Strong convergence of approximated iterations for asymptoticallypseudocontractive mappings

Abstract

The asymptotically nonexpansive mappings have been introduced by Goebel and Kirkin 1972. Since then, a large number of authors have studied the weak and strongconvergence problems of the iterative algorithms for such a class of mappings.It is well known that the asymptotically nonexpansive mappings is a propersubclass of the class of asymptotically pseudocontractive mappings. In thepresent paper, we devote our study to the iterative algorithms for finding thefixed points of asymptotically pseudocontractive mappings in Hilbert spaces. Wesuggest an iterative algorithm and prove that it converges strongly to the fixedpoints of asymptotically pseudocontractive mappings.

MSC: 47J25, 47H09, 65J15.

1 Introduction

Let H be a real Hilbert space with inner product $〈\cdot ,\cdot 〉$ and norm $\parallel \cdot \parallel$, respectively. Let C be a nonempty, closed,and convex subset of H. Let $T:C\to C$ be a nonlinear mapping. We use$F(T)$ to denote the fixed point set of T.

Recall that T is said to be L-Lipschitzian if there exists$L>0$ such that

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

for all $x,y\in C$. In this case, if $L<1$, then we call T anL-contraction. If $L=1$, we call T nonexpansive. T is said to be asymptotically nonexpansiveif there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ such that

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

(1.1)

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

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] in 1972. They proved that, if C is a nonempty bounded, closed,and convex subset of a uniformly convex Banach space E, then everyasymptotically nonexpansive self-mapping T of C has a fixed point.Further, the set $F(T)$ of fixed points of T is closed and convex.

Since then, a large number of authors have studied the following algorithms for theiterative approximation of fixed points of asymptotically nonexpansive mappings(see, e.g., [2–29] and the references therein).

• (A) The modified Mann iterative algorithm. For arbitrary $x_0\in C$, the modified Mann iteration generates a sequence $\{x_n\}$ by

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

  (1.2)

• (B) The modified Ishikawa iterative algorithm. For arbitrary $x_0\in C$, the modified Ishikawa iteration generates a sequence $\{x_n\}$ by

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

  (1.3)

• (C) The CQ algorithm. For arbitrary $x_0\in C$, the CQ algorithm generates a sequence $\{x_n\}$ by

  $$\{\begin{array}{c}{y}_n={\alpha }_n{x}_n+(1-{\alpha }_n)T^n{x}_n,\hfill \\ C_n=\{z\in C:{\parallel y_n-z\parallel }^2\le {\parallel x_n-z\parallel }^2+{\theta }_n\},\hfill \\ Q_n=\{z\in C:〈x_n-z,x_0-x_n〉\ge 0\},\hfill \\ x_{n+1}=P_{C_n\cap Q_n}(x_0),\mathrm{\forall }n\ge 1.\hfill \end{array}$$

  (1.4)

An important class of asymptotically pseudocontractive mappings generalizing theclass of asymptotically nonexpansive mapping has been introduced and studied by Schuin 1991; see [19].

Recall that $T:C\to C$ is called an asymptotically pseudocontractivemapping if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ for which the following inequality holds:

$$〈T^{n}x-T^{n}y,x-y〉\le k_n{\parallel x-y\parallel }^2$$

(1.5)

for all $x,y\in C$ and all $n\ge 1$. It is clear that (1.5) is equivalent to

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

(1.6)

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

Recall also that T is called uniformly L-Lipschitzian if there exists $L>0$ such that

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

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

Now, we know that the class of asymptotically nonexpansive mappings is a propersubclass of the class of asymptotically pseudocontractive mappings. If we define amapping $T:[0,1]\to [0,1]$ by the formula $Tx={(1-x^{\frac{2}{3}})}^{\frac{3}{2}}$, then we can verify that T is asymptoticallypseudocontractive but it is not asymptotically nonexpansive.

In order to approximate the fixed point of asymptotically pseudocontractive mappings,the following two results are interesting.

One is due to Schu [19], who proved the following convergence theorem.

