Algorithms with variant anchors for pseudocontractive mappings

Abstract

The purpose of this paper is to find the fixed points of pseudocontractive mappings by using the iterative technique. Two algorithms with variant anchors have been introduced. Strong convergence results are given. Especially, we can find the minimum-norm fixed point of pseudocontractive mappings.

MSC:47H05, 47H10, 47H17.

1 Introduction

In this paper, we assume that H is a real Hilbert space and $C\subset H$ is a nonempty closed convex subset. Recall that a mapping $T:C\to C$ is said to be Lipschitzian if

$$\parallel Tu-T{u}^{†}\parallel \le \kappa \parallel u-u^{†}\parallel ,\mathrm{\forall }u,u^{†}\in C,$$

where $\kappa >0$ is a constant, which is in general called the Lipschitz constant. If $\kappa =1$, T is called nonexpansive.

A mapping $T:C\to C$ is said to be pseudocontractive if

$$〈Tu-T{u}^{†},u-u^{†}〉\le {\parallel u-u^{†}\parallel }^2,\mathrm{\forall }u,u^{†}\in C.$$

We use $Fix(T)$ to denote the set of fixed points of T.

In the literature, there are a large number references associated with the fixed point algorithms for the pseudocontractive mappings. See, for instance, [1–24]. (The interest of pseudocontractions lies in their connection with monotone operators; namely, T is a pseudocontraction if and only if the complement $I-T$ is a monotone operator.)

Now there exists an example which shows that Mann iteration does not converge for the pseudocontractive mappings [2]. At present, it is still an interesting topic to construct algorithms for finding the fixed points of the pseudocontractive mappings.

On the other hand, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. Recently, in order to find the fixed points of the nonexpansive mappings, Yao and Shahzad [25] introduced the following algorithms with perturbations and obtained the strong convergence results.

Algorithm 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a nonexpansive mapping. For given $x_0\in C$, define a sequence $\{x_m\}$ in the following manner:

$$x_m={proj}_C[{\alpha }_m{u}_m+(1-{\alpha }_m)T{x}_m],m\ge 0,$$

(1.1)

where $\{{\alpha }_m\}$ is a sequence in $[0,1]$ and the sequence $\{u_m\}\subset H$ is a small perturbation for the m-step iteration satisfying $\parallel u_m\parallel \to 0$ as $m\to \mathrm{\infty }$.

Theorem 1.2 Suppose $Fix(T)\ne \mathrm{\varnothing }$. Then, as ${\alpha }_m\to 0$, the sequence $\{x_m\}$ generated by the implicit method (1.1) converges to $\tilde{x}\in Fix(T)$, which is the minimum-norm fixed point of T.

Algorithm 1.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a nonexpansive mapping. For given $x_0\in C$, define a sequence $\{x_n\}$ in the following manner:

$$x_{n+1}=(1-{\beta }_n)x_n+{\beta }_n{proj}_C[{\alpha }_n{u}_n+(1-{\alpha }_n)T{x}_n],n\ge 0,$$

(1.2)

where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are two sequences in $(0,1)$ and the sequence $\{u_n\}\subset H$ is a perturbation for the n-step iteration.

Theorem 1.4 Suppose that $Fix(T)\ne \mathrm{\varnothing }$. Assume the following conditions are satisfied:

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

• (ii) $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$;

• (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n\parallel u_n\parallel <\mathrm{\infty }$.

Then the sequence $\{x_n\}$ generated by the explicit iterative method (1.2) converges to $\tilde{x}\in Fix(T)$, which is the minimum-norm fixed point of T.

Note that the idea of the iterative algorithms with perturbations has been extended to the other topics, see, for example, [26].

Motivated by the above ideas and the results in the literature, in the present paper, we present two algorithms with variant anchors for finding the fixed points of the pseudocontractive mappings in Hilbert spaces. Strong convergence results are given. As special cases, we can find the minimum-norm fixed point of the pseudocontractive mappings.

2 Preliminaries

Recall that the metric projection ${proj}_C:H\to C$ is defined by

$${proj}_{C}x:=arg\underset{\mathrm{\forall }y\in C}{min}\parallel x-y\parallel ,x\in H.$$

It is obvious that ${proj}_C$ satisfies

$$\parallel x-{proj}_{C}x\parallel \le \parallel x-y\parallel ,\mathrm{\forall }y\in C,$$

and is characterized by

$${proj}_{C}x\in C,〈x-{proj}_{C}x,y-{proj}_{C}x〉\le 0,\mathrm{\forall }y\in C.$$

The following two lemmas will be useful for our main results.

