Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces

Abstract

In this paper, we establish strong convergence for the iterative scheme introduced by Sahu and Petruşel associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

MSC:47H10, 47J25.

1 Introduction

Let H be a Hilbert space, and let $T:H\to H$ be a mapping. The mapping T is called Lipshitzian if there exists $L>0$ such that

$$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in H.$$

If $L=1$, then T is called nonexpansive and if $0\le L<1$, then T is called contractive.

The mapping $T:H\to H$ is said to be pseudocontractive (see, for example, [1, 2]) if

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+{\parallel (I-T)x-(I-T)y\parallel }^2,\mathrm{\forall }x,y\in H$$

(1.1)

and it is said to be strongly pseudocontractive if there exists $k\in (0,1)$ such that

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel (I-T)x-(I-T)y\parallel }^2,\mathrm{\forall }x,y\in H.$$

(1.2)

Let $F(T):=\{x\in H:Tx=x\}$, and let K be a nonempty subset of H. A mapping $T:K\to K$ is called hemicontractive if $F(T)\ne \mathrm{\varnothing }$ and

$${\parallel Tx-x^{\ast }\parallel }^2\le {\parallel x-x^{\ast }\parallel }^2+{\parallel x-Tx\parallel }^2,\mathrm{\forall }x\in H,x^{\ast }\in F(T).$$

It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractions. For the importance of fixed points of pseudocontractions, the reader may consult [1].

In 1974, Ishikawa [3] proved the following result.

Theorem 1.1 Let K be a compact convex subset of a Hilbert space H, and let $T:K\to K$ be a Lipschitzian pseudocontractive mapping.

For arbitrary $x_1\in K$, let $\{x_n\}$ be a sequence defined iteratively by the Ishikawa iterative scheme

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

(1.3)

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences satisfying the conditions

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

  ;

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

  ;

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

  .

Then the sequence $\{x_n\}$ converges strongly to a fixed point of T.

Another iterative scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings is the S-iterative scheme introduced by Sahu and Petruşel [4] in 2011.

In this paper, we establish strong convergence for the S-iterative scheme associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

2 Main results

We need the following lemma.

Lemma 2.1 [5]

For all $x,y\in H$ and $\lambda \in [0,1]$, the following well-known identity holds:

$${\parallel (1-\lambda )x+\lambda y\parallel }^2=(1-\lambda ){\parallel x\parallel }^2+\lambda {\parallel y\parallel }^2-\lambda (1-\lambda ){\parallel x-y\parallel }^2.$$

Now we prove our main results.

Theorem 2.2 Let K be a compact convex subset of a real Hilbert space H, and let $T:K\to K$ be a Lipschitzian hemicontractive mapping satisfying

Let $\{{\beta }_n\}$ be a sequence in $[0,1]$ satisfying

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

  ;

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

  .

For arbitrary $x_1\in K$, let $\{x_n\}$ be a sequence defined iteratively by the S-iterative scheme

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

(2.1)

Then the sequence $\{x_n\}$ converges strongly to the fixed point of T.

Proof From Schauder’s fixed point theorem, $F(T)$ is nonempty since K is a compact convex set and T is continuous. Let $x^{\ast }\in F(T)$. Using the fact that T is hemicontractive, we obtain

$${\parallel T{x}_n-x^{\ast }\parallel }^2\le {\parallel x_n-x^{\ast }\parallel }^2+{\parallel x_n-T{x}_n\parallel }^2$$

(2.2)

and

$${\parallel T{y}_n-x^{\ast }\parallel }^2\le {\parallel y_n-x^{\ast }\parallel }^2+{\parallel y_n-T{y}_n\parallel }^2.$$

(2.3)

With the help of (2.1), (2.2) and Lemma 2.1, we obtain the following estimates:

(2.4)

(2.5)

