Strong convergence of a relaxed three-step iterative algorithm for countable families of pseudocontractions

Abstract

An up-to-date method for the approximation of common fixed points of countable families of nonlinear operators is introduced, by which a relaxed three-step iterative algorithm is developed for the class of pseudocontractive mappings, and a strong convergence theorem is established in the framework of Hilbert spaces. Since there is no need to impose uniformity assumption on the involved Lipschitzian and closed mappings, the results improve and extend those announced by Cheng et al. (Fixed Point Theory Appl. 2013:100, 2013) and other authors with the related interest.

MSC:47H05, 47H09, 47H10.

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product $〈\cdot ,\cdot 〉$ and the corresponding norm $\parallel \cdot \parallel$. A mapping $T:C\to H$ is said to be nonexpansive if

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

(1.1)

A mapping $T:C\to H$ is called pseudocontractive or a pseudocontraction if

$$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in C.$$

(1.2)

Note that inequality (1.2) can be equivalently written as

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

(1.3)

where I denotes the identity operator. A mapping A with the domain $D(A)$ and the range $R(A)$ in H is called monotone if the inequality

$$\parallel x-y\parallel \le \parallel (x-y)+s(Ax-Ay)\parallel$$

(1.4)

holds for any $x,y\in D(A)$ and for all $s\ge 0$.

Not only from its being an important generalization of nonexpansive mappings, but also from the firm connection with the important class of nonlinear monotone mappings stems the interest in the class of pseudocontractions. We observe that A is monotone if and only if $T:=I-A$ is pseudocontractive, and hence a zero $x^{\ast }$ of A, that is, $x^{\ast }\in \mathcal{N}(A):=\{x\in D(A):Ax=0\}$, is just a fixed point of T. It is well known (see, e.g., [1]) that if A is monotone, then the solutions to the equation $Ax=0$ correspond to the equilibrium points of some evolution systems. Considerable efforts have then been devoted to developing iterative techniques for approximating fixed points of pseudocontractive mappings (see, for example, [2–4] and the references contained therein).

In 2013, Cheng et al. [5] constructed the following three-step iteration method and obtain the convergence theorem for a countable family of Lipschitz pseudocontractive mappings in Hilbert space H. For the iteration format,

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

(1.5)

they proved that the sequence $\{x_n\}$ generated from above converges strongly to a common fixed point of ${\{T_n\}}_{n\ge 1}$. But it is worth mentioning that the involved mappings were assumed to be uniformly closed and uniformly Lipschitz pseudocontractive, which are obviously two quite strong conditions for countable families of nonlinear operators. Recall that a countable family of mapping ${\{T_n\}}_{n=1}^{\mathrm{\infty }}:C\to H$ is called uniformly Lipschitz with Lipschitz constant $L_n\ge 0$, if there exists an $L:={sup}_{n\ge 1}{L}_n>0$ such that

$$\parallel T_{n}x-T_{n}y\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,n\ge 1.$$

(1.6)

A countable family of mapping ${\{T_n\}}_{n=1}^{\mathrm{\infty }}:C\to H$ is called uniformly closed if as $n\to \mathrm{\infty }$, $x_n\to x^{\ast }$ and $\parallel x_n-T_n{x}_n\parallel \to 0$ imply $x^{\ast }\in {\bigcap }_{n=1}^{\mathrm{\infty }}F(T_n)$.

Inspired and motivated by the studies mentioned above, in this paper, we introduced an up-to-date method for the approximation of common fixed points of countable families of nonlinear operators, by which a relaxed three-step iterative algorithm is developed for the class of pseudocontractive mappings, and a strong convergence theorem is established in the framework of Hilbert spaces. No compactness assumption is imposed either on the involved mappings or on the set C. The results are more applicable than those of other authors with the related interest.

2 Preliminaries

In the sequel, we shall need the following definitions. Let H be a real Hilbert space. The function $\varphi :H\times H\to \mathbb{R}$, defined by

$$\varphi (x,y):={\parallel x-y\parallel }^2={\parallel x\parallel }^2-2〈x,y〉+{\parallel y\parallel }^2,$$

(2.1)

is studied by Alber [6], Kamimura and Takahashi [7] and Reich [8]. It is obvious from the definition of the function ϕ that

$${(\parallel x\parallel -\parallel y\parallel )}^2\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^2.$$

(2.2)

The function ϕ has also the following property

$$\varphi (y,x)=\varphi (z,x)+\varphi (y,z)+2〈z-y,x-z〉.$$

(2.3)

In what follows, we shall make use of the following lemmas.

