Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings

Abstract

It is well-known that Mann’s algorithm fails to converge for Lipschitzian pseudocontractions and strong convergence of Ishikawa’s algorithm for Lipschitzian pseudocontractions have not been achieved without compactness assumption on pseudocontractive mapping T or underlying space C. A new algorithm, which couples Ishikawa algorithms with hybrid techniques for finding the fixed points of a Lipschitzian pseudocontractive mapping, is constructed in this paper. Strong convergence of the presented algorithm is shown without any compactness assumption.

MSC:47H05, 47H10, 47H17.

1 Introduction

In the present article, we are devoted to finding the fixed points of pseudocontractive mappings. Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling [1], that if T is an accretive operator, then the solutions of the equations $Tx=0$ correspond to the equilibrium points of some evolution systems. This explains why a considerable research effort has been devoted to iterative methods for approximating solutions of the equation above, when T is accretive or corresponding to the iterative approximation of fixed points of pseudocontractions. Results of this kind have been obtained firstly in Hilbert spaces, but only for Lipschitz operators, and then they have been extended to more general Banach spaces (thanks to several geometric inequalities for general Banach spaces developed) and to more general classes of operators. There are still no results for the case of arbitrary Lipschitzian and pseudocontractive operators, even when the domain of the operator is a compact and convex subset of a Hilbert space. It is now well known that Mann’s algorithm [2] fails to converge for Lipschitzian pseudocontractions. This explains the importance, from this point of view, of the improvement brought by the Ishikawa iteration, which was introduced by Ishikawa [3] in 1974.

The original result of Ishikawa involves a Lipschitzian pseudocontractive self-mapping T on a convex and compact subset C of a Hilbert space. It establishes sufficient conditions such that Ishikawa iteration converges strongly to a fixed point of T.

However, a strong convergence has not been achieved without a compactness assumption on T or C. Consequently, considerable research efforts, especially within the past 40 years or so, have been devoted to iterative methods for approximating fixed points of T, when T is pseudocontractive (see, for example, [4–17] and the references therein). On the other hand, some convergence results are obtained by using the hybrid method in mathematical programming, see, for example, [14, 18–20]. Especially, Zegeye et al. [21] assumed that the interior of $Fix(T)$ is nonempty ($intFix(T)\ne \mathrm{\varnothing }$) to achieve a strong convergence, when T is a self-mapping of a nonempty closed convex subset of a real Hilbert space. This appears very restrictive, since even in ℝ with the usual norm, Lipschitz pseudocontractive maps with finite number of fixed points do not enjoy this condition that $intFix(T)\ne \mathrm{\varnothing }$.

The purpose of this article is to construct a new algorithm, which couples Ishikawa algorithms with hybrid techniques for finding the fixed points of a Lipschitzian pseudocontractive mapping. Strong convergence of the presented algorithm is given without any compactness assumption.

2 Preliminaries

Let H be a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$, respectively. Let C be a nonempty closed convex subset of H. Recall that a mapping $T:C\to C$ is called pseudocontractive (or a pseudocontraction) if

$$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2$$

for all $x,y\in C$.

It is easily seen that T is pseudocontractive if and only if T satisfies the condition

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+{\parallel (I-T)x-(I-T)y\parallel }^2$$

(2.1)

for all $x,y\in C$.

A mapping $T:C\to C$ is called L-Lipschitzian if there exists $L>0$ such that

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

for all $x,y\in C$.

We will use $Fix(T)$ to denote the set of fixed points of T, that is,

$$Fix(T)=\{x\in C:x=Tx\}.$$

The original result of Ishikawa is stated in the following.

Theorem 2.1 Let C be a convex and compact subset of a Hilbert space H, and let $T:C\to C$ be a Lipschitzian pseudocontractive mapping. Given $x_1\in C$, then the Ishikawa iteration $\{x_n\}$ defined by

$$\{\begin{array}{c}{y}_n=(1-{\beta }_n)x_n+{\beta }_{n}T{x}_n,\hfill \\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{y}_n,\hfill \end{array}$$

(2.2)

for all $n\in \mathbb{N}$, where $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences of positive numbers satisfying

• (a) $0\le {\alpha }_n\le {\beta }_n\le 1$,

• (b) ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$,

• (c) ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n{\beta }_n=\mathrm{\infty }$,

converges strongly to a fixed point of T.

To make our exposition self-contained, we have to recall that the (nearest point or metric) projection from H onto C, denoted by $P_C$, assigns to each $x\in H$ the unique point $P_C(x)\in C$ with the property

$$\parallel x-P_C(x)\parallel =inf\{\parallel x-y\parallel :y\in C\}.$$

It is well known that the metric projection $P_C$ of H onto C is characterized by

$$〈x-P_C(x),y-P_C(x)〉\le 0$$

(2.3)

for all $x\in H$, $y\in C$. Also, it is well known that in a real Hilbert space H, the following equality holds

$${\parallel tx+(1-t)y\parallel }^2=t{\parallel x\parallel }^2+(1-t){\parallel y\parallel }^2-t(1-t){\parallel x-y\parallel }^2$$

(2.4)

for all $x,y\in H$ and $t\in [0,1]$.