Lemma 2.1 ([24])

Let C be a closed convex subset of a Hilbert space H. Let $T:C\to C$ be a Lipschitzian pseudocontractive mapping. Then $Fix(T)$ is a closed convex subset of C and the mapping $I-T$ is demiclosed at 0, i.e., whenever $\{x_n\}\subset C$ is such that $x_n\rightharpoonup x$ and $(I-T)x_n\to 0$, then $(I-T)x=0$.

Lemma 2.2 ([27])

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

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\gamma }_n{\delta }_n,n\ge 0,$$

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

• (i) ${\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$;

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

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

3 Main results

In the sequel, we assume that C is a nonempty closed convex subset of a real Hilbert space H and $T:C\to C$ is a κ-Lipschitzian pseudocontractive mapping with nonempty fixed points set $Fix(T)$.

The first result is on the convergence of the path for the pseudocontractive mappings. Now, we define our path as follows.

For fixed $\zeta ,t\in (0,1)$ and $u_t\in H$, we define a mapping $G_t:C\to C$ by

$$G_{t}x=(1-\zeta ){proj}_C[t{u}_t+(1-t)x]+\zeta Tx,\mathrm{\forall }x\in C,$$

where ${proj}_C:H\to C$ is the metric projection from H on C.

Next, we show that the mapping $G_t$ is strongly pseudocontractive. Indeed, for $x,y\in C$, we have

$$\begin{array}{rcl}〈G_{t}x-G_{t}y,x-y〉& =& (1-\zeta )〈{proj}_C[t{u}_t+(1-t)x]-{proj}_C[t{u}_t+(1-t)y],x-y〉\\ +\zeta 〈Tx-Ty,x-y〉\\ \le & (1-\zeta )\parallel {proj}_C[t{u}_t+(1-t)x]-{proj}_C[t{u}_t+(1-t)y]\parallel \parallel x-y\parallel \\ +\zeta {\parallel x-y\parallel }^2\\ \le & (1-\zeta )(1-t){\parallel x-y\parallel }^2+\zeta {\parallel x-y\parallel }^2\\ =& [1-(1-\zeta )t]{\parallel x-y\parallel }^2.\end{array}$$

Since $\zeta ,t\in (0,1)$, $1-(1-\zeta )t\in (0,1)$. Hence, $G_t$ is a strongly pseudocontractive mapping. By [2], $G_t$ has a unique fixed point $x_t\in C$. That is, $x_t$ satisfies

$$x_t=(1-\zeta ){proj}_C[t{u}_t+(1-t)x_t]+\zeta T{x}_t,\mathrm{\forall }t\in (0,1).$$

(3.1)

Remark 3.1 $u_t\in H$ can be seen as a perturbation.

Next, we prove the convergence of the path (3.1).

Theorem 3.2 If ${lim}_{t\to 0}{u}_t=u\in H$, then the path $\{x_t\}$ defined by (3.1) converges strongly to ${proj}_{Fix(T)}(u)$.

Proof Let $p\in Fix(T)$. We get from (3.1) that

$$\begin{array}{rcl}{\parallel x_t-p\parallel }^2& =& (1-\zeta )〈{proj}_C[t{u}_t+(1-t)x_t]-p,x_t-p〉+\zeta 〈T{x}_t-p,x_t-p〉\\ \le & (1-\zeta )\parallel {proj}_C[t{u}_t+(1-t)x_t]-p\parallel \parallel x_t-p\parallel +\zeta {\parallel x_t-p\parallel }^2\\ \le & (1-\zeta )\parallel t(u_t-p)+(1-t)(x_t-p)\parallel \parallel x_t-p\parallel +\zeta {\parallel x_t-p\parallel }^2\\ \le & (1-\zeta )[(1-t)\parallel x_t-p\parallel +t\parallel u_t-p\parallel ]\parallel x_t-p\parallel +\zeta {\parallel x_t-p\parallel }^2.\end{array}$$

It follows that

$$\parallel x_t-p\parallel \le \parallel u_t-p\parallel .$$

Since ${lim}_{t\to 0}{u}_t=u\in H$, there exists a constant $M>0$ such that ${sup}_{t\in (0,1)}\parallel u_t-u\parallel \le M$. So,

$$\parallel x_t-p\parallel \le \parallel u_t-p\parallel \le \parallel u_t-u\parallel +\parallel u-p\parallel \le M+\parallel u-p\parallel .$$

Thus, $\{x_t\}$ is bounded.

