Strong convergence of a CQ method for k-strictly asymptotically pseudocontractive mappings

Abstract

Let E be a real q-uniformly smooth Banach space, which is also uniformly convex (for example, $L_p$ or ${\ell }_p$ spaces, $1<p<\mathrm{\infty }$), and C be a nonempty bounded closed convex subset of E. Let $T:C\to C$ be a k-strictly asymptotically pseudocontractive map with a nonempty fixed point set. A hybrid algorithm is constructed to approximate fixed points of such maps. Furthermore, strong convergence of the proposed algorithm is established.

1 Introduction

Let E be a real Banach space and $E^{\ast }$ be the dual of E. We denote the value of $x^{\ast }\in E^{\ast }$ at $x\in E$ by $〈x,x^{\ast }〉$. The normalized duality mapping J from E to $2^{E^{\ast }}$ is defined by

$$J(x)=\{x^{\ast }\in E^{\ast }:〈x,x^{\ast }〉={\parallel x\parallel }^2={\parallel x^{\ast }\parallel }^2\}$$

for all $x\in E$. It is known that a Banach space E is smooth if and only if the normalized duality mapping J is single valued. Some properties of the duality mapping have been given in [1, 2].

Let C be a nonempty subset of E. The mapping $T:C\to C$ is called nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel$$

for all $x,y\in C$. Also, T is called uniformly L-Lipschitz if there exists a constant $L>0$ such that

$$\parallel T^{n}x-T^{n}y\parallel \le L\parallel x-y\parallel$$

for all $x,y\in C$ and each $n\ge 1$. The mapping $T:C\to C$ is called k-strictly asymptotically pseudocontractive if there exist a sequence $\{k_n\}$ in $[1,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ and a constant $k\in [0,1)$, and for any $x,y\in C$, there exists $j(x-y)\in J(x-y)$ such that

$$〈T^{n}x-T^{n}y,j(x-y)〉\le \frac{1}{2}(1+k_n){\parallel x-y\parallel }^2-\frac{1}{2}(1-k){\parallel x-T^{n}x-(y-T^{n}y)\parallel }^2$$

(1.1)

for each $n\ge 1$. If I denotes the identity operator, then (1.1) can be written in the form

$$\begin{array}{rcl}〈(I-T^n)x-(I-T^n)y,j(x-y)〉& \ge & \frac{1}{2}(1-k){\parallel (I-T^n)x-(I-T^n)y\parallel }^2\\ -\frac{1}{2}(k_n-1){\parallel x-y\parallel }^2.\end{array}$$

(1.2)

The class of k-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, j is the identity and it is shown [4] that (1.1) (and hence (1.2)) is equivalent to the inequality

$${\parallel T^{n}x-T^{n}y\parallel }^2\le k_n{\parallel x-y\parallel }^2+k{\parallel (I-T^n)x-(I-T^n)y\parallel }^2$$

which is the inequality considered by Qihou [3]. In the same paper, the author proved strong convergence of the modified Mann iteration processes for k-strictly asymptotically pseudocontractive mappings in Hilbert spaces. The modified Mann iteration scheme was introduced by Schu [5, 6] and has been used by several authors (see, for example, [7–12]). In [13] Osilike extended Qihou’s result from Hilbert spaces to much more general real q-uniformly smooth Banach spaces, $1<q<\mathrm{\infty }$.

The classes of nonexpansive and asymptotically nonexpansive mappings are important classes of mappings because they have applications to solutions of differential equations which have been studied by several authors (see, e.g., [14–16] and references contained therein). It would be of interest to study the class of k-strictly asymptotically pseudocontractive mappings in view of the fact that it is closely related to the above two classes.