Lemma 2.1 [7]

Let H be a real Hilbert space, let C be a closed convex subset of H. Let $T:C\to C$ be a continuous pseudocontractive mapping. Then

• (i) $Fix(T)$ is a closed convex subset of C.

• (ii) $(I-T)$ is demiclosed at zero.

In the sequel, we shall use the following notations:

• ${\omega }_w(x_n)=\{x:\mathrm{\exists }{x}_{n_j}\to x\text{ weakly}\}$ denote the weak ω-limit set of $\{x_n\}$;

• $x_n\rightharpoonup x$ stands for the weak convergence of $\{x_n\}$ to x;

• $x_n\to x$ stands for the strong convergence of $\{x_n\}$ to x.

Lemma 2.2 [18]

Let C be a closed convex subset of H. Let $\{x_n\}$ be a sequence in H, and let $u\in H$. Let $q=P_{C}u$. If $\{x_n\}$ is such that ${\omega }_w(x_n)\subset C$ and satisfies the condition

$$\parallel x_n-u\parallel \le \parallel u-q\parallel \mathit{\text{for all }}n\in \mathbb{N},$$

then $x_n\to q$.

3 Main results

In this section, we state our main results.

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let $T:C\to C$ be an L-Lipschitzian pseudocontractive mapping with $Fix(T)\ne \mathrm{\varnothing }$.

Firstly, we present our new algorithm, which couples Ishikawa’s algorithm (2.2) with the hybrid projection algorithm.

Algorithm 3.1 Let $x_0\in H$. For $C_1=C$ and $x_1=P_{C_1}(x_0)$, define a sequence $\{x_n\}$ of C as follows:

$$\{\begin{array}{c}{y}_n=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,\hfill \\ z_n={\beta }_n{x}_n+(1-{\beta }_n)T{y}_n,\hfill \\ C_{n+1}=\{x^{\ast }\in C_n,\parallel z_n-x^{\ast }\parallel \le \parallel x_n-x^{\ast }\parallel \},\hfill \\ x_{n+1}=P_{C_{n+1}}(x_0),\hfill \end{array}$$

(3.1)

for all $n\ge 1$, where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are two sequences in $[0,1]$.

In the sequel, we assume that the sequences $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ satisfy the following conditions

$$0<k\le 1-{\beta }_n\le {\alpha }_n<\frac{1}{\sqrt{1+L^2}+1}$$

for all $n\in \mathbb{N}$.

Remark 3.1 Without loss of generality, we can assume that the Lipschitz constant $L>1$. If not, then T is nonexpansive. In this case, algorithm (3.1) is trivial. So, in this article, we assume $L>1$. It is obvious that $\frac{1}{\sqrt{1+L^2}+1}<\frac{1}{L}$ for all $n\ge 1$.

We prove the following several lemmas, which will support our main theorem below.

Lemma 3.1 $Fix(T)\subset C_n$ for $n\ge 1$ and $\{x_n\}$ is well defined.

Proof We use mathematical induction to prove $Fix(T)\subset C_n$ for all $n\in \mathbb{N}$.

• (i) $Fix(T)\subset C_1=C$ is obvious.

• (ii) Suppose that $Fix(T)\subset C_k$ for some $k\in \mathbb{N}$. Take $u\in Fix(T)\subset C_k$. From (3.1), by using (2.4), we have

  $$\begin{array}{rcl}{\parallel z_n-u\parallel }^2& =& {\parallel {\beta }_n(x_n-u)+(1-{\beta }_n)(T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)-u)\parallel }^2\\ =& {\beta }_n{\parallel x_n-u\parallel }^2+(1-{\beta }_n){\parallel T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)-u\parallel }^2\\ -{\beta }_n(1-{\beta }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2.\end{array}$$

  (3.2)

Since $u\in Fix(T)$, from (2.1), we have

$${\parallel Tx-u\parallel }^2\le {\parallel x-u\parallel }^2+{\parallel x-Tx\parallel }^2$$

(3.3)

for all $x\in C$.