By (3.1), we have

$$\begin{array}{rcl}\parallel x_t-T{x}_t\parallel & =& \parallel (1-\zeta ){proj}_C[t{u}_t+(1-t)x_t]+\zeta T{x}_t-T{x}_t\parallel \\ \le & (1-\zeta )\parallel {proj}_C[t{u}_t+(1-t)x_t]-T{x}_t\parallel \\ \le & (1-\zeta )[\parallel x_t-T{x}_t\parallel +t\parallel u_t-x_t\parallel ].\end{array}$$

Therefore,

$$\begin{array}{r}\parallel x_t-T{x}_t\parallel \le \frac{(1-\zeta )t}{\zeta }\parallel u_t-x_t\parallel \le \frac{(1-\zeta )t}{\zeta }(\parallel u_t-u\parallel +\parallel x_t-u\parallel )\to 0\\ (\text{as }t\to 0).\end{array}$$

(3.2)

Let $\{t_n\}\subset (0,1)$ be a sequence satisfying $t_n\to 0^{+}$ as $n\to \mathrm{\infty }$. Put $x_n:=x_{t_n}$. By (3.2), we get

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

(3.3)

By (3.1), we obtain

$$\begin{array}{rcl}{\parallel x_t-p\parallel }^2& =& (1-\zeta )〈{proj}_C[t{u}_t+(1-t)x_t]-p,x_t-p〉+\zeta 〈T{x}_t-p,x_t-p〉\\ \le & (1-\zeta )\parallel {proj}_C[t{u}_t+(1-t)x_t]-p\parallel \parallel x_t-p\parallel +\zeta {\parallel x_t-p\parallel }^2\\ \le & \frac{1-\zeta }{2}({\parallel {proj}_C[t{u}_t+(1-t)x_t]-p\parallel }^2+{\parallel x_t-p\parallel }^2)+\zeta {\parallel x_t-p\parallel }^2.\end{array}$$

Hence,

$$\begin{array}{rcl}{\parallel x_t-p\parallel }^2& \le & {\parallel {proj}_C[t{u}_t+(1-t)x_t]-p\parallel }^2\\ \le & {\parallel x_t-p+t(u_t-x_t)\parallel }^2\\ =& {\parallel x_t-p\parallel }^2+2t〈u_t-x_t,x_t-p〉+t^2{\parallel u_t-x_t\parallel }^2\\ =& {\parallel x_t-p\parallel }^2-2t〈x_t-p,x_t-p〉+2t〈u_t-p,x_t-p〉+t^2{\parallel u_t-x_t\parallel }^2\\ =& (1-2t){\parallel x_t-p\parallel }^2+2t〈u_t-p,x_t-p〉+t^2{\parallel u_t-x_t\parallel }^2.\end{array}$$

It follows that

$$\begin{array}{rcl}{\parallel x_t-p\parallel }^2& \le & 〈u_t-p,x_t-p〉+\frac{t}{2}{\parallel u_t-x_t\parallel }^2\\ \le & 〈u_t-p,x_t-p〉+t{M}_1.\end{array}$$

(3.4)

Here $M_1>0$ is a constant such that ${sup}_{t\in (0,1)}\frac{{\parallel u_t-x_t\parallel }^2}{2}\le M_1$. In particular, we obtain

$${\parallel x_n-p\parallel }^2\le 〈u_n-p,x_n-p〉+t_n{M}_1,\mathrm{\forall }p\in Fix(T).$$

(3.5)

Since $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ satisfying $x_{n_i}\to x^{\ast }\in C$ weakly. By (3.3), we get

$$\underset{i\to \mathrm{\infty }}{lim}\parallel x_{n_i}-T{x}_{n_i}\parallel =0.$$

(3.6)

Applying Lemma 2.1 to (3.6) to deduce $x^{\ast }\in Fix(T)$.

By (3.5), we derive

$${\parallel x_{n_i}-x^{\ast }\parallel }^2\le 〈u_{n_i}-x^{\ast },x_{n_i}-x^{\ast }〉+t_{n_i}{M}_1.$$

(3.7)

Since $u_{n_i}-x^{\ast }\to u-x^{\ast }$ and $t_{n_i}\to 0$, we deduce that $x_{n_i}\to x^{\ast }$ by (3.7). By (3.5), we have

$${\parallel x^{\ast }-p\parallel }^2\le 〈u-p,x^{\ast }-p〉,\mathrm{\forall }p\in Fix(T).$$

(3.8)

Assume that there exists another subsequence $\{x_{n_j}\}$ of $\{x_n\}$ satisfying $x_{n_j}\to x^{†}$ weakly. Similarly, we can prove that $x_{n_j}\to x^{†}\in Fix(T)$, which satisfies

$${\parallel x^{†}-p\parallel }^2\le 〈u-p,x^{†}-p〉,\mathrm{\forall }p\in Fix(T).$$

(3.9)

