Iterative process for a strictly pseudo-contractive mapping in uniformly convex Banach spaces

Abstract

This paper is concerned with a new method to prove the weak convergence of a strictly pseudo-contractive mapping in a p-uniformly convex Banach space with more relaxed restrictions on the parameters. Our results extend and improve the corresponding earlier results.

MSC:41A65, 47H17, 47J20.

1 Introduction and preliminaries

In 1967, Browder and Petryshyn [1] gave the classical definition for strictly pseudo-contractive mappings in Hilbert spaces for the first time.

Definition 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. $T:C\to H$ is called a Browder-Petryshyn-type k-strictly pseudo-contractive mapping. Then there exists $k\in [0,1)$ such that for every $x,y\in C$

$$〈Tx-Ty,j(x-y)〉\le {\parallel x-y\parallel }^2-k{\parallel (I-T)x-(I-T)y\parallel }^2.$$

(1.1)

In 2010, Zhou [2] gave a new definition for k-strictly pseudo-contractive mappings in q-uniformly smooth Banach spaces.

Definition 1.2 Let C be a nonempty closed convex subset of a q-uniformly smooth Banach space X. $T:C\to C$ is called a Zhou-type k-strictly pseudo-contractive mapping, if there exists $k\in [0,1)$ such that for every $x,y\in C$

$$〈Tx-Ty,j_q(x-y)〉\le {\parallel x-y\parallel }^q-\frac{1-k}{2}{\parallel (I-T)x-(I-T)y\parallel }^q.$$

(1.2)

In 2009, Hu and Wang [3] gave another definition for k-strictly pseudo-contractive mappings in p-uniformly convex Banach spaces.

Definition 1.3 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X. $T:C\to C$ is called a Hu-type k-strictly pseudo-contractive mapping, if there exists $k\in [0,1)$ such that for every $x,y\in C$

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

(1.3)

Remark 1.1 The mappings defined by (1.1) and (1.2) are pseudo-contractive mappings, but the mapping defined by (1.3) may not be pseudo-contractive in general Banach spaces.

Remark 1.2 If and only if $q=2$, the mappings defined by (1.1) and (1.2) are equivalent.

Remark 1.3 If $p=q=2$, the mappings defined by (1.1), (1.2), and (1.3) are equivalent in Hilbert space.

In 1979, Reich [4] established a weak convergence theorem via a Mann-type iterative process for nonexpansive mapping in a uniformly convex Banach space with Fréchet differentiable norm.

Theorem R Let C be a closed convex subset of a uniformly convex Banach space X with a Fréchet differentiable norm and $T:C\to C$ a nonexpansive mapping with $F(T)\ne \mathrm{\varnothing }$. For any $x_1\in C$, the iterative sequence $\{x_n\}$ is defined by $x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n$, where the real sequence $\{{\alpha }_n\}\subset [0,1]$ and ${\sum }_{n=1}^{\mathrm{\infty }}(1-{\alpha }_n){\alpha }_n=\mathrm{\infty }$. Then the sequence $\{x_n\}$ converges weakly to a fixed point of T.

In 2007, Marino and Xu [5] improved Reich’s [4] result and gave several weak convergence theorems via the normal Mann iterative algorithm for strictly pseudo-contractive mappings in Hilbert spaces. Further, they proposed an open problem: Do the main results of [5]still hold true in the framework of Banach spaces which are uniformly convex and have a Fréchet differentiable norm?

In 2009, Hu and Wang [3] considered above problem in a p-uniformly convex Banach space and established the following theorem.

Theorem H Let C be a closed convex subset of a p-uniformly convex Banach space X with a Fréchet differentiable norm and $T:C\to C$ be a k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients $p,k<min\{1,2^{-(p-2)}{c}_p\}$ and $F(T)\ne \mathrm{\varnothing }$. For any $x_1\in C$ and $n>1$, the iterative sequence $\{x_n\}$ is defined by $x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n$, where the real sequence $\{{\alpha }_n\}\subset [0,1]$ and $0<\epsilon \le {\alpha }_n\le 1-\epsilon <1-\frac{2^{p-2}k}{c_p}$. Then the sequence $\{x_n\}$ converges weakly to a fixed point of T.