From (2.4) and (3.3), we obtain

$$\begin{array}{r}{\parallel T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)-u\parallel }^2\\ \le {\parallel (1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2+{\parallel (1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n-u\parallel }^2\\ ={\parallel (1-{\alpha }_n)(x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n))+{\alpha }_n(T{x}_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n))\parallel }^2\\ +{\parallel (1-{\alpha }_n)(x_n-u)+{\alpha }_n(T{x}_n-u)\parallel }^2\\ =(1-{\alpha }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2+{\alpha }_n{\parallel T{x}_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2\\ -{\alpha }_n(1-{\alpha }_n){\parallel x_n-T{x}_n\parallel }^2+(1-{\alpha }_n){\parallel x_n-u\parallel }^2+{\alpha }_n{\parallel T{x}_n-u\parallel }^2\\ -{\alpha }_n(1-{\alpha }_n){\parallel x_n-T{x}_n\parallel }^2\\ \le (1-{\alpha }_n){\parallel x_n-u\parallel }^2+{\alpha }_n({\parallel x_n-u\parallel }^2+{\parallel x_n-T{x}_n\parallel }^2)-{\alpha }_n(1-{\alpha }_n){\parallel x_n-T{x}_n\parallel }^2\\ +(1-{\alpha }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2+{\alpha }_n{\parallel T{x}_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2\\ -{\alpha }_n(1-{\alpha }_n){\parallel x_n-T{x}_n\parallel }^2.\end{array}$$

Note that T is L-Lipschitzian. It follows that

$$\begin{array}{r}{\parallel T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)-u\parallel }^2\\ \le (1-{\alpha }_n){\parallel x_n-u\parallel }^2+{\alpha }_n({\parallel x_n-u\parallel }^2+{\parallel x_n-T{x}_n\parallel }^2)-{\alpha }_n(1-{\alpha }_n){\parallel x_n-T{x}_n\parallel }^2\\ +(1-{\alpha }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2+{\alpha }_n^3{L}^2{\parallel x_n-T{x}_n\parallel }^2\\ -{\alpha }_n(1-{\alpha }_n){\parallel x_n-T{x}_n\parallel }^2\\ ={\parallel x_n-u\parallel }^2+(1-{\alpha }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2\\ -{\alpha }_n(1-2{\alpha }_n-{\alpha }_n^2{L}^2){\parallel x_n-T{x}_n\parallel }^2.\end{array}$$

(3.4)

By condition ${\alpha }_n<\frac{1}{\sqrt{1+L^2}+1}$, we have $1-2{\alpha }_n-{\alpha }_n^2{L}^2>0$. Substituting (3.4) to (3.2), we obtain

$$\begin{array}{rcl}{\parallel z_n-u\parallel }^2& =& {\beta }_n{\parallel x_n-u\parallel }^2+(1-{\beta }_n){\parallel T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)-u\parallel }^2\\ -{\beta }_n(1-{\beta }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2\\ \le & {\beta }_n{\parallel x_n-u\parallel }^2+(1-{\beta }_n)[{\parallel x_n-u\parallel }^2\\ +(1-{\alpha }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2]\\ -{\beta }_n(1-{\beta }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2\\ =& {\parallel x_n-u\parallel }^2+(1-{\beta }_n)(1-{\alpha }_n-{\beta }_n){\parallel x_n-T((1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n)\parallel }^2.\end{array}$$

Since ${\alpha }_n+{\beta }_n\ge 1$, we deduce

$$\parallel z_n-u\parallel \le \parallel x_n-u\parallel .$$

(3.5)

Hence $u\in C_{k+1}$. This implies that

$$Fix(T)\subset C_n$$

for all $n\in \mathbb{N}$.

Next, we show that $C_n$ is closed and convex for all $n\in \mathbb{N}$.

It is obvious that $C_1=C$ is closed and convex.

Suppose that $C_k$ is closed and convex for some $k\in \mathbb{N}$. For $u\in C_k$, it is obvious that $\parallel z_k-u\parallel \le \parallel x_k-u\parallel$ is equivalent to ${\parallel z_k-x_k\parallel }^2+2〈z_k-x_k,x_k-u〉\le 0$. So, $C_{k+1}$ is closed and convex. Then, for any $n\in \mathbb{N}$, the set $C_n$ is closed and convex. This implies that $\{x_n\}$ is well defined. □

Lemma 3.2 The sequence $\{x_n\}$ is bounded.

Proof Using the characterization inequality (2.3) of metric projection, from $x_n=P_{C_n}(x_0)$, we have

$$〈x_0-x_n,x_n-y〉\ge 0\text{for all }y\in C_n.$$

Since $Fix(T)\subset C_n$, we also have

$$〈x_0-x_n,x_n-u〉\ge 0\text{for all }u\in Fix(T).$$

So, for $u\in Fix(T)$, we obtain

$$\begin{array}{rl}0& \le 〈x_0-x_n,x_n-u〉\\ =〈x_0-x_n,x_n-x_0+x_0-u〉\\ =-{\parallel x_0-x_n\parallel }^2+〈x_0-x_n,x_0-u〉\\ \le -{\parallel x_0-x_n\parallel }^2+\parallel x_0-x_n\parallel \parallel x_0-u\parallel .\end{array}$$

Hence,

$$\parallel x_0-x_n\parallel \le \parallel x_0-u\parallel \text{for all }u\in Fix(T).$$

(3.6)

This implies that the sequence $\{x_n\}$ is bounded. □

Lemma 3.3 ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$.