Theorem 1.1 Let H be a Hilbert space, C be a nonempty closed bounded and convex subset of H. Let T be a completely continuous, uniformly L-Lipschitzian and asymptotically pseudocontractiveself-mapping of C with $\{k_n\}\subset [1,\mathrm{\infty })$ and ${\sum }_{n=1}^{\mathrm{\infty }}(q_n^2-1)<\mathrm{\infty }$, where $q_n=(2k_n-1)$ for all $n\ge 1$. Let $\{{\alpha }_n\}\subset [0,1]$ and $\{{\beta }_n\}\subset [0,1]$ be two sequences satisfying $0<ϵ\le {\alpha }_n\le {\beta }_n\le b<L^{-2}(\sqrt{1+L^2}-1)$ for all $n\ge 1$. Then the sequence $\{x_n\}$ generated by the modified Ishikawa iteration (1.3) converges stronglyto some fixed point of T.

Another one is due to Chidume and Zegeye [30] who introduced the following algorithm in 2003.

Let a sequence $\{x_n\}$ be generated from $x_1\in C$ by

$$x_{n+1}={\lambda }_n{\theta }_n{x}_1+(1-{\lambda }_n-{\lambda }_n{\theta }_n)x_n+{\lambda }_n{T}^n{x}_n,\mathrm{\forall }n\ge 1,$$

(1.7)

where the sequences $\{{\lambda }_n\}$ and $\{{\theta }_n\}$ satisfy

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n{\theta }_n=\mathrm{\infty }$ and ${\lambda }_n(1+{\theta }_n)\le 1$;

• (ii) $\frac{{\lambda }_n}{{\theta }_n}\to 0$, ${\theta }_n\to 0$ and $\frac{(\frac{{\theta }_{n-1}}{{\theta }_n}-1)}{{\lambda }_n{\theta }_n}\to 0$;

• (iii) $\frac{k_n-k_{n-1}}{{\lambda }_n{\theta }_n^2}\to 0$;

• (iv) $\frac{k_n-1}{{\theta }_n}\to 0$.

They gave the strong convergence analysis for the above algorithm (1.7) with somefurther assumptions on the mapping T in Banach spaces.

Remark 1.2 Note that there are some additional assumptions imposed on theunderlying space C and the mapping T in the above two results. In(1.7), the parameter control is also restricted.

Inspired by the results above, the main purpose of this article is to construct aniterative method for finding the fixed points of asymptotically pseudocontractivemappings. We construct an algorithm which is based on the algorithms (1.2) and(1.7). Under some mild conditions, we prove that the suggested algorithm convergesstrongly to the fixed point of asymptotically pseudocontractive mappingT.

2 Preliminaries

It is well known that in a real Hilbert space H, the following inequalityand equality hold:

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉,\mathrm{\forall }x,y\in H$$

(2.1)

and

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

(2.2)

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

Lemma 2.1 ([31])

Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a uniformly L-Lipschtzian and asymptotically pseudocontraction. Then $I-T$ is demiclosed at zero.

Lemma 2.2 ([32])

Let $\{r_n\}$ be a sequence of real numbers. Assume $\{r_n\}$ does not decrease at infinity, that is, there exists at leasta subsequence $\{r_{n_k}\}$ of $\{r_n\}$ such that $r_{n_k}\le r_{n_k+1}$ for all $k\ge 0$. For every $n\ge N$, define an integer sequence $\{\tau (n)\}$ as

$$\tau (n)=max\{i\le n:r_{n_i}<r_{n_i+1}\}.$$

Then $\tau (n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$, and for all $n\ge N$

$$max\{r_{\tau (n)},r_n\}\le r_{\tau (n)+1}.$$

Lemma 2.3 ([33])

Assume that $\{a_n\}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\delta }_n)a_n+{\xi }_n,$$

where $\{{\delta }_n\}$ is a sequence in $(0,1)$ and $\{{\xi }_n\}$ is a sequence such that

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\delta }_n=\mathrm{\infty }$;

• (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\xi }_n}{{\delta }_n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\xi }_n|<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

3 Main results

Now we introduce the following iterative algorithm for asymptoticallypseudocontractive mappings.

Let C be a nonempty, closed, and convex subset of a real Hilbert spaceH. Let $T:C\to C$ be a uniformly L-Lipschitzian asymptoticallypseudocontractive mapping satisfying ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$. Let $f:C\to C$ be a ρ-contractive mapping. Let$\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be three real number sequences in$[0,1]$.