Lemma 2.1 [9]

Let H be a Hilbert space. Then for all $x,y\in H$ and ${\alpha }_i\in [0,1]$ for $i=0,1,2,\dots ,n$ such that ${\sum }_{i=0}^n{\alpha }_i=1$ the following equality holds

$${\parallel \sum _{i=0}^n{\alpha }_i{x}_i\parallel }^2=\sum _{i=0}^n{\alpha }_i{\parallel x_i\parallel }^2-\sum _{0\le i,j\le n}{\alpha }_i{\alpha }_j{\parallel x_i-x_j\parallel }^2.$$

(2.4)

Lemma 2.2 [10]

Let $\{a_n\}$, $\{{\delta }_n\}$, and $\{b_n\}$ be the sequences of nonnegative real numbers satisfying

$$a_{n+1}\le (1+{\delta }_n)a_n+b_n,\mathrm{\forall }n\ge 1.$$

(2.5)

If ${\sum }_{n=1}^{\mathrm{\infty }}{\delta }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists.

Lemma 2.3 [11]

The unique solutions to the positive integer equation

$$n=i+\frac{(m-1)m}{2},m\ge i,n=1,2,3,\dots$$

(2.6)

are

$$i=n-\frac{(m-1)m}{2},m=-[\frac{1}{2}-\sqrt{2n+\frac{1}{4}}],n=1,2,3,\dots ,$$

(2.7)

where $[x]$ denotes the maximal integer that is not larger than x.

3 Main results

Recall that an operator T on a Hilbert space is closed if $x_n\to x$ and $T{x}_n\to y$ as $n\to \mathrm{\infty }$, then $Tx=y$.

Theorem 3.1 Let H be a real Hilbert space, and let C be a closed convex nonempty subset of H. Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a sequence of closed and Lipschitz pseudocontractive mappings with Lipschitzian constants $L_i$ for each $i\ge 1$ and the interior of $F:={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)\ne \mathrm{\varnothing }$. Starting from an arbitrary $x_1\in C$, define $\{x_n\}$ by

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

(3.1)

where $\{{\alpha }_i\},\{{\beta }_i\},\{{\gamma }_i\}\subset (0,1)$ satisfying the following conditions: (i) ${\alpha }_i\le {\beta }_i\le {\gamma }_i$ and (ii) ${\gamma }_i^3{L}_i^4+2{\gamma }_i^2{L}_i^3+{\gamma }_i^2{L}_i^2+{\gamma }_i{L}_i^2+2{\gamma }_i<1$ for each $i\ge 1$; $i_n$ is the solutions to the positive integer equation: $n=i+\frac{(m-1)m}{2}$ ($m\ge i$, $n=1,2,\dots$), that is, for each $n\ge 1$, there exists a unique $i_n$ such that

$$\begin{array}{c}{i}_1=1,i_2=1,i_3=2,i_4=1,i_5=2,i_6=3,\hfill \\ i_7=1,i_8=2,i_9=3,i_{10}=4,i_{11}=1,\dots .\hfill \end{array}$$

Then $\{x_n\}$ converges strongly to an $x^{\ast }\in F$.

Proof Let $p\in F$. Using the similar argument presented in the proof of [[1], Theorem 3.1], we have from (3.1) and Lemma 2.1,

$$\begin{array}{rl}{\parallel x_{n+1}-p\parallel }^2\le & (1-{\alpha }_{i_n}){\parallel x_n-p\parallel }^2+{\alpha }_{i_n}{\parallel y_n-p\parallel }^2+{\alpha }_{i_n}{\parallel y_n-T_{i_n}{y}_n\parallel }^2\\ -{\alpha }_{i_n}(1-{\alpha }_{i_n}){\parallel x_n-T_{i_n}{y}_n\parallel }^2,\end{array}$$

(3.2)

$$\begin{array}{rl}{\parallel y_n-p\parallel }^2\le & (1-{\beta }_{i_n}){\parallel x_n-p\parallel }^2+{\beta }_{i_n}{\parallel z_n-p\parallel }^2+{\beta }_{i_n}{\parallel z_n-T_{i_n}{z}_n\parallel }^2\\ -{\beta }_{i_n}(1-{\beta }_{i_n}){\parallel x_n-T_{i_n}{z}_n\parallel }^2\end{array}$$