Question Can one relax the restriction on the parameters ${\alpha }_n$ in Theorem H and simplify its proof?

The purpose of this paper is to solve the question mentioned above. To prove our results, we need the following lemmas.

Lemma 1.1 (see [3])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X and $T:C\to C$ be a Hu-type strictly pseudo-contractive mapping in the light of (1.3). For $\alpha \in (0,1)$, define $T_{\alpha }:C\to C$ by $T_{\alpha }=(1-\alpha )x+\alpha Tx$, for $x\in C$. If $\alpha \in (0,1-(k{2}^{p-2})/c_p)$, then $T_{\alpha }$ is a nonexpansive mapping and $F(T_{\alpha })=F(T)$.

Lemma 1.2 (see [3])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X and $T:C\to C$ be a Hu-type strictly pseudo-contractive mapping in the light of (1.3). For $\mu \in (0,1)$, $T_{\mu }:C\to C$ is defined by $T_{\mu }=(1-\mu )x+\mu Tx$, for $x\in C$. Then the following inequality holds:

$${\parallel T_{\mu }x-T_{\mu }y\parallel }^p\le {\parallel x-y\parallel }^p-(W_p(\mu )c_p-\mu \lambda ){\parallel (I-T)x-(I-T)y\parallel }^p,\mathrm{\forall }x,y\in C,$$

where $W_p(\mu )={\mu }^p(1-\mu )+\mu {(1-\mu )}^p$.

Lemma 1.3 (see [6])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X and $T:C\to C$ be a nonexpansive mapping, then $I-T$ is demiclosed at zero.

Lemma 1.4 (see [7])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X which satisfies the Opial condition and $T:C\to C$ be a quasi-nonexpansive mapping with $F(T)\ne \mathrm{\varnothing }$. If $I-T$ is demiclosed at zero, then for any $x_0\in C$, the normal Mann iteration $\{x_n\}$ defined by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,\mathrm{\forall }n\ge 0,$$

converges weakly to a fixed point of T, where $\{{\alpha }_n\}\subset [0,1]$ and ${\sum }_{n=0}^{\mathrm{\infty }}min\{{\alpha }_n,(1-{\alpha }_n)\}=\mathrm{\infty }$.

Lemma 1.5 (see [7])

Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X whose dual space $X^{\ast }$ satisfies Kadec-Klee property and $T:C\to C$ be a nonexpansive mapping with $F(T)\ne \mathrm{\varnothing }$. Then, for any $x_0\in C$, the normal Mann iteration $\{x_n\}$ defined by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,\mathrm{\forall }n\ge 0,$$

converges weakly to a fixed point of T, where $\{{\alpha }_n\}\subset [0,1]$ and ${\sum }_{n=0}^{\mathrm{\infty }}min\{{\alpha }_n,(1-{\alpha }_n)\}=\mathrm{\infty }$.

Now we are in a position to state and prove the main results in this paper.

2 Main results

Theorem 2.1 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X with Fréchet differential norm. Let $T:C\to C$ be a Hu-type k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients $p,k<min\{1,2^{-(p-2)}{c}_n\}$ and $F(T)\ne \mathrm{\varnothing }$. Assume that a real sequence $\{{\alpha }_n\}$ in $[0,1]$ satisfies the conditions:

• (i) $0\le {\alpha }_n\le \alpha =1-(k{2}^{p-2}/c_p)$, $n\ge 0$;

• (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n[(1-{\alpha }_n)2^{2-p}{c}_p-k]=\mathrm{\infty }$.

For any $x_0\in C$, the normal Mann iterative sequence $\{x_n\}$ is defined by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,n\ge 0.$$

(2.1)

Then the sequence $\{x_n\}$ defined by (2.1) converges weakly to a fixed point of T.

Proof Let $T_{\alpha }$ be given as in Lemma 1.1. Then $T_{\alpha }:C\to C$ is a nonexpansive mapping with $F(T_{\alpha })=F(T)$. Set ${\beta }_n=\frac{\alpha -{\alpha }_n}{\alpha }$. Then (2.1) reduces to $x_{n+1}={\beta }_n{x}_n+(1-{\beta }_n)T_{\alpha }{x}_n$.