In (3.8), we pick up $p=x^{†}$ to get

$${\parallel x^{\ast }-x^{†}\parallel }^2\le 〈u-x^{†},x^{\ast }-x^{†}〉.$$

(3.10)

In (3.9), we pick up $p=x^{\ast }$ to get

$${\parallel x^{†}-x^{\ast }\parallel }^2\le 〈u-x^{\ast },x^{†}-x^{\ast }〉.$$

(3.11)

Adding (3.10) and (3.11), we deduce

$${\parallel x^{†}-x^{\ast }\parallel }^2\le 0.$$

Thus, $x^{\ast }=x^{†}$. This indicates that the weak limit set of $\{x_n\}$ is singleton and the path $\{x_t\}$ converges strongly to $x^{\ast }={proj}_{Fix(T)}(u)$ by (3.8). This completes the proof. □

Corollary 3.3 The path $\{x_t\}$ defined by

$$x_t=(1-\zeta ){proj}_C[(1-t)x_t]+\zeta T{x}_t,\mathrm{\forall }t\in (0,1),$$

converges strongly to ${proj}_{Fix(T)}(0)$, which is the minimum-norm fixed point of T.

Now, we introduce another algorithm, which is an explicit manner.

Algorithm 3.4 Let $\{{\varsigma }_n\}$ and $\{{\zeta }_n\}$ be two real number sequences in $(0,1)$. Let $\{u_n\}\subset H$ be a sequence. For $x_0\in C$ arbitrarily, let the sequence $\{x_n\}$ be generated by

$$x_{n+1}=(1-{\zeta }_n){proj}_C[{\varsigma }_n{u}_n+(1-{\varsigma }_n)x_n]+{\zeta }_{n}T{x}_n,n\ge 0.$$

(3.12)

Theorem 3.5 Assume the following conditions are satisfied:

• (C1) ${lim}_{n\to \mathrm{\infty }}{\varsigma }_n={lim}_{n\to \mathrm{\infty }}\frac{{\varsigma }_n}{{\zeta }_n}={lim}_{n\to \mathrm{\infty }}\frac{{\zeta }_n^2}{{\varsigma }_n}=0$;

• (C2) ${lim}_{n\to \mathrm{\infty }}{u}_n=u\in H$.

Then we have

• (1) the sequence $\{x_n\}$ is bounded;

• (2) the sequence $\{x_n\}$ is asymptotically regular, that is, ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$.

Further, if ${\sum }_{n=0}^{\mathrm{\infty }}{\varsigma }_n=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\zeta }_n}=0$, then the sequence $\{x_n\}$ converges strongly to ${proj}_{Fix(T)}(u)$.

Proof By the condition (C1), we can find a sufficiently large positive integer m such that

$$1-\frac{1}{1/2-{\zeta }_m}(\kappa +1)(\kappa +2)({\varsigma }_m+2{\zeta }_m+\frac{{\zeta }_m^2}{{\varsigma }_m})>0.$$

(3.13)

Let $p\in Fix(T)$. For fixed m, we pick up a constant $M_2>0$ such that

$$max\{\parallel x_0-p\parallel ,\parallel x_1-p\parallel ,\dots ,\parallel x_{m-1}-p\parallel ,4\parallel x_m-p\parallel +4\parallel u_m-p\parallel \}\le M_2.$$

(3.14)

Next, we show that $\parallel x_{m+1}-p\parallel \le M_2$. Set $y_n={proj}_C[{\varsigma }_n{u}_n+(1-{\varsigma }_n)x_n]$ for all $n\ge 0$. Thus, we have $x_{n+1}=(1-{\zeta }_n)y_n+{\zeta }_{n}T{x}_n$ for all $n\ge 0$.