(3.3)

and

$$\begin{array}{rl}{\parallel z_n-p\parallel }^2\le & (1-{\gamma }_{i_n}){\parallel x_n-p\parallel }^2+{\gamma }_{i_n}{\parallel x_n-p\parallel }^2+{\gamma }_{i_n}{\parallel x_n-T_{i_n}{x}_n\parallel }^2\\ -{\gamma }_{i_n}(1-{\gamma }_{i_n}){\parallel x_n-T_{i_n}{x}_n\parallel }^2\\ =& {\parallel x_n-p\parallel }^2+{\gamma }_{i_n}^2{\parallel x_n-T_{i_n}{x}_n\parallel }^2.\end{array}$$

(3.4)

In addition, from (3.1), we also have

$$\begin{array}{rcl}{\parallel z_n-T_{i_n}{z}_n\parallel }^2& \le & (1-{\gamma }_{i_n}){\parallel x_n-T_{i_n}{z}_n\parallel }^2+{\gamma }_{i_n}{L}_{i_n}^2{\parallel x_n-z_n\parallel }^2\\ -{\gamma }_{i_n}(1-{\gamma }_{i_n}){\parallel x_n-T_{i_n}{x}_n\parallel }^2\\ =& (1-{\gamma }_{i_n}){\parallel x_n-T_{i_n}{z}_n\parallel }^2\\ +{\gamma }_{i_n}({\gamma }_{i_n}^2{L}_{i_n}^2+{\gamma }_{i_n}-1){\parallel x_n-T_{i_n}{x}_n\parallel }^2.\end{array}$$

(3.5)

Substituting (3.4) and (3.5) into (3.3), we obtain that

$$\begin{array}{rl}{\parallel y_n-p\parallel }^2\le & {\parallel x_n-p\parallel }^2+{\beta }_{i_n}{\gamma }_{i_n}({\gamma }_{i_n}^2{L}_{i_n}^2+2{\gamma }_{i_n}-1){\parallel x_n-T_{i_n}{x}_n\parallel }^2\\ +{\beta }_{i_n}({\beta }_{i_n}-{\gamma }_{i_n}){\parallel x_n-T_{i_n}{z}_n\parallel }^2.\end{array}$$

(3.6)

Since

$$\begin{array}{rl}{\parallel y_n-T_{i_n}{y}_n\parallel }^2\le & (1-{\beta }_{i_n}){\parallel x_n-T_{i_n}{y}_n\parallel }^2+{\beta }_{i_n}{L}_{i_n}^2{\parallel z_n-y_n\parallel }^2\\ -{\beta }_{i_n}(1-{\beta }_{i_n}){\parallel x_n-T_{i_n}{z}_n\parallel }^2\end{array}$$

(3.7)

and

$$\begin{array}{rl}\parallel z_n-y_n\parallel & \le ({\gamma }_{i_n}-{\beta }_{i_n})\parallel x_n-T_{i_n}{x}_n\parallel +{\beta }_{i_n}{L}_{i_n}\parallel x_n-z_n\parallel \\ =({\gamma }_{i_n}-{\beta }_{i_n})\parallel x_n-T_{i_n}{x}_n\parallel +{\beta }_{i_n}{\gamma }_{i_n}{L}_{i_n}\parallel x_n-T_{i_n}{x}_n\parallel \\ =({\gamma }_{i_n}-{\beta }_{i_n}+{\beta }_{i_n}{\gamma }_{i_n}{L}_{i_n})\parallel x_n-T_{i_n}{x}_n\parallel ,\end{array}$$

(3.8)

it then follows from (3.7) and (3.8) that

$$\begin{array}{rl}{\parallel y_n-T_{i_n}{y}_n\parallel }^2\le & (1-{\beta }_{i_n}){\parallel x_n-T_{i_n}{y}_n\parallel }^2\\ +{\beta }_{i_n}{L}_{i_n}^2{({\gamma }_{i_n}-{\beta }_{i_n}+{\beta }_{i_n}{\gamma }_{i_n}{L}_{i_n})}^2{\parallel x_n-T_{i_n}{x}_n\parallel }^2\\ -{\beta }_{i_n}(1-{\beta }_{i_n}){\parallel x_n-T_{i_n}{z}_n\parallel }^2.\end{array}$$

(3.9)