On the other hand, using the metric projection, Matsushita and Takahashi [17] introduced the following iterative algorithm for nonexpansive mappings: $x_0=x\in C$ and

$$\{\begin{array}{c}{C}_n=\overline{co}\{z\in C:\parallel z-Tz\parallel \le t_n\parallel x_n-T{x}_n\parallel \},\hfill \\ D_n=\{z\in C:〈x_n-z,J(x-x_n)〉\ge 0\},\hfill \\ x_{n+1}=P_{C_n\cap D_n}x,n=0,1,2,\dots ,\hfill \end{array}$$

(1.3)

where $\overline{co}D$ denotes the convex closure of the set D, J is the normalized duality mapping, $\{t_n\}$ is a sequence in $(0,1)$ with $t_n\to 0$, and $P_{C_n\cap D_n}$ is the metric projection from E onto $C_n\cap D_n$. Then, they proved that $\{x_n\}$ generated by (1.3) converges strongly to a fixed point of the mapping T.

In this paper, motivated by these facts, we introduce the following iterative algorithm for finding fixed points of a k-strictly asymptotically pseudocontractive mapping T in a uniformly convex and q-uniformly smooth Banach space: $x_1=x\in C$, $C_0=D_0=C$ and

$$\{\begin{array}{c}{C}_n=\overline{co}\{z\in C_{n-1}:\parallel z-T^{n}z\parallel \le t_n\parallel x_n-T^n{x}_n\parallel \},\hfill \\ D_n=\{z\in D_{n-1}:〈x_n-z,J(x-x_n)〉\ge 0\},\hfill \\ x_{n+1}=P_{C_n\cap D_n}x,n=1,2,\dots ,\hfill \end{array}$$

(1.4)

where $\overline{co}D$ denotes the convex closure of the set D, J is the normalized duality mapping, $\{t_n\}$ is a sequence in $(0,1)$ with $t_n\to 0$, and $P_{C_n\cap D_n}$ is the metric projection from E onto $C_n\cap D_n$.

The purpose of this paper is to establish a strong convergence theorem of the iterative algorithm (1.4) for k-strictly asymptotically pseudocontractive mappings in a uniformly convex and q-uniformly smooth Banach space.

2 Preliminaries

The modulus of smoothness of a Banach space E is the function ${\rho }_E:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ defined by

$${\rho }_E(t)=sup\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:\parallel x\parallel \le 1,\parallel y\parallel \le t\}.$$

E is uniformly smooth if and only if ${lim}_{t\to 0^{+}}{\rho }_E(t)/t=0$. Let $q>1$. The Banach space E is said to be q-uniformly smooth if there exists a constant $c>0$ such that ${\rho }_E(t)\le c{t}^q$. Hilbert spaces, $L_p$ (or ${\ell }_p$) spaces, $1<p<\mathrm{\infty }$, and the Sobolev spaces, $W_m^p$, $1<p<\mathrm{\infty }$, are q-uniformly smooth.

When $\{x_n\}$ is a sequence in E, we denote strong convergence of $\{x_n\}$ to $x\in E$ by $x_n\to x$ and weak convergence by $x_n\rightharpoonup x$. The Banach space E is said to have the Kadec-Klee property if for every sequence $\{x_n\}$ in E, $x_n\rightharpoonup x$ and $\parallel x_n\parallel \to \parallel x\parallel$ imply that $x_n\to x$. Every uniformly convex Banach space has the Kadec-Klee property [1].

Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E. Then for any $x\in E$, there exists a unique point $x_0\in C$ such that

$$\parallel x_0-x\parallel =\underset{y\in C}{min}\parallel y-x\parallel .$$

The mapping $P_C:E\to C$ defined by $P_{C}x=x_0$ is called the metric projection from E onto C. Let $x\in E$ and $u\in C$. Then it is known that $u=P_{C}x$ if and only if

$$〈u-y,J(x-u)〉\ge 0$$

(2.1)

for all $y\in C$ ( see [1, 18]).

In the sequel, we need the following results.