Algorithm 3.1 For $x_0\in C$, define the sequence $\{x_n\}$ by

$$\{\begin{array}{c}{y}_n=(1-{\gamma }_n)x_n+{\gamma }_n{T}^n{x}_n,\hfill \\ x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n-{\beta }_n)x_n+{\beta }_n{T}^n{y}_n,\mathrm{\forall }n\ge 1.\hfill \end{array}$$

(3.1)

Next, we prove our main result as follows.

Theorem 3.2 Suppose that $F(T)\ne \mathrm{\varnothing }$. Assume the sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ satisfy the following conditions:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) ${\alpha }_n+{\beta }_n\le {\gamma }_n$ and $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n$;

• (iii) $0<a\le {\gamma }_n\le b<\frac{2}{\sqrt{{(1+k_n)}^2+4L^2}+1+k_n}$ for all $n\ge 1$.

Then the sequence $\{x_n\}$ defined by (3.1) converges strongly to $u=P_{F(T)}f(u)$, which is the unique solution of the variationalinequality $〈(I-f)x^{\ast },x-x^{\ast }〉\ge 0$ for all $x\in F(T)$.

Proof From (3.1), we have

$$\begin{array}{c}\parallel x_{n+1}-u\parallel \hfill \\ =\parallel {\alpha }_{n}f(x_n)+(1-{\alpha }_n-{\beta }_n)x_n+{\beta }_n{T}^n{y}_n-u\parallel \hfill \\ \le \parallel {\alpha }_n(f(x_n)-u)+(1-{\alpha }_n-{\beta }_n)(x_n-u)+{\beta }_n(T^n{y}_n-u)\parallel \hfill \\ =\parallel {\alpha }_n(f(x_n)-u)+(1-{\alpha }_n)(\frac{1-{\alpha }_n-{\beta }_n}{1-{\alpha }_n}(x_n-u)+\frac{{\beta }_n}{1-{\alpha }_n}(T^n{y}_n-u))\parallel \hfill \\ \le (1-{\alpha }_n)\parallel \frac{(1-{\alpha }_n-{\beta }_n)(x_n-u)}{1-{\alpha }_n}+\frac{{\beta }_n(T^n{y}_n-u)}{1-{\alpha }_n}\parallel +{\alpha }_n\parallel f(x_n)-u\parallel .\hfill \end{array}$$

(3.2)

Using the equality (2.2), we get

$$\begin{array}{c}{\parallel \frac{(1-{\alpha }_n-{\beta }_n)(x_n-u)}{1-{\alpha }_n}+\frac{{\beta }_n(T^n{y}_n-u)}{1-{\alpha }_n}\parallel }^2\hfill \\ =\frac{1-{\alpha }_n-{\beta }_n}{1-{\alpha }_n}{\parallel x_n-u\parallel }^2+\frac{{\beta }_n}{1-{\alpha }_n}{\parallel T^n{y}_n-u\parallel }^2\hfill \\ -\frac{{\beta }_n(1-{\alpha }_n-{\beta }_n)}{{(1-{\alpha }_n)}^2}{\parallel x_n-T^n{y}_n\parallel }^2.\hfill \end{array}$$

(3.3)

Picking up $y=u$ in (1.6) we deduce

$${\parallel T^{n}x-u\parallel }^2\le k_n{\parallel x-u\parallel }^2+{\parallel x-T^{n}x\parallel }^2$$

(3.4)

for all $x\in C$.

From (3.1), (3.4), and (2.2), we obtain

$$\begin{array}{c}{\parallel T^n{y}_n-u\parallel }^2\hfill \\ \le k_n{\parallel y_n-u\parallel }^2+{\parallel y_n-T^n{y}_n\parallel }^2\hfill \\ =k_n{\parallel (1-{\gamma }_n)x_n+{\gamma }_n{T}^n{x}_n-u\parallel }^2+{\parallel (1-{\gamma }_n)x_n+{\gamma }_n{T}^n{x}_n-T^n{y}_n\parallel }^2\hfill \\ =k_n{\parallel (1-{\gamma }_n)(x_n-u)+{\gamma }_n(T^n{x}_n-u)\parallel }^2\hfill \\ +{\parallel (1-{\gamma }_n)(x_n-T^n{y}_n)+{\gamma }_n(T^n{x}_n-T^n{y}_n)\parallel }^2\hfill \\ =k_n[(1-{\gamma }_n){\parallel x_n-u\parallel }^2+{\gamma }_n{\parallel T^n{x}_n-u\parallel }^2-{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2]\hfill \\ +(1-{\gamma }_n){\parallel x_n-T^n{y}_n\parallel }^2+{\gamma }_n{\parallel T^n{x}_n-T^n{y}_n\parallel }^2-{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2\hfill \\ \le k_n[(1-{\gamma }_n){\parallel x_n-u\parallel }^2+{\gamma }_n(k_n{\parallel x_n-u\parallel }^2+{\parallel x_n-T^n{x}_n\parallel }^2)\hfill \\ -{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2]+(1-{\gamma }_n){\parallel x_n-T^n{y}_n\parallel }^2\hfill \\ +{\gamma }_n{\parallel T^n{x}_n-T^n{y}_n\parallel }^2-{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2.\hfill \end{array}$$