Since $I-T$ is monotone, we have

$$〈(I-T)x_{m+1},x_{m+1}-p〉=〈(I-T)x_{m+1}-(I-T)p,x_{m+1}-p〉\ge 0.$$

By (3.12), we obtain

$$\begin{array}{rcl}{\parallel x_{m+1}-p\parallel }^2& =& (1-{\zeta }_m)〈y_m-p,x_{m+1}-p〉+{\zeta }_m〈T{x}_m-p,x_{m+1}-p〉\\ =& (1-{\zeta }_m)〈y_m-{\varsigma }_m{u}_m-(1-{\varsigma }_m)x_m,x_{m+1}-p〉\\ +(1-{\zeta }_m)〈{\varsigma }_m{u}_m+(1-{\varsigma }_m)x_m-p,x_{m+1}-p〉\\ +{\zeta }_m〈T{x}_m-p,x_{m+1}-p〉\\ =& (1-{\zeta }_m)〈y_m-{\varsigma }_m{u}_m-(1-{\varsigma }_m)x_m,x_{m+1}-p〉\\ +(1-{\zeta }_m)〈x_m-p,x_{m+1}-p〉+(1-{\zeta }_m){\varsigma }_m〈u_m-x_m,x_{m+1}-p〉\\ +{\zeta }_m〈T{x}_m-p,x_{m+1}-p〉\\ =& (1-{\zeta }_m)〈y_m-{\varsigma }_m{u}_m-(1-{\varsigma }_m)x_m,x_{m+1}-p〉\\ +〈x_m-p,x_{m+1}-p〉-(1-{\zeta }_m){\varsigma }_m〈x_{m+1}-p,x_{m+1}-p〉\\ -(1-{\zeta }_m){\varsigma }_m〈x_m-x_{m+1},x_{m+1}-p〉-(1-{\zeta }_m){\varsigma }_m〈p-u_m,x_{m+1}-p〉\\ +{\zeta }_m〈T{x}_m-T{x}_{m+1},x_{m+1}-p〉+{\zeta }_m〈x_{m+1}-x_m,x_{m+1}-p〉\\ -{\zeta }_m〈x_{m+1}-T{x}_{m+1},x_{m+1}-p〉.\end{array}$$

Note that

$$\begin{array}{rl}\parallel y_m-{\varsigma }_m{u}_m-(1-{\varsigma }_m)x_m\parallel & \le \parallel y_m-x_m\parallel +{\varsigma }_m\parallel x_m-u_m\parallel \\ =\parallel {proj}_C[{\varsigma }_m{u}_m+(1-{\varsigma }_m)x_m]-x_m\parallel +{\varsigma }_m\parallel x_m-u_m\parallel \\ \le 2{\varsigma }_m\parallel x_m-u_m\parallel .\end{array}$$

Then we have

$$\begin{array}{rcl}{\parallel x_{m+1}-p\parallel }^2& \le & (1-{\zeta }_m)\parallel y_m-{\varsigma }_m{u}_m-(1-{\varsigma }_m)x_m\parallel \parallel x_{m+1}-p\parallel \\ +\parallel x_m-p\parallel \parallel x_{m+1}-p\parallel -(1-{\zeta }_m){\varsigma }_m{\parallel x_{m+1}-p\parallel }^2\\ +(1-{\zeta }_m){\varsigma }_m(\parallel x_{m+1}-x_m\parallel +\parallel u_m-p\parallel )\parallel x_{m+1}-p\parallel \\ +{\zeta }_m(\parallel T{x}_m-T{x}_{m+1}\parallel +\parallel x_{m+1}-x_m\parallel )\parallel x_{m+1}-p\parallel \\ \le & 2(1-{\zeta }_m){\varsigma }_m\parallel x_m-u_m\parallel \parallel x_{m+1}-p\parallel +\parallel x_m-p\parallel \parallel x_{m+1}-p\parallel \\ +(1-{\zeta }_m){\varsigma }_m(\parallel x_{m+1}-x_m\parallel +\parallel u_m-p\parallel )\parallel x_{m+1}-p\parallel \\ -(1-{\zeta }_m){\varsigma }_m{\parallel x_{m+1}-p\parallel }^2+{\zeta }_m(\kappa +1)\parallel x_{m+1}-x_m\parallel \parallel x_{m+1}-p\parallel \\ \le & \parallel x_m-p\parallel \parallel x_{m+1}-p\parallel +2(1-{\zeta }_m){\varsigma }_m(\parallel x_m-p\parallel +\parallel u_m-p\parallel )\parallel x_{m+1}-p\parallel \\ -(1-{\zeta }_m){\varsigma }_m{\parallel x_{m+1}-p\parallel }^2+({\varsigma }_m+{\zeta }_m)(\kappa +1)\parallel x_{m+1}-x_m\parallel \parallel x_{m+1}-p\parallel .\end{array}$$

Hence,

$$\begin{array}{rcl}[1+(1-{\zeta }_m){\varsigma }_m]\parallel x_{m+1}-p\parallel & \le & \parallel x_m-p\parallel +2{\varsigma }_m(\parallel x_m-p\parallel +\parallel u_m-p\parallel )\\ +(\kappa +1)({\varsigma }_m+{\zeta }_m)\parallel x_{m+1}-x_m\parallel .\end{array}$$

(3.15)

