An Averaged Halpern Type Algorithm for Solving Fixed Point Problems and Variational Inequality Problems

1. Introduction

Let $\mathcal{H}$ be a real Hilbert space with norm $\parallel ·\parallel$ and inner product $\langle ·,·\rangle .$ Let $D\subset \mathcal{H}$ be a closed convex set, and consider the self-mapping $G:D\to D.$ Throughout this paper the set of all fixed points of G in D is denoted by

$$Fix\left(G\right)=\{u\in D:Gu=u\}.$$

The mapping G is said to be

(a)

nonexpansive if

$$\parallel Gu-Gv\parallel \le \parallel u-v\parallel ,\mathrm{for}\mathrm{all}u,v\in D;$$

(1)

(b)

quasi-nonexpansive if $Fix\left(G\right)\ne \varnothing$ and

$$\parallel Gu-v\parallel \le \parallel u-v\parallel ,\mathrm{for}\mathrm{all}u\in D\mathrm{and}v\in Fix\left(G\right);$$

(2)

(c)

β-demicontractive if $Fix\left(G\right)\ne \varnothing$ and there exists a positive number $\beta <1$ such that

$${\parallel Gu-v\parallel }^2\le {\parallel u-v\parallel }^2+\beta {\parallel u-Gu\parallel }^2,$$

(3)

for all $u\in D$ and $v\in Fix\left(G\right)$.

(d)

strongly quasi-nonexpansive if $Fix\left(G\right)\ne \varnothing ,$ G is quasi-nonexpansive and $u_p-G{u}_p\to 0$ whenever $\left\{u_p\right\}$ is a bounded sequence such that $\parallel u_p-u^{*}\parallel -\parallel G{u}_p-u^{*}\parallel \to 0$ for some $u^{*}\in Fix\left(G\right).$

By the previous definitions, it is obvious that any nonexpansive mapping G with $Fix\left(G\right)\ne \varnothing$ is quasi-nonexpansive, any strongly quasi-nonexpansive is quasi-nonexpansive, and that any quasi-nonexpansive mapping is demicontractive, too, but the reverses are no more true, as illustrated by the next example.

Example 1

([4]). Let $\mathcal{H}$ be the real line with the usual norm and $D=[0,1].$ Define F on D as

$$F\left(u\right)=\left\{\begin{array}{cc}7/8,\hfill & if0\le u<1\hfill \\ 1/4,\hfill & \mathrm{if}u=1.\hfill \end{array}\right.$$

(4)

Then F is ${\displaystyle \frac{2}{3}}$-demicontractive but F is neither nonexpansive nor quasi-nonexpasive (and hence not strongly quasi-nonexpansive).

There are several papers in literature which are devoted to the approximation of common fixed points of nonexpansive type mappings. For example, in order to approximate the common fixed points of a pair of nonexpansive self mappings $(T_1,T_2)$ with $Fix\left(T_1\right)\cap Fix\left(T_2\right)\ne \varnothing$, Takahashi and Tamura [21] considered the following iterative procedure:

$$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{T}_1\left[(1-{\beta }_n)x_n+{\beta }_n{T}_2{x}_n\right],n\ge 1,\hfill \end{array}\right.$$

(5)

for which they established a weak convergence theorem.

Moudafi [15] considered a slightly different Krasnoselsij-Mann iterative procedure for the same problem, that he called "hierarchical fixed-point problem":

$$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n\left[(1-{\beta }_n)T_2{x}_n+{\beta }_n{T}_1{x}_n\right],n\ge 1,\hfill \end{array}\right.$$

(6)

where $Fix\left(T_1\right)$ and $Fix\left(T_2\right)$ are assumed to be nonempty.

Further, Iemoto and Takahashi [21] considered the problem of approximating the common fixed points of a nonexpansive mapping T and of a nonspreading mappings S in a Hilbert space, and utilized the iterative scheme

$$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n\left[(1-{\beta }_n)T{x}_n+{\beta }_{n}S{x}_n\right],n\ge 1,\hfill \end{array}\right.$$

(7)

for which they formulated and proved some weak convergence theorems.