(3.5)

By (3.1), we have

$$\parallel x_n-y_n\parallel ={\gamma }_n\parallel x_n-T^n{x}_n\parallel .$$

(3.6)

Noting that T is uniformly L-Lipschitzian, from (3.5) and (3.6),we deduce

$$\begin{array}{c}{\parallel T^n{y}_n-u\parallel }^2\hfill \\ \le k_n[(1-{\gamma }_n){\parallel x_n-u\parallel }^2+{\gamma }_n(k_n{\parallel x_n-u\parallel }^2+{\parallel x_n-T^n{x}_n\parallel }^2)\hfill \\ -{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2]+(1-{\gamma }_n){\parallel x_n-T^n{y}_n\parallel }^2\hfill \\ +{\gamma }_n{L}^2{\parallel x_n-y_n\parallel }^2-{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2\hfill \\ =k_n[(1-{\gamma }_n){\parallel x_n-u\parallel }^2+{\gamma }_n(k_n{\parallel x_n-u\parallel }^2+{\parallel x_n-T^n{x}_n\parallel }^2)\hfill \\ -{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2]+(1-{\gamma }_n){\parallel x_n-T^n{y}_n\parallel }^2\hfill \\ +{\gamma }_n^3{L}^2{\parallel x_n-T^n{x}_n\parallel }^2-{\gamma }_n(1-{\gamma }_n){\parallel x_n-T^n{x}_n\parallel }^2\hfill \\ =[1+(k_n{\gamma }_n+1)(k_n-1)]{\parallel x_n-u\parallel }^2+(1-{\gamma }_n){\parallel x_n-T^n{y}_n\parallel }^2\hfill \\ -{\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2.\hfill \end{array}$$

(3.7)

By condition (iii), we know that ${\gamma }_n\le b<\frac{2}{\sqrt{{(1+k_n)}^2+4L^2}+k_n+1}$ for all n. Then we deduce that$1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2>0$ for all $n\ge 0$.

Therefore, from (3.7), we derive

$$\begin{array}{rcl}{\parallel T^n{y}_n-u\parallel }^2& \le & [1+(k_n{\gamma }_n+1)(k_n-1)]{\parallel x_n-u\parallel }^2\\ +(1-{\gamma }_n){\parallel x_n-T^n{y}_n\parallel }^2.\end{array}$$

(3.8)

Note that ${\beta }_n\le 1-{\alpha }_n$ and substituting (3.8) to (3.3), we obtain