By (3.12), we have

$$\begin{array}{rcl}\parallel x_{m+1}-x_m\parallel & \le & (1-{\zeta }_m)\parallel {proj}_C[{\varsigma }_m{u}_m+(1-{\varsigma }_m)x_m]-x_m\parallel +{\zeta }_m\parallel T{x}_m-x_m\parallel \\ \le & (1-{\zeta }_m){\varsigma }_m(\parallel x_m-p\parallel +\parallel u_m-p\parallel )+{\zeta }_m(\parallel T{x}_m-p\parallel +\parallel p-x_m\parallel )\\ \le & {\varsigma }_m(\parallel x_m-p\parallel +\parallel u_m-p\parallel )+{\zeta }_m(\kappa +1)\parallel x_m-p\parallel \\ \le & (\kappa +1)({\varsigma }_m+{\zeta }_m)\parallel x_m-p\parallel +{\varsigma }_m\parallel u_m-p\parallel \\ \le & (\kappa +2)({\varsigma }_m+{\zeta }_m)M_2.\end{array}$$

(3.16)

From condition (C1), we deduce ${\varsigma }_m\to 0$ and ${\zeta }_m\to 0$ as $m\to \mathrm{\infty }$. Therefore, we get

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

That is, the sequence $\{x_m\}$ is asymptotically regular.

By (3.15) and (3.16), we have

$$\begin{array}{r}[1+(1-{\zeta }_m){\varsigma }_m]\parallel x_{m+1}-p\parallel \\ \le \parallel x_m-p\parallel +{\varsigma }_m(2\parallel x_m-p\parallel +2\parallel u_m-p\parallel )+(\kappa +1)(\kappa +2){({\varsigma }_m+{\zeta }_m)}^2{M}_2\\ \le (1+\frac{1}{2}{\varsigma }_m)M_2+(\kappa +1)(\kappa +2){({\varsigma }_m+{\zeta }_m)}^2{M}_2.\end{array}$$

This together with (3.13) and (3.14) imply that

$$\begin{array}{rcl}\parallel x_{m+1}-p\parallel & \le & [1-\frac{(1/2-{\zeta }_m){\varsigma }_m-(\kappa +1)(\kappa +2){({\varsigma }_m+{\zeta }_m)}^2}{1+(1-{\zeta }_m){\varsigma }_m}]M_2\\ =& \{1-\frac{(1/2-{\zeta }_m){\varsigma }_m[1-\frac{1}{1/2-{\zeta }_m}(\kappa +1)(\kappa +2)({\varsigma }_m+2{\zeta }_m+({\zeta }_m^2/{\varsigma }_m))]}{1+(1-{\zeta }_m){\varsigma }_m}\}{M}_2\\ \le & M_2.\end{array}$$

By induction, we get

$$\parallel x_n-p\parallel \le M_2,\mathrm{\forall }n\ge 0.$$

So $\{x_n\}$ is bounded.

By (3.12), we have

$$\begin{array}{rcl}\parallel x_n-T{x}_n\parallel & \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T{x}_n\parallel \\ \le & \parallel x_n-x_{n+1}\parallel +(1-{\zeta }_n)\parallel {proj}_C[{\varsigma }_n{u}_n+(1-{\varsigma }_n)x_n]-T{x}_n\parallel \\ \le & \parallel x_n-x_{n+1}\parallel +(1-{\zeta }_n)\parallel x_n-T{x}_n\parallel +{\varsigma }_n\parallel x_n-u_n\parallel .\end{array}$$

It follows that

$$\parallel x_n-T{x}_n\parallel \le \frac{1}{{\zeta }_n}\parallel x_n-x_{n+1}\parallel +\frac{{\varsigma }_n}{{\zeta }_n}\parallel x_n-u_n\parallel .$$

By the condition ${lim}_{n\to \mathrm{\infty }}\frac{{\varsigma }_n}{{\zeta }_n}=0$ and the assumption ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\zeta }_n}=0$, we deduce

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

(3.17)

Let the net $\{z_t\}$ be defined by $z_t=(1-\zeta ){proj}_C[t{u}_t+(1-t)z_t]+\zeta T{z}_t$. By Theorem 3.2, we know that $z_t$ converges strongly to ${proj}_{Fix(T)}(u)$. Next, we prove

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈{proj}_{Fix(T)}(u)-u_n,{proj}_{Fix(T)}(u)-y_n〉\le 0.$$

By the definition of $\{z_t\}$, we have

$$z_t-x_n=(1-\zeta )({proj}_C[t{u}_t+(1-t)z_t]-x_n)+\zeta (T{z}_t-T{x}_n)+\zeta (T{x}_n-x_n).$$