Substituting (2.4) and (2.5) in (2.3) we obtain

$$\begin{array}{rcl}{\parallel T{y}_n-x^{\ast }\parallel }^2& \le & {\parallel x_n-x^{\ast }\parallel }^2+(1-{\beta }_n){\parallel x_n-T{y}_n\parallel }^2+{\beta }_n{\parallel T{x}_n-T{y}_n\parallel }^2\\ -{\beta }_n(1-2{\beta }_n){\parallel x_n-T{x}_n\parallel }^2.\end{array}$$

(2.6)

Also, with the help of condition (C) and (2.6), we have

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& =& {\parallel T{y}_n-x^{\ast }\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2+(1-{\beta }_n){\parallel x_n-T{y}_n\parallel }^2+{\beta }_n{\parallel T{x}_n-T{y}_n\parallel }^2\\ -{\beta }_n(1-2{\beta }_n){\parallel x_n-T{x}_n\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2+{\parallel T{x}_n-T{y}_n\parallel }^2-{\beta }_n(1-2{\beta }_n){\parallel x_n-T{x}_n\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2+L^2{\parallel x_n-y_n\parallel }^2-{\beta }_n(1-2{\beta }_n){\parallel x_n-T{x}_n\parallel }^2\\ =& {\parallel x_n-x^{\ast }\parallel }^2+L^2{\beta }_n^2{\parallel x_n-T{x}_n\parallel }^2-{\beta }_n(1-2{\beta }_n){\parallel x_n-T{x}_n\parallel }^2\\ =& {\parallel x_n-x^{\ast }\parallel }^2-{\beta }_n(1-(2+L^2){\beta }_n){\parallel x_n-T{x}_n\parallel }^2.\end{array}$$

(2.7)

Now, by ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$, there exists $n_0\in \mathbb{N}$ such that for all $n\ge n_0$,

$${\beta }_n\le \frac{1}{2(2+L^2)},$$

(2.8)

and with the help of (2.8), (2.7) yields

$${\parallel x_{n+1}-x^{\ast }\parallel }^2\le {\parallel x_n-x^{\ast }\parallel }^2-\frac{1}{2}{\beta }_n{\parallel x_n-T{x}_n\parallel }^2,$$

which implies

$$\frac{1}{2}{\beta }_n{\parallel x_n-T{x}_n\parallel }^2\le {\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2,$$

so that

$$\frac{1}{2}\sum _{j=N}^n{\beta }_j{\parallel x_j-T{x}_j\parallel }^2\le {\parallel x_N-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2.$$

The rest of the argument follows exactly as in the proof of theorem of [3]. This completes the proof. □

Theorem 2.3 Let K be a compact convex subset of a real Hilbert space H, and let $T:K\to K$ be a Lipschitzian hemicontractive mapping satisfying condition (C). Let $\{{\beta }_n\}$ be a sequence in $[0,1]$ satisfying conditions (iv) and (v).

Assume that $P_K:H\to K$ is the projection operator of H onto K. Let $\{x_n\}$ be a sequence defined iteratively by

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

Then the sequence $\{x_n\}$ converges strongly to a fixed point of T.

Proof The operator $P_K$ is nonexpansive (see, e.g., [2]). K is a Chebyshev subset of H so that $P_K$ is a single-valued mapping. Hence, we have the following estimate:

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& =& {\parallel P_K(T{y}_n)-P_K{x}^{\ast }\parallel }^2\\ \le & {\parallel T{y}_n-x^{\ast }\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-{\beta }_n(1-(2+L^2){\beta }_n){\parallel x_n-T{x}_n\parallel }^2.\end{array}$$

The set $K=K\cup T(K)$ is compact, and so the sequence $\{\parallel x_n-T{x}_n\parallel \}$ is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.2. This completes the proof. □

Remark 2.4 In Theorem 1.1, putting ${\alpha }_n=1$, $0\le {\alpha }_n\le {\beta }_n\le 1$ implies ${\beta }_n=1$, which contradicts ${lim}_{n\to \mathrm{\infty }}{\beta }_n=0$. Hence the S-iterative scheme is not the special case of Ishikawa iterative scheme.

Remark 2.5 In Theorems 2.2 and 2.3, condition (C) is not new; it is due to Liu et al. [6].