$$\begin{array}{c}{\parallel \frac{(1-{\alpha }_n-{\beta }_n)(x_n-u)}{1-{\alpha }_n}+\frac{{\beta }_n(T^n{y}_n-u)}{1-{\alpha }_n}\parallel }^2\hfill \\ \le \frac{{\beta }_n}{1-{\alpha }_n}\{[1+(k_n{\gamma }_n+1)(k_n-1)]{\parallel x_n-u\parallel }^2+(1-{\gamma }_n){\parallel x_n-T^n{y}_n\parallel }^2\}\hfill \\ +\frac{1-{\alpha }_n-{\beta }_n}{1-{\alpha }_n}{\parallel x_n-u\parallel }^2-\frac{{\beta }_n(1-{\alpha }_n-{\beta }_n)}{{(1-{\alpha }_n)}^2}{\parallel x_n-T^n{y}_n\parallel }^2\hfill \\ =[1+\frac{{\beta }_n}{1-{\alpha }_n}(k_n{\gamma }_n+1)(k_n-1)]{\parallel x_n-u\parallel }^2+\frac{{\beta }_n({\alpha }_n+{\beta }_n-{\gamma }_n)}{{(1-{\alpha }_n)}^2}{\parallel x_n-T^n{y}_n\parallel }^2\hfill \\ \le [1+\frac{{\beta }_n}{1-{\alpha }_n}(k_n{\gamma }_n+1)(k_n-1)]{\parallel x_n-u\parallel }^2\hfill \\ \le [1+(k_n{\gamma }_n+1)(k_n-1)]{\parallel x_n-u\parallel }^2.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\parallel \frac{(1-{\alpha }_n-{\beta }_n)(x_n-u)}{1-{\alpha }_n}+\frac{{\beta }_n(T^n{y}_n-u)}{1-{\alpha }_n}\parallel \hfill \\ \le \sqrt{1+(k_n{\gamma }_n+1)(k_n-1)}\parallel x_n-u\parallel \hfill \\ \le [1+(k_n{\gamma }_n+1)(k_n-1)]\parallel x_n-u\parallel .\hfill \end{array}$$

(3.9)

Since $k_n\to 1$, without loss of generality, we assume that$k_n\le 2$ for all $n\ge 1$. It follows from (3.2) and (3.9) that

$$\begin{array}{c}\parallel n{x}_{n+1}-u\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-u\parallel +(1-{\alpha }_n)[1+(k_n{\gamma }_n+1)(k_n-1)]\parallel n{x}_n-u\parallel \hfill \\ \le {\alpha }_n\parallel f(x_n)-f(u)\parallel +{\alpha }_n\parallel f(u)-u\parallel \hfill \\ +(1-{\alpha }_n)[1+(k_n{\gamma }_n+1)(k_n-1)]\parallel x_n-u\parallel \hfill \\ \le {\alpha }_n\rho \parallel x_n-u\parallel +{\alpha }_n\parallel f(u)-u\parallel \hfill \\ +(1-{\alpha }_n)[1+(k_n{\gamma }_n+1)(k_n-1)]\parallel x_n-u\parallel \hfill \\ \le {\alpha }_n\parallel f(u)-u\parallel +[1-(1-\rho ){\alpha }_n]\parallel x_n-u\parallel \hfill \\ +(k_n{\gamma }_n+1)(k_n-1)\parallel x_n-u\parallel \hfill \\ \le (1-\rho ){\alpha }_n\frac{\parallel f(u)-u\parallel }{1-\rho }+[1-(1-\rho ){\alpha }_n]\parallel x_n-u\parallel +3(k_n-1)\parallel x_n-u\parallel .\hfill \end{array}$$

An induction induces

$$\begin{array}{rl}\parallel x_{n+1}-u\parallel & \le [1+3(k_n-1)]max\{\parallel x_n-u\parallel ,\frac{\parallel f(u)-u\parallel }{1-\rho }\}\\ \le \prod _{j=1}^n[1+3(k_j-1)]max\{\parallel x_0-u\parallel ,\frac{\parallel f(u)-u\parallel }{1-\rho }\}.\end{array}$$

This implies that the sequence $\{x_n\}$ is bounded by the condition ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$.

From (2.1) and (3.1), we have

$$\begin{array}{c}{\parallel x_{n+1}-u\parallel }^2\hfill \\ ={\parallel (1-{\alpha }_n)(x_n-u)-{\beta }_n(x_n-T^n{y}_n)+{\alpha }_n(f(x_n)-u)\parallel }^2\hfill \\ \le {\parallel (1-{\alpha }_n)(x_n-u)-{\beta }_n(x_n-T^n{y}_n)\parallel }^2+2{\alpha }_n〈f(x_n)-u,x_{n+1}-u〉\hfill \\ ={\parallel (1-{\alpha }_n)(x_n-u)\parallel }^2-2{\beta }_n(1-{\alpha }_n)〈x_n-T^n{y}_n,x_n-u〉\hfill \\ +{\beta }_n^2{\parallel x_n-T^n{y}_n\parallel }^2+2{\alpha }_n〈f(x_n)-u,x_{n+1}-u〉.\hfill \end{array}$$

(3.10)