We note that

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\beta }_n(1-{\beta }_n)& =\frac{1}{{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n(\alpha -{\alpha }_n)\\ =\frac{1}{{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n(1-{\alpha }_n-\frac{k{2}^{p-2}}{c_p})\\ =\frac{2^{p-2}}{{\alpha }^2{c}_p}\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n[(1-{\alpha }_n)2^{p-2}{c}_p-k]\\ =\mathrm{\infty }.\end{array}$$

By using Theorem R, we conclude that $\{x_n\}$ converges weakly to a fixed point of $T_{\alpha }$, and of T. The proof is complete. □

Remark 2.2 Theorem 2.1 relaxes the iterative parameters in Theorem H and our proof method is also quite concise.

Theorem 2.3 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X which satisfies the Opial condition. Let $T:C\to C$ be a Hu-type k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients $p,k<min\{1,2^{-(p-2)}{c}_p\}$ and $F(T)\ne \mathrm{\varnothing }$. Assume that the real sequence $\{{\alpha }_n\}$ in $[0,1]$ satisfies the conditions:

• (i) $0\le {\alpha }_n\le \alpha =1-(k{2}^{p-2}/c_p)$, $n\ge 0$;

• (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n[(1-{\alpha }_n)2^{2-p}{c}_p-k]=\mathrm{\infty }$.

For any $x_0\in C$, the normal Mann iteration $\{x_n\}$ is defined by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,n\ge 0.$$

(2.2)

Then the sequence $\{x_n\}$ defined by (2.2) converges weakly to the fixed point of T.

Proof Let $T_{\alpha }$ be given as in Lemma 1.1. Then $T_{\alpha }:C\to C$ is a nonexpansive mapping with $F(T_{\alpha })=F(T)$. Set ${\beta }_n=\frac{\alpha -{\alpha }_n}{\alpha }$. Then (2.2) reduces to $x_{n+1}={\beta }_n{x}_n+(1-{\beta }_n)T_{\alpha }{x}_n$. As shown in Theorem 2.1, ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n(1-{\beta }_n)=\mathrm{\infty }$. By Lemma 1.3, $I-T_{\alpha }$ is demiclosed at zero. By Lemma 1.4, we conclude that $\{x_n\}$ converges weakly to a fixed point of $T_{\alpha }$, and of T. The proof is complete. □

Theorem 2.4 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X with the dual space $X^{\ast }$ satisfying the Kadec-Klee property. Let $T:C\to C$ be a Hu-type k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients $p,k<min\{1,2^{-(p-2)}{c}_p\}$ and $F(T)\ne \mathrm{\varnothing }$. Assume that the real sequence $\{{\alpha }_n\}$ in $[0,1]$ satisfies the conditions:

• (i) $0\le {\alpha }_n\le \alpha =1-(k{2}^{p-2}/c_p)$, $n\ge 0$;

• (ii) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n[(1-{\alpha }_n)2^{2-p}{c}_p-k]=\mathrm{\infty }$.

For any $x_0\in C$, the normal Mann iteration $\{x_n\}$ is defined by

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,n\ge 0.$$

(2.3)

Then the sequence $\{x_n\}$ defined by (2.3) converges weakly to a fixed point of T.

Proof Let $T_{\alpha }$ be given as in Lemma 1.1. Then $T_{\alpha }:C\to C$ is a nonexpansive mapping with $F(T_{\alpha })=F(T)$. Set ${\beta }_n=\frac{\alpha -{\alpha }_n}{\alpha }$. Then (2.3) reduces to $x_{n+1}={\beta }_n{x}_n+(1-{\beta }_n)T_{\alpha }{x}_n$. As shown in Theorem 2.1, ${\sum }_{n=0}^{\mathrm{\infty }}{\beta }_n(1-{\beta }_n)=\mathrm{\infty }$. By using Lemma 1.5, $\{x_n\}$ defined by (2.3) converges weakly to a fixed point of $T_{\alpha }$, and of T. The proof is complete. □