Substituting (3.6) and (3.9) into (3.2), we obtain that

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & {\parallel x_n-p\parallel }^2+[{\alpha }_{i_n}{\beta }_{i_n}{\gamma }_{i_n}({\gamma }_{i_n}^2{L}_{i_n}^2+2{\gamma }_{i_n}-1)\\ +{\alpha }_{i_n}{\beta }_{i_n}{L}_{i_n}^2{({\gamma }_{i_n}-{\beta }_{i_n}+{\beta }_{i_n}{\gamma }_{i_n}{L}_{i_n})}^2]{\parallel x_n-T_{i_n}{x}_n\parallel }^2\\ +{\alpha }_{i_n}({\alpha }_{i_n}-{\beta }_{i_n}){\parallel x_n-T_{i_n}{y}_n\parallel }^2\\ +{\alpha }_{i_n}{\beta }_{i_n}(2{\beta }_{i_n}-{\gamma }_{i_n}-1){\parallel x_n-T_{i_n}{z}_n\parallel }^2,\end{array}$$

(3.10)

which, together with condition (i), that is, ${\alpha }_{i_n}({\alpha }_{i_n}-{\beta }_{i_n})\le 0$ and ${\alpha }_{i_n}{\beta }_{i_n}(2{\beta }_{i_n}-{\gamma }_{i_n}-1)\le 0$, yields that

$${\parallel x_{n+1}-p\parallel }^2\le {\parallel x_n-p\parallel }^2-{\delta }_{i_n}{\parallel x_n-T_{i_n}{x}_n\parallel }^2,$$

(3.11)

where ${\delta }_{i_n}:={\alpha }_{i_n}{\beta }_{i_n}{\gamma }_{i_n}(1-{\gamma }_{i_n}^2{L}_{i_n}^2-2{\gamma }_{i_n})+{\alpha }_{i_n}{\beta }_{i_n}{L}_{i_n}^2{({\beta }_{i_n}-{\gamma }_{i_n}-{\beta }_{i_n}{\gamma }_{i_n}{L}_{i_n})}^2$. Noting that, in the light of condition (ii), ${\delta }_{i_n}>0$, we have

$$\parallel x_{n+1}-p\parallel \le \parallel x_n-p\parallel .$$

(3.12)

So, by Lemma 2.2, we conclude that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists.

Furthermore, from (2.3), we also have that

$$\varphi (p,x_n)=\varphi (x_{n+1},x_n)+\varphi (p,x_{n+1})+2〈x_{n+1}-p,x_n-x_{n+1}〉,\mathrm{\forall }p\in H.$$

This implies that

$$〈x_{n+1}-p,x_n-x_{n+1}〉+\frac{1}{2}\varphi (x_{n+1},x_n)=\frac{1}{2}(\varphi (p,x_n)-\varphi (p,x_{n+1})).$$

(3.13)

Moreover, since the interior of F is nonempty, there exists a $p^{\ast }\in F$ and $r>0$ such that $(p^{\ast }+rh)\in F$ whenever $\parallel h\parallel \le 1$. Thus, from (3.12) and (3.13), we obtain that

$$0\le 〈x_{n+1}-(p^{\ast }+rh),x_n-x_{n+1}〉+\frac{1}{2}\varphi (x_{n+1},x_n).$$

(3.14)

Then from (3.13) and (3.14), we obtain that

$$\begin{array}{rcl}r〈h,x_n-x_{n+1}〉& \le & 〈x_{n+1}-p^{\ast },x_n-x_{n+1}〉+\frac{1}{2}\varphi (x_{n+1},x_n)\\ =& \frac{1}{2}(\varphi (p^{\ast },x_n)-\varphi (p^{\ast },x_{n+1})),\end{array}$$

and hence

$$〈h,x_n-x_{n+1}〉\le \frac{1}{2r}(\varphi (p^{\ast },x_n)-\varphi (p^{\ast },x_{n+1})).$$

Since h with $\parallel h\parallel \le 1$ is arbitrary, we have

$$\parallel x_n-x_{n+1}\parallel \le \frac{1}{2r}(\varphi (p^{\ast },x_n)-\varphi (p^{\ast },x_{n+1})).$$

So, if $n>m$, then we have that

$$\begin{array}{rl}\parallel x_m-x_n\parallel & \le \sum _{j=m}^{n-1}\parallel x_j-x_{j+1}\parallel \\ \le \frac{1}{2r}\sum _{j=m}^{n-1}(\varphi (p^{\ast },x_j)-\varphi (p^{\ast },x_{j+1}))\\ =\frac{1}{2r}(\varphi (p^{\ast },x_m)-\varphi (p^{\ast },x_n)).\end{array}$$