From (3.7), we deduce

$$\begin{array}{c}2〈x_n-T^n{y}_n,x_n-u〉\hfill \\ \ge {\gamma }_n{\parallel x_n-T^n{y}_n\parallel }^2+{\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2\hfill \\ -(k_n{\gamma }_n+1)(k_n-1){\parallel x_n-u\parallel }^2\hfill \\ \ge {\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2+{\gamma }_n{\parallel x_n-T^n{y}_n\parallel }^2.\hfill \end{array}$$

(3.11)

By condition (ii), we have $1\ge {\gamma }_n\ge {\alpha }_n+{\beta }_n\ge {\alpha }_n{\gamma }_n+{\beta }_n$ for all $n\ge 1$. Hence, by (3.10) and (3.11), we get

$$\begin{array}{r}{\parallel x_{n+1}-u\parallel }^2\\ \le (1-{\alpha }_n){\parallel x_n-u\parallel }^2-{\beta }_n(1-{\alpha }_n){\gamma }_n{\parallel x_n-T^n{y}_n\parallel }^2+{\beta }_n^2{\parallel x_n-T^n{y}_n\parallel }^2\\ -{\beta }_n(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2\\ +2{\alpha }_n〈f(x_n)-u,x_{n+1}-u〉\\ \le (1-{\alpha }_n){\parallel x_n-u\parallel }^2+2{\alpha }_n〈f(x_n)-u,x_{n+1}-u〉\\ -{\beta }_n(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2.\end{array}$$

(3.12)

It follows that

$$\begin{array}{c}{\parallel x_{n+1}-u\parallel }^2-{\parallel x_n-u\parallel }^2+{\beta }_n(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2\hfill \\ \le {\alpha }_n(2〈f(x_n)-u,x_{n+1}-u〉-{\parallel x_n-u\parallel }^2).\hfill \end{array}$$

Since $\{x_n\}$ and $\{f(x_n)\}$ are bounded, there exists $M>0$ such that ${sup}_n\{2〈f(x_n)-u,x_{n+1}-u〉-{\parallel x_n-u\parallel }^2\}\le M$. So,

$$\begin{array}{c}{\beta }_n(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2\hfill \\ +{\parallel x_{n+1}-u\parallel }^2-{\parallel x_n-u\parallel }^2\le {\alpha }_{n}M.\hfill \end{array}$$

(3.13)

Next, we consider two possible cases.

Case 1. Assume there exists some integer $m>0$ such that $\{\parallel x_n-u\parallel \}$ is decreasing for all $n\ge m$.

In this case, we know that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-u\parallel$ exists. From (3.13), we deduce

$$\begin{array}{c}{\beta }_n(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2){\parallel x_n-T^n{x}_n\parallel }^2\hfill \\ \le {\parallel x_n-u\parallel }^2-{\parallel x_{n+1}-u\parallel }^2+M{\alpha }_n.\hfill \end{array}$$

(3.14)

By conditions (ii) and (iii), we have ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n(1-{\alpha }_n){\gamma }_n(1-{\gamma }_n-k_n{\gamma }_n-{\gamma }_n^2{L}^2)>0$. Thus, from (3.14), we get

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

(3.15)

It follows from (3.6) and (3.15) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-y_n\parallel =0.$$

(3.16)

Since T is uniformly L-Lipschitzian, we have$\parallel T^n{y}_n-T^n{x}_n\parallel \le L\parallel x_n-y_n\parallel$. This together with (3.16) implies that

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

(3.17)

Note that

$$\parallel x_n-T^n{y}_n\parallel \le \parallel x_n-T^n{x}_n\parallel +\parallel T^n{x}_n-T^n{y}_n\parallel .$$

(3.18)

Combining (3.15), (3.17), and (3.18), we have

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

(3.19)

From (3.1), we deduce

$$\parallel x_{n+1}-x_n\parallel \le {\alpha }_n\parallel f(x_n)-x_n\parallel +{\beta }_n\parallel T^n{y}_n-x_n\parallel .$$

Therefore,

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

(3.20)