Proof From $x_n=P_{C_n}(x_0)$ and $x_{n+1}=P_{C_{n+1}}(x_0)\in C_{n+1}\subset C_n$, we have

$$〈x_0-x_n,x_n-x_{n+1}〉\ge 0.$$

Hence,

$$\begin{array}{rl}0& \le 〈x_0-x_n,x_n-x_{n+1}〉\\ =〈x_0-x_n,x_n-x_0+x_0-x_{n+1}〉\\ =-{\parallel x_0-x_n\parallel }^2+〈x_0-x_n,x_0-x_{n+1}〉\\ \le -{\parallel x_0-x_n\parallel }^2+\parallel x_0-x_n\parallel \parallel x_0-x_{n+1}\parallel ,\end{array}$$

and, therefore,

$$\parallel x_0-x_n\parallel \le \parallel x_0-x_{n+1}\parallel ,$$

which implies that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists. Thus,

$$\begin{array}{rl}{\parallel x_{n+1}-x_n\parallel }^2& ={\parallel (x_{n+1}-x_0)-(x_n-x_0)\parallel }^2\\ ={\parallel x_{n+1}-x_0\parallel }^2-{\parallel x_n-x_0\parallel }^2-2〈x_{n+1}-x_n,x_n-x_0〉\\ \le {\parallel x_{n+1}-x_0\parallel }^2-{\parallel x_n-x_0\parallel }^2\\ \to 0.\end{array}$$

 □

Theorem 3.2 The sequence $\{x_n\}$ defined by (3.1) converges strongly to $P_{Fix(T)}(x_0)$.

Remark 3.3 Note that $Fix(T)$ is closed and convex. Thus, the projection $P_{Fix(T)}$ is well defined.

Proof Since $x_{n+1}\in C_{n+1}\subset C_n$, we have

$$\parallel z_n-x_{n+1}\parallel \le \parallel x_n-x_{n+1}\parallel \to 0.$$

Further, we obtain

$$\parallel z_n-x_n\parallel \le \parallel z_n-x_{n+1}\parallel +\parallel x_{n+1}-x_n\parallel \to 0.$$

From (3.1), we get

$$\begin{array}{rcl}\parallel x_n-T{x}_n\parallel & \le & \parallel x_n-z_n\parallel +\parallel z_n-T{x}_n\parallel \\ \le & \parallel x_n-z_n\parallel +{\beta }_n\parallel x_n-T{x}_n\parallel +(1-{\beta }_n)\parallel T{y}_n-T{x}_n\parallel \\ \le & \parallel x_n-z_n\parallel +{\beta }_n\parallel x_n-T{x}_n\parallel +(1-{\beta }_n)L{\alpha }_n\parallel x_n-T{x}_n\parallel \\ =& \parallel x_n-z_n\parallel +[{\beta }_n+(1-{\beta }_n)L{\alpha }_n]\parallel x_n-T{x}_n\parallel .\end{array}$$

Since $0<k\le 1-{\beta }_n\le {\alpha }_n<\frac{1}{\sqrt{1+L^2}+1}$ and $1-[{\beta }_n+(1-{\beta }_n)L{\alpha }_n]>k(1-\frac{L}{\sqrt{1+L^2}+1})>0$, it follows that

$$\begin{array}{rcl}\parallel x_n-T{x}_n\parallel & \le & \frac{1}{1-[{\beta }_n+(1-{\beta }_n)L{\alpha }_n]}\parallel x_n-z_n\parallel \\ \le & \frac{1}{k(1-\frac{L}{\sqrt{1+L^2}+1})}\parallel x_n-z_n\parallel \to 0.\end{array}$$

(3.7)

Now, (3.7) and Lemma 2.1 guarantee that every weak limit point of $\{x_n\}$ is a fixed point of T. That is, ${\omega }_w(x_n)\subset Fix(T)$. This fact, inequality (3.6) and Lemma 2.2 ensure the strong convergence of $\{x_n\}$ to $P_{Fix(T)}(x_0)$. This completes the proof. □

Remark 3.4 It is easily seen that all of the results above hold for nonexpansive mappings.

Remark 3.5 It is nowadays quite clear that, for large classes of contractive type operators, it suffices to consider the simpler Mann iteration, even if the Ishikawa iteration, which is more general but also computationally more complicated than the Mann iteration, could always be used. But if T is only a pseudocontractive mapping, then generally, the Mann iterative process does not converge to the fixed point, and strong convergence of the Ishikawa iteration has not been achieved without the compactness assumption on T or C. However, our algorithm (3.1) has a strong convergence without the compactness assumption.