q}}{q}\big).$ For each $t\in (0,1)$ and $\gamma \in (0,\frac{\tau}{\alpha})$ define a map $T_t:E\to E$ by \begin{eqnarray*} T_tx=t\gamma f(x)+(I-t\mu A)Tx,~x\in E. \end{eqnarray*} Then, $T_t$ is a strict contraction. Furthermore \begin{eqnarray*} ||T_tx-T_ty||\leq[1-t(\tau-\gamma\alpha)]||x-y||. \end{eqnarray*} \end{lemma} \begin{proof} Without loss of generality, assume $\eta <\frac{1}{q}.$ Then, as $\mu<(\frac{q\eta}{d_{q}\kappa^{q}})^\frac{1}{(q-1)},$ we have $0 0$. By Lemma \ref{xu2}, we have $\underset{n\rightarrow \infty}\lim ||y_{n+1}-y_n||=0$. Furthermore, we obtain \begin{eqnarray}\label{demiclose} ||y_n-Ty_n||&\leq&||y_n-y_{n+1}||+||y_{n+1}-Ty_n||\\ &=&||y_n-y_{n+1}||+\alpha_n||\gamma f(y_n)-\mu ATy_n||\rightarrow 0~{\rm as}~n\to \infty.\nonumber \end{eqnarray} \noindent Let $\{y_{n_{j}}\}$ be a subsequence of $\{y_{n}\}$ such that $$\underset{n\to\infty}{limsup}\langle(\gamma f-\mu A)x^{*},j(y_{n}-x^{*})\rangle=\underset{j\to\infty}{lim}\langle(\gamma f-\mu A)x^{*},j(y_{n_{j}}-x^{*})\rangle. $$ Assume also $y_{n_{j}}\rightharpoonup z$ as $ j\to\infty,$ for some $z\in E.$ Then, using this, \eqref{demiclose} and the demiclosedness of $(I-T)$ at zero, we have $z\in F(T).$ Since $j$ is weakly sequentially continuous, we have \begin{eqnarray*} \underset{n\to\infty}{limsup}\langle(\gamma f-\mu A)x^{*},j(y_{n}-x^{*})\rangle&=&\underset{j\to\infty}{lim}\langle(\gamma f-\mu A)x^{*},j(y_{n_{j}}-x^{*})\rangle\\ &=& \langle(\gamma f-\mu A)x^{*},j(z-x^{*})\rangle \le 0. \end{eqnarray*} \noindent Finally, we show that $y_n\to x^{*}$. From the recursion formula \eqref{namdi}, let $$T_ny_{n}:=\alpha_{n}\gamma f(y_{n})+(I-\alpha_{n}\mu A)Ty_{n},$$ \noindent and from Lemma \ref{l1}, we have \begin{eqnarray*} ||y_{n+1}-x^{*}||^{2}&=&||T_ny_{n}-T_nx^{*}+T_nx^{*}-x^{*}||^{2}\\ &=& ||T_ny_{n}-T_nx^{*}+\alpha_{n}(\gamma f-\mu A)x^{*}||^{2}\\ &\leq&||T_ny_{n}-T_nx^{*}||^2+2\alpha_n\langle(\gamma f-\mu A)x^{*},j(y_{n+1}-x^{*})\rangle\\ &\leq&[1-\alpha_{n}(\tau-\gamma\alpha)]||y_{n}-x^{*}||^2\\ & &+2\alpha_{n}(\tau-\gamma\alpha)\frac{\langle(\gamma f-\mu A)x^{*},j(y_{n+1}-x^{*})\rangle}{\tau-\gamma\alpha} \end{eqnarray*} and by Lemma \ref{xu2} we have that $ y_{n} \to x^{*}~~{\rm as}~~n\to \infty.$ This completes the proof. \end{proof} \noindent We have the following corollaries. \begin{corollary} Let $E=l_{p}$ space, $(1