Since T is uniformly L-Lipschitzian, we derive

$$\begin{array}{c}\parallel x_n-T{x}_n\parallel \hfill \\ \le \parallel x_n-T^n{x}_n\parallel +\parallel T^n{x}_n-T{x}_n\parallel \hfill \\ \le \parallel x_n-T^n{x}_n\parallel +L\parallel T^{n-1}{x}_n-x_n\parallel \hfill \\ \le \parallel x_n-T^n{x}_n\parallel +L\parallel T^{n-1}{x}_n-T^{n-1}{x}_{n-1}\parallel \hfill \\ +L\parallel T^{n-1}{x}_{n-1}-x_{n-1}\parallel +L\parallel x_{n-1}-x_n\parallel \hfill \\ \le \parallel x_n-T^n{x}_n\parallel +(L^2+L)\parallel x_n-x_{n-1}\parallel +L\parallel T^{n-1}{x}_{n-1}-x_{n-1}\parallel .\hfill \end{array}$$

(3.21)

By (3.15), (3.20), and (3.21), we have immediately

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

(3.22)

Since $\{x_n\}$ is bounded, there exists a subsequence$\{x_{n_k}\}$ of $\{x_n\}$ satisfying

$$x_{n_k}\to \tilde{x}\in C$$

and

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f(u)-u,x_n-u〉=\underset{k\to \mathrm{\infty }}{lim}〈f(u)-u,x_{n_k}-u〉.$$

Thus, we use the demiclosed principle of T (Lemma 2.1) and (3.22) todeduce

$$\tilde{x}\in F(T).$$

So,

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f(u)-u,x_n-u〉& =\underset{k\to \mathrm{\infty }}{lim}〈f(u)-u,x_{n_k}-u〉\\ =〈f(u)-u,\tilde{x}-u〉\\ \le 0.\end{array}$$

Returning to (3.12) to obtain

$$\begin{array}{rcl}{\parallel x_{n+1}-u\parallel }^2& \le & (1-{\alpha }_n){\parallel x_n-u\parallel }^2+2{\alpha }_n〈f(x_n)-u,x_{n+1}-u〉\\ =& (1-{\alpha }_n){\parallel x_n-u\parallel }^2+2{\alpha }_n〈f(x_n)-f(u),x_{n+1}-u〉\\ +2{\alpha }_n〈f(u)-u,x_{n+1}-u〉\\ \le & (1-{\alpha }_n){\parallel x_n-u\parallel }^2+2{\alpha }_n\rho \parallel x_n-u\parallel \parallel x_{n+1}-u\parallel \\ +2{\alpha }_n〈f(u)-u,x_{n+1}-u〉\\ \le & (1-{\alpha }_n){\parallel x_n-u\parallel }^2+{\alpha }_n\rho ({\parallel x_n-u\parallel }^2+{\parallel x_{n+1}-u\parallel }^2)\\ +2{\alpha }_n〈f(u)-u,x_{n+1}-u〉.\end{array}$$

It follows that

$$\begin{array}{rcl}{\parallel x_{n+1}-u\parallel }^2& \le & [1-(1-\rho ){\alpha }_n]{\parallel x_n-u\parallel }^2\\ +\frac{2{\alpha }_n}{1-{\alpha }_n\rho }〈f(u)-u,x_{n+1}-u〉.\end{array}$$

(3.23)

In Lemma 2.3, we take $a_n={\parallel x_{n+1}-u\parallel }^2$, ${\delta }_n=(1-\rho ){\alpha }_n$, and ${\xi }_n=\frac{2{\alpha }_n}{1-{\alpha }_n\rho }〈f(u)-u,x_{n+1}-u〉$. We can easily check that ${\sum }_{n=1}^{\mathrm{\infty }}{\delta }_n=\mathrm{\infty }$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\xi }_n}{{\delta }_n}\le 0$. Thus, we deduce that $x_n\to u$.

Case 2. Assume there exists an integer $n_0$ such that $\parallel x_{n_0}-u\parallel \le \parallel x_{n_0+1}-u\parallel$. At this case, we set ${\omega }_n=\{\parallel x_n-u\parallel \}$. Then we have ${\omega }_{n_0}\le {\omega }_{n_0+1}$. Define an integer sequence $\{{\tau }_n\}$ for all $n\ge n_0$ as follows:

$$\tau (n)=max\{l\in \mathbb{N}|n_0\le l\le n,{\omega }_l\le {\omega }_{l+1}\}.$$