It follows that

$$\begin{array}{rcl}{\parallel z_t-x_n\parallel }^2& =& (1-\zeta )〈{proj}_C[t{u}_t+(1-t)z_t]-x_n,z_t-x_n〉+\zeta 〈T{z}_t-T{x}_n,z_t-x_n〉\\ +\zeta 〈T{x}_n-x_n,z_t-x_n〉\\ =& (1-\zeta )〈{proj}_C[t{u}_t+(1-t)z_t]-t{u}_t-(1-t)z_t,z_t-x_n〉\\ +(1-\zeta )〈t{u}_t+(1-t)z_t-x_n,z_t-x_n〉+\zeta 〈T{z}_t-T{x}_n,z_t-x_n〉\\ +\zeta 〈T{x}_n-x_n,z_t-x_n〉.\end{array}$$

Since $x_n\in C$, by the characteristic inequality of metric projection, we have

$$〈{proj}_C[t{u}_t+(1-t)z_t]-t{u}_t-(1-t)z_t,z_t-x_n〉\le 0.$$

Then

$$\begin{array}{rcl}{\parallel z_t-x_n\parallel }^2& \le & (1-\zeta )〈t{u}_t+(1-t)z_t-x_n,z_t-x_n〉+\zeta {\parallel z_t-x_n\parallel }^2\\ +\zeta \parallel T{x}_n-x_n\parallel \parallel z_t-x_n\parallel \\ =& (1-\zeta ){\parallel z_t-x_n\parallel }^2-(1-\zeta )t〈z_t-u_t,z_t-x_n〉+\zeta {\parallel z_t-x_n\parallel }^2\\ +\zeta \parallel T{x}_n-x_n\parallel \parallel z_t-x_n\parallel ,\end{array}$$

which implies that

$$〈z_t-u_t,z_t-x_n〉\le \frac{\zeta }{(1-\zeta )t}\parallel T{x}_n-x_n\parallel \parallel z_t-x_n\parallel .$$

By (3.17), we deduce

$$\underset{t\to 0}{lim\hspace{0.17em}sup}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈z_t-u_t,z_t-x_n〉\le 0.$$

(3.18)

Note the fact that the two limits ${lim\hspace{0.17em}sup}_{t\to 0}$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}$ are interchangeable. This together with $z_t\to {proj}_{Fix(T)}(u)$, $u_t\to u$ and (3.18) implies that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈{proj}_{Fix(T)}(u)-u,{proj}_{Fix(T)}(u)-x_n〉\le 0.$$

Note that $\parallel y_n-x_n\parallel \to 0$ and $u_n-u\to 0$. We derive

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈{proj}_{Fix(T)}(u)-u_n,{proj}_{Fix(T)}(u)-y_n〉\le 0.$$

Finally, we prove that $x_n\to {proj}_{Fix(T)}(u)$. Note that

$$\begin{array}{r}〈T{x}_n-{proj}_{Fix(T)}(u),x_{n+1}-{proj}_{Fix(T)}(u)〉\\ =〈T{x}_n-{proj}_{Fix(T)}(u),x_n-{proj}_{Fix(T)}(u)〉+〈T{x}_n-{proj}_{Fix(T)}(u),x_{n+1}-x_n〉\\ \le {\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2+\parallel T{x}_n-{proj}_{Fix(T)}(u)\parallel \parallel x_{n+1}-x_n\parallel \end{array}$$

(3.19)

and

$$\begin{array}{r}{\parallel y_n-{proj}_{Fix(T)}(u)\parallel }^2\\ =〈y_n-{\varsigma }_n{u}_n-(1-{\varsigma }_n)x_n,y_n-{proj}_{Fix(T)}(u)〉\\ +〈{\varsigma }_n{u}_n+(1-{\varsigma }_n)x_n-{proj}_{Fix(T)}(u),y_n-{proj}_{Fix(T)}(u)〉\\ \le 〈{\varsigma }_n{u}_n+(1-{\varsigma }_n)x_n-{proj}_{Fix(T)}(u),y_n-{proj}_{Fix(T)}(u)〉\\ =(1-{\varsigma }_n)〈x_n-{proj}_{Fix(T)}(u),y_n-{proj}_{Fix(T)}(u)〉\\ -{\varsigma }_n〈{proj}_{Fix(T)}(u)-u_n,y_n-{proj}_{Fix(T)}(u)〉\\ \le \frac{(1-{\varsigma }_n)}{2}{\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2+\frac{1}{2}{\parallel y_n-{proj}_{Fix(T)}(u)\parallel }^2\\ -{\varsigma }_n〈{proj}_{Fix(T)}(u)-u_n,y_n-{proj}_{Fix(T)}(u)〉.\end{array}$$

Then

$$\begin{array}{rl}{\parallel y_n-{proj}_{Fix(T)}(u)\parallel }^2\le & (1-{\varsigma }_n){\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2\\ -2{\varsigma }_n〈{proj}_{Fix(T)}(u)-u_n,y_n-{proj}_{Fix(T)}(u)〉.\end{array}$$