Proposition 2.1 (See [19])

Let C be a bounded closed convex subset of a uniformly convex Banach space E. Then there exists a strictly increasing convex continuous function $\gamma :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\gamma (0)=0$ depending only on the diameter of C such that

$$\gamma \left(\parallel \sum _{i=1}^n{\lambda }_{i}T{x}_i-T\left(\sum _{i=1}^n{\lambda }_i{x}_i\right)\parallel \right)\le \underset{1\le i<j\le n}{max}(\parallel x_i-x_j\parallel -\parallel T{x}_i-T{x}_j\parallel )$$

holds for any nonexpansive mapping $T:C\to E$, any elements $x_1,\dots ,x_n$ in C, and any numbers ${\lambda }_1,\dots ,{\lambda }_n\ge 0$ with ${\lambda }_1+\cdots +{\lambda }_n=1$.

Corollary 2.2 [[20], Corollary 1.2]

Under the same suppositions as in Proposition  2.1, there exists a strictly increasing convex continuous function $\gamma :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\gamma (0)=0$ depending only on the diameter of C such that

$$\gamma \left(\parallel \sum _{i=1}^n{\lambda }_i{x}_i-T\left(\sum _{i=1}^n{\lambda }_i{x}_i\right)\parallel \right)\le \underset{1\le i\le n}{max}(\parallel x_i-T{x}_i\parallel )$$

holds for any nonexpansive mapping $T:C\to E$, any elements $x_1,\dots ,x_n$ in C, and any numbers ${\lambda }_1,\dots ,{\lambda }_n\ge 0$ with ${\lambda }_1+\cdots +{\lambda }_n=1$. (Note that γ does not depend on T.)

In order to utilize Corollary 2.2 for k-strictly asymptotically pseudocontractive mappings, we need the following lemmas.

Lemma 2.3 [4]

Let E be a real Banach space, C be a nonempty subset of E, and $T:C\to C$ be a k-strictly asymptotically pseudocontractive mapping. Then T is uniformly L-Lipschitzian.

Lemma 2.4 [[21], Lemma 3.1]

Let E be a real q-uniformly smooth Banach space and C be a nonempty convex subset of E. Let $T:C\to C$ be a k-strictly asymptotically pseudocontractive map, and let $\{{\alpha }_n\}$ be a real sequence in $[0,1]$. Define $S_n:C\to C$ by $S_{n}x:=(1-{\alpha }_n)x+{\alpha }_n{T}^{n}x$ for all $x\in C$. Then for all $x,y\in C$, we have

$$\begin{array}{rcl}{\parallel S_{n}x-S_{n}y\parallel }^q& \le & (1+\frac{q}{2}{\alpha }_n(k_n-1)){\parallel x-y\parallel }^q\\ -{\alpha }_n(\frac{q}{2}(1-k){(1+L)}^{-(q-2)}-c_q{\alpha }_n^{q-1}){\parallel (I-T^n)x-(I-T^n)y\parallel }^q,\end{array}$$

where L is the uniformly Lipschitzian constant of T and $c_q>0$ is the constant which appeared in [[21], Theorem  2.1].

Let $\beta =min\{1,{[\frac{q}{2}(1-k){(1+L)}^{-(q-2)}/c_q]}^{1/(q-1)}\}$ and choose $\alpha \in (0,\beta )$. Set ${\alpha }_n=\alpha$ for all $n\ge 1$ in Lemma 2.4 and observe that ${\parallel S_{n}x-S_{n}y\parallel }^q\le (1+\frac{q}{2}\alpha (k_n-1)){\parallel x-y\parallel }^q$. Thus,

$$\parallel S_{n}x-S_{n}y\parallel \le {(1+\frac{q}{2}\alpha (k_n-1))}^{1/q}\parallel x-y\parallel$$

(2.2)

for all $x,y\in C$ and each $n\ge 1$.