It is clear that $\tau (n)$ is a non-decreasing sequence satisfying

$$\underset{n\to \mathrm{\infty }}{lim}\tau (n)=\mathrm{\infty }$$

and

$${\omega }_{\tau (n)}\le {\omega }_{\tau (n)+1}$$

for all $n\ge n_0$. From (3.22), we get

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{\tau (n)}-T{x}_{\tau (n)}\parallel =0.$$

This implies that ${\omega }_w(x_{\tau (n)})\subset F(T)$. Thus, we obtain

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈f(u)-u,x_{\tau (n)}-u〉\le 0.$$

(3.24)

Since ${\omega }_{\tau (n)}\le {\omega }_{\tau (n)+1}$, we have from (3.23) that

$${\omega }_{\tau (n)}^2\le {\omega }_{\tau (n)+1}^2\le [1-(1-\rho ){\alpha }_{\tau (n)}]{\omega }_{\tau (n)}^2+\frac{2{\alpha }_{\tau (n)}}{1-{\alpha }_{\tau (n)}\rho }〈f(u)-u,x_{\tau (n)+1}-u〉.$$

It follows that

$${\omega }_{\tau (n)}^2\le \frac{2}{(1-{\alpha }_{\tau (n)}\rho )(1-\rho )}〈f(u)-u,x_{\tau (n)+1}-u〉.$$

(3.25)

Combining (3.24) and (3.25), we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\omega }_{\tau (n)}\le 0,$$

and hence

$$\underset{n\to \mathrm{\infty }}{lim}{\omega }_{\tau (n)}=0.$$

(3.26)

From (3.23), we obtain

$$\begin{array}{rcl}{\parallel x_{\tau (n)+1}-u\parallel }^2& \le & [1-(1-\rho ){\alpha }_{\tau }(n)]{\parallel x_{\tau (n)}-u\parallel }^2\\ +\frac{2{\alpha }_{\tau (n)}}{1-{\alpha }_{\tau (n)}\rho }〈f(u)-u,x_{\tau (n)+1}-u〉.\end{array}$$

It follows that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\omega }_{\tau (n)+1}\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\omega }_{\tau (n)}.$$

This together with (3.26) imply that

$$\underset{n\to \mathrm{\infty }}{lim}{\omega }_{\tau (n)+1}=0.$$

Applying Lemma 2.2 to get

$$0\le {\omega }_n\le max\{{\omega }_{\tau (n)},{\omega }_{\tau (n)+1}\}.$$

Therefore, ${\omega }_n\to 0$. That is, $x_n\to u$. The proof is completed. □

Since the class of asymptotically nonexpansive mappings is a proper subclass of theclass of asymptotically pseudocontractive mappings and asymptotically nonexpansivemapping T is L-Lipschitzian with $L={sup}_n{k}_n$. Thus, from Theorem 3.2, we get the followingcorollary.

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be an asymptotically nonexpansive mapping satisfying ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$. Suppose that $F(T)\ne \mathrm{\varnothing }$. Let $f:C\to C$ be a ρ-contractive mapping. Let $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ be three real number sequences in $[0,1]$. Assume the sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, and $\{{\gamma }_n\}$ satisfy the following conditions:

• (i) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) ${\alpha }_n+{\beta }_n\le {\gamma }_n$ and $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n$;

• (iii) $0<a\le {\gamma }_n\le b<\frac{2}{\sqrt{{(1+L)}^2+4L^2}+1+L}$ for all $n\ge 1$, where $L={sup}_n{k}_n$.

Then the sequence $\{x_n\}$ defined by (3.1) converges strongly to $u=P_{F(T)}f(u)$, which is the unique solution of the variationalinequality $〈(I-f)x^{\ast },x-x^{\ast }〉\ge 0$ for all $x\in F(T)$.

Remark 3.4 Our Theorem 3.2 does not impose any boundedness or compactnessassumption on the space C or the mapping T. The parameter controlconditions (i)-(iii) are mild.

Remark 3.5 Our Corollary 3.3 is also a new result.

4 Conclusion

This work contains our dedicated study to develop and improve iterative algorithmsfor finding the fixed points of asymptotically pseudocontractive mappings in Hilbertspaces. We introduced our iterative algorithm for this class of problems, and wehave proven its strong convergence. This study is motivated by relevant applicationsfor solving classes of real-world problems, which give rise to mathematical modelsin the sphere of nonlinear analysis.