(3.15)

Since $\{\varphi (p^{\ast },x_n)\}$ converges, it then follows from (3.15) that $\{x_n\}$ is a Cauchy sequence, and hence there exists an $x^{\ast }\in H$ such that

$$x_n\to x^{\ast }\in H(n\to \mathrm{\infty }).$$

(3.16)

Next, set ${\mathbb{N}}_i=\{k\in \mathbb{N}:k=i+\frac{(m-1)m}{2},m\ge i,m\in \mathbb{N}\}$ for each $i\ge 1$. For example, by Lemma 2.3 and the definition of ${\mathbb{N}}_1$, we have ${\mathbb{N}}_1=\{1,2,4,7,11,16,\dots \}$ and $i_1=i_2=i_4=i_7=i_{11}=i_{16}=\cdots =1$. Note that $T_{i_k}=T_i$, ${\delta }_{i_k}={\delta }_i$ whenever $k\in {\mathbb{N}}_i$ for each $i\ge 1$. We have, from (3.11),

$${\delta }_i{\parallel x_k-T_i{x}_k\parallel }^2\le {\parallel x_k-p\parallel }^2-{\parallel x_{k+1}-p\parallel }^2,\mathrm{\forall }k\in {\mathbb{N}}_i.$$

(3.17)

Since ${\{x_k\}}_{k\in {\mathbb{N}}_i}$ and ${\{x_{k+1}\}}_{k\in {\mathbb{N}}_i}$ are subsequences of $\{x_n\}$, the existence of ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ implies that

$$\underset{{\mathbb{N}}_i\ni k\to \mathrm{\infty }}{lim}\parallel x_k-T_i{x}_k\parallel =0,\mathrm{\forall }i\ge 1.$$

(3.18)

Note that, from (3.16), $x_k\to x^{\ast }$ as ${\mathbb{N}}_i\ni k\to \mathrm{\infty }$. It immediately follows from (3.18) and the closedness of $T_i$ that $x^{\ast }\in F(T_i)$ for each $i\ge 1$, and hence $x^{\ast }\in F$. This completes the proof. □

We now give an example, to which the results of Cheng et al. [5] cannot be applied.

Example 3.2 Let $H={\mathbb{R}}^1$ and $C=[-1,1]$. Let ${\{T_i\}}_{i=1}^{\mathrm{\infty }}:C\to C$ be a sequence of nonlinear mappings defined by

$$T_{i}x=\{\begin{array}{cc}(\frac{i}{2i+1}+\frac{1}{2})x,\hfill & x\in C_1:=[0,1],\hfill \\ x,\hfill & x\in C_2:=[-1,0).\hfill \end{array}$$

It is clear that $F:={\bigcap }_{i=1}^{\mathrm{\infty }}F(T_i)=[-1,0]$, and hence the interior of the common fixed points is nonempty. We show that ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ is a countable family of pseudocontractive mappings. If $x\in C_2$ and $y\in C_1$, then

$$\begin{array}{rcl}〈T_{i}x-T_{i}y,x-y〉& =& 〈x-(\frac{i}{2i+1}+\frac{1}{2})y,x-y〉\\ =& 〈(x-y)+(\frac{1}{2}-\frac{i}{2i+1})y,x-y〉\\ =& {|x-y|}^2-(\frac{i}{2i+1}-\frac{1}{2})(x-y)y.\end{array}$$

Noting that $(\frac{i}{2i+1}-\frac{1}{2})(x-y)y>0$, we have

$$〈T_{i}x-T_{i}y,x-y〉\le {|x-y|}^2.$$

The rest is trivial, and it is easy to show that each $T_i$ is Lipschitz and closed. However, ${\{T_i\}}_{i=1}^{\mathrm{\infty }}$ is not uniformly closed. In fact, for any $[0,1]\supset \{x_i\}\to x^{\ast }\in (0,1]$ as $i\to \mathrm{\infty }$, we have

$$|x_i-T_i{x}_i|=|x_i-(\frac{i}{2i+1}+\frac{1}{2})x_i|\to 0(i\to \mathrm{\infty }),$$

while $x^{\ast }$ is obviously not a member of F.

Remark 3.3 By using a specific way of choosing the indexes of the involved mappings and parameters, we propose an up-to-date iterative approach to approximating common fixed points of countable families of pseudocontractive mappings. The results extend previous results, announced by the authors with the related research interest.