Theorem 2.5 [[21], Theorem 3.1]

Let E be a real q-uniformly smooth Banach space which is also uniformly convex. Let C be a nonempty closed convex subset of E and $T:C\to C$ be a k-strictly asymptotically pseudocontractive mapping with a nonempty fixed point set. Then $(I-T)$ is demiclosed at zero, i.e., if $x_n\rightharpoonup x$ and $x_n-T{x}_n\to 0$, then $x\in F(T)$, where $F(T)$ is the set of all fixed points of T.

3 Strong convergence theorem

In this section, we study the iterative algorithm (1.4) for finding fixed points of k-strictly asymptotically pseudocontractive mappings in a uniformly convex and q-uniformly smooth Banach space. We first prove that the sequence $\{x_n\}$ generated by (1.4) is well defined. Then, we prove that $\{x_n\}$ converges strongly to $P_{F(T)}x$, where $P_{F(T)}$ is the metric projection from E onto $F(T)$.

Lemma 3.1 Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E, and let $T:C\to C$ be a mapping. If $F(T)\ne \mathrm{\varnothing }$, then the sequence $\{x_n\}$ generated by (1.4) is well defined.

Proof It is easy to check that $C_n\cap D_n$ is closed and convex and $F(T)\subset C_n$ for each $n\in \mathbb{N}$. Moreover, $D_1=C$ and so $F(T)\subset C_1\cap D_1$. Suppose $F(T)\subset C_k\cap D_k$ for $k\in \mathbb{N}$. Then there exists a unique element $x_{k+1}\in C_k\cap D_k$ such that $x_{k+1}=P_{C_k\cap D_k}x$. If $u\in F(T)$, then it follows from (2.1) that

$$〈x_{k+1}-u,J(x-x_{k+1})〉\ge 0,$$

which implies $u\in D_{k+1}$. Therefore, $F(T)\subset C_{k+1}\cap D_{k+1}$. By the mathematical induction, we obtain that $F(T)\subset C_n\cap D_n$ for all $n\in \mathbb{N}$. Therefore, $\{x_n\}$ is well defined. □

In order to prove our main result, the following lemma is needed.

Lemma 3.2 Let C be a nonempty bounded closed convex subset of a real q-uniformly smooth and uniformly convex Banach space E. Let $T:C\to C$ be a k-strictly asymptotically pseudocontractive mapping with $\{k_n\}$ such that $F(T)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by (1.4), then for any $j\in \mathbb{N}$,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T^{n-j}{x}_n\parallel =0.$$

Proof Fix $j\in \mathbb{N}$ and put $m=n-j$. Since $x_n=P_{C_{n-1}\cap D_{n-1}}x$, we have $x_n\in C_{n-1}\subseteq \cdots \subseteq C_m$. Since $t_m>0$, there exist $y_1,\dots ,y_N\in C$ and ${\lambda }_1,\dots ,{\lambda }_N\ge 0$ with ${\lambda }_1+\cdots +{\lambda }_N=1$ such that

$$\parallel x_n-\sum _{i=1}^N{\lambda }_i{y}_i\parallel <t_m,$$

(3.1)

and $\parallel y_i-T^m{y}_i\parallel \le t_m\parallel x_m-T^m{x}_m\parallel$ for all $i\in \{1,\dots ,N\}$. It follows from Lemma 2.3 that T is uniformly L-Lipschitzian. Put $M={sup}_{x\in C}\parallel x\parallel$, $u=P_{F(T)}x$ and $r_0={sup}_{n\ge 1}(1+L)\parallel x_n-u\parallel$. Thus,

$$\parallel y_i-T^m{y}_i\parallel \le t_m\parallel x_m-T^m{x}_m\parallel \le t_m(1+L)\parallel x_m-u\parallel \le r_0{t}_m$$