(3.20)

By (3.12), (3.16), and (3.20), we get

$$\begin{array}{r}{\parallel x_{n+1}-{proj}_{Fix(T)}(u)\parallel }^2\\ ={\parallel (1-{\zeta }_n)(y_n-{proj}_{Fix(T)}(u))+{\zeta }_n(T{x}_n-{proj}_{Fix(T)}(u))\parallel }^2\\ \le {\parallel (1-{\zeta }_n)(y_n-{proj}_{Fix(T)}(u))\parallel }^2\\ +2{\zeta }_n〈T{x}_n-{proj}_{Fix(T)}(u),x_{n+1}-{proj}_{Fix(T)}(u)〉\\ \le {(1-{\zeta }_n)}^2(1-{\varsigma }_n){\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2+2{\zeta }_n{\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2\\ -2{\varsigma }_n{(1-{\zeta }_n)}^2〈{proj}_{Fix(T)}(u)-u_n,y_n-{proj}_{Fix(T)}(u)〉\\ +2{\zeta }_n\parallel T{x}_n-{proj}_{Fix(T)}(u)\parallel \parallel x_{n+1}-x_n\parallel \\ \le [1-(1-2{\zeta }_n){\varsigma }_n]{\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2+{\zeta }_n^2{\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2\\ +2{\varsigma }_n{(1-{\zeta }_n)}^2〈{proj}_{Fix(T)}(u)-u_n,{proj}_{Fix(T)}(u)-y_n〉\\ +2{\zeta }_n\parallel T{x}_n-{proj}_{Fix(T)}(u)\parallel (\kappa +2)({\varsigma }_n+{\zeta }_n)M_2\\ =(1-{\gamma }_n){\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2+{\gamma }_n{\delta }_n,\end{array}$$

(3.21)

where ${\gamma }_n=(1-2{\zeta }_n){\varsigma }_n$ and

$$\begin{array}{rl}{\delta }_n=& \frac{2{(1-{\zeta }_n)}^2}{1-2{\zeta }_n}〈{proj}_{Fix(T)}(u)-u_n,{proj}_{Fix(T)}(u)-y_n〉+\frac{{\zeta }_n^2}{(1-2{\zeta }_n){\varsigma }_n}{\parallel x_n-{proj}_{Fix(T)}(u)\parallel }^2\\ +\frac{2{\zeta }_n}{1-2{\zeta }_n}\parallel T{x}_n-{proj}_{Fix(T)}(u)\parallel (\kappa +2)M_2\\ +\frac{2{\zeta }_n^2}{(1-2{\zeta }_n){\varsigma }_n}\parallel T{x}_n-{proj}_{Fix(T)}(u)\parallel (\kappa +2)M_2.\end{array}$$

It is clear that ${\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$. We can therefore apply Lemma 2.2 to (3.21) and conclude that $x_n\to {proj}_{Fix(T)}(u)$ as $n\to \mathrm{\infty }$. This completes the proof. □

Corollary 3.6 Let $\{{\varsigma }_n\}$ and $\{{\zeta }_n\}$ be two real number sequences in $(0,1)$. For $x_0\in C$ arbitrarily, let the sequence $\{x_n\}$ be generated by

$$x_{n+1}=(1-{\zeta }_n){proj}_C[(1-{\varsigma }_n)x_n]+{\zeta }_{n}T{x}_n,n\ge 0.$$

(3.22)

Assume ${lim}_{n\to \mathrm{\infty }}{\varsigma }_n={lim}_{n\to \mathrm{\infty }}\frac{{\varsigma }_n}{{\zeta }_n}={lim}_{n\to \mathrm{\infty }}\frac{{\zeta }_n^2}{{\varsigma }_n}=0$. Then we have

• (1) the sequence $\{x_n\}$ is bounded;

• (2) the sequence $\{x_n\}$ is asymptotically regular, that is, ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$.

Further, if ${\sum }_{n=0}^{\mathrm{\infty }}{\varsigma }_n=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}\frac{\parallel x_{n+1}-x_n\parallel }{{\zeta }_n}=0$, then the sequence $\{x_n\}$ converges strongly to ${proj}_{Fix(T)}(0)$, which is the minimum-norm fixed point of T.

Proof Letting $u_n=u=0$ in (3.12), we obtain (3.22). Consequently, by Theorem 3.5, we find that the sequence $\{x_n\}$ generated by (3.22) converges strongly to ${proj}_{Fix(T)}(0)$, which is the minimum-norm fixed point of T. □