(3.2)

for all $i\in \{1,\dots ,N\}$. Define $H_m:C\to E$ by

$$H_{m}x=\frac{1}{a_m}{S}_{m}x$$

for all $x\in C$, where $a_m={(1+\frac{q}{2}\alpha (k_m-1))}^{1/q}$ and $S_m$ is as in (2.2). It follows from (2.2) that $H_m$ is nonexpansive. Using (3.2) and the fact that $\parallel y_i-S_m{y}_i\parallel =\alpha \parallel y_i-T^m{y}_i\parallel$, we have

$$\parallel y_i-H_m{y}_i\parallel \le (1-\frac{1}{a_m})\parallel y_i\parallel +\frac{1}{a_m}\parallel y_i-S_m{y}_i\parallel \le (1-\frac{1}{a_m})M+\alpha r_0{t}_m$$

(3.3)

for all $i\in \{1,\dots ,N\}$. It follows from Corollary 2.2, (3.1), and (3.3) that

$$\begin{array}{rcl}\parallel x_n-H_m{x}_n\parallel & \le & \parallel x_n-\sum _{i=1}^N{\lambda }_i{y}_i\parallel +\parallel \sum _{i=1}^N{\lambda }_i{y}_i-H_m\left(\sum _{i=1}^N{\lambda }_i{y}_i\right)\parallel \\ +\parallel H_m\left(\sum _{i=1}^N{\lambda }_i{y}_i\right)-H_m{x}_n\parallel \\ \le & 2t_m+{\gamma }^{-1}(\underset{1\le i\le N}{max}\parallel y_i-H_m{y}_i\parallel )\\ \le & 2t_m+{\gamma }^{-1}((1-\frac{1}{a_m})M+\alpha r_0{t}_m).\end{array}$$

Since and , it follows from the last inequality that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-H_m{x}_n\parallel =0$. Thus, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-S_m{x}_n\parallel =0$ and so ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T^m{x}_n\parallel =0$. This completes the proof. □

Theorem 3.3 Let C be a nonempty bounded closed convex subset of a real q-uniformly smooth and uniformly convex Banach space E. Let $T:C\to C$ be a k-strictly asymptotically pseudocontractive mapping with $\{k_n\}$ such that $F(T)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by (1.4). Then $\{x_n\}$ converges strongly to the element $P_{F(T)}x$ of $F(T)$, where $P_{F(T)}$ is the metric projection from E onto $F(T)$.

Proof Put $u=P_{F(T)}x$. Since $F(T)\subset C_n\cap D_n$ and $x_{n+1}=P_{C_n\cap D_n}x$, we have that

$$\parallel x-x_{n+1}\parallel \le \parallel x-u\parallel$$

(3.4)

for all $n\in \mathbb{N}$. By Lemma 3.2, we have

$$\begin{array}{rcl}\parallel x_n-T{x}_n\parallel & \le & \parallel x_n-T^{n-1}{x}_n\parallel +\parallel T^{n-1}{x}_n-T{x}_n\parallel \\ \le & \parallel x_n-T^{n-1}{x}_n\parallel +L\parallel T^{n-2}{x}_n-x_n\parallel \to 0\text{as }n\to \mathrm{\infty }.\end{array}$$

Since $\{x_n\}$ is bounded, there exists $\{x_{n_i}\}\subset \{x_n\}$ such that $x_{n_i}\rightharpoonup v$. It follows from Theorem 2.5 (demiclosedness of T) that $v\in F(T)$. From the weakly lower semicontinuity of norm and (3.4), we obtain

$$\parallel x-u\parallel \le \parallel x-v\parallel \le \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x-x_{n_i}\parallel \le \underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x-x_{n_i}\parallel \le \parallel x-u\parallel .$$

This together with the uniqueness of $P_{F(T)}x$ implies $u=v$, and hence $x_{n_i}\rightharpoonup u$. Therefore, we obtain $x_n\rightharpoonup u$. Furthermore, we have that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x-x_n\parallel =\parallel x-u\parallel .$$

Since E is uniformly convex, using the Kadec-Klee property, we have that $x-x_n\to x-u$. It follows that $x_n\to u$. This completes the proof. □