Starting from this background, our aim in this paper is to solve the common fixed point problem in the setting of Hilbert spaces for the case of the larger class of demicontractive mappings, thus extending and unifying the main results in Cianciaruso et al. [7], Falset et al. [8], Iemoto and Takahashi [9] and many others.

Our main result (Theorem 1) provides a convergence theorem for an averaged iterative Halpern type algorithm used to approximate a solution of the common fixed point problem for a pair consisting of a nonexpansive mapping and a demicontractive mapping, which also solves a certain variational inequality problem.

2. Preliminaries

We recall some important lemmas used in the proofs of our main results. The following two lemmas are taken from Berinde [3].

Lemma 1.

[3] Let $\mathcal{H}$ be a real Hilbert space and $D\subset \mathcal{H}$ a closed and convex set. If $G:D\to D$ is β-demicontractive, then the Krasnoselskij perturbation $G_{\nu }=(1-\nu )I+\nu G$ of G is $(1+\beta /\nu -1/\nu )$-demicontractive.

Lemma 2.

[3] Let $\mathcal{H}$ be a real Hilbert space and $D\subset \mathcal{H}$ a closed and convex set. If $G:D\to D$ is β-demicontractive, then for any $\nu \in (0,1-\beta )$

$$G_{\nu }=(1-\nu )I+\nu G$$

is quasi-nonexpansive.

Lemma 3

(Zhou [30]). Let C be a nonempty subset of a real Hilbert space and let $T:C\to C$ be a k-strictly pseudocontractive mapping. Then the averaged mapping $T_{\lambda }=(1-\lambda )I+\lambda T$ is nonexpansive for any $\lambda \in (0,1-k)$.

Lemma 4.

Let $\mathcal{H}$ be a real Hilbert space, $D\subset \mathcal{H}$ a closed and convex set and $F:D\to D$ a mapping. Then, for any $\nu \in (0,1),$ we have $Fix\left(F_{\nu }\right)=Fix\left(F\right).$

Lemma 5.

[24] Let $\left\{{\alpha }_n\right\}$ be a sequence of nonnegative numbers such that

$${\alpha }_n\le (1-c_n){\alpha }_n+c_n{\mu }_n+{\delta }_n,n\ge 0,$$

where $\left\{c_n\right\}$ is a sequence in $[0,1]$ and $\left\{{\mu }_n\right\}$ is a sequence in $\mathbb{R}$ such that

$$\sum _{n=1}^{\infty }{c}_n=\infty ,\underset{n\to \infty }{lim\; sup}{\mu }_n\le 0,{\delta }_n\ge 0,\mathit{and}\sum _{n=1}^{\infty }{\delta }_n<\infty .$$

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

Lemma 6.

Let $\left\{{\gamma }_p\right\}$ be a sequence of real numbers that has a subsequence $\left\{{\gamma }_{p_k}\right\}$ which satisfies ${\gamma }_{p_k}<{\gamma }_{p_k+1}$ for all $k\in \mathbb{N}.$ There exists an increasing sequence of integers ${\left\{\tau \left(p\right)\right\}}_{p\ge p_0}$ satisfying:

$$\underset{p\to \infty }{lim}\tau \left(p\right)=\infty ,{\gamma }_{\tau \left(p\right)}\le {\gamma }_{\tau \left(p\right)+1},{\gamma }_p\le {\gamma }_{\tau \left(p\right)+1},\forall p\ge p_0.$$

3. Fixed Points and Variational Inequalities

In this section we state and prove our main results. To do this, we first consider the following property.

A mapping G satisfies Condition A if $u_p-G{u}_p\to 0$ whenever $\left\{u_p\right\}$ is a bounded sequence such that

$$\parallel u_p-u^{*}\parallel -\parallel (1-\nu )u_p+\nu G{u}_p-u^{*}\parallel \to 0$$

for some $u^{*}\in Fix\left(G\right)$ and $\nu \in [0,1].$

Theorem 1.

Let $\mathcal{H}$ be a Hilbert space and C be a closed convex subset of $\mathcal{H}.$ Let $F:C\to C$ be a nonexpansive mapping and $G:C\to C$ be a β-demicontractive mapping satisfying Condition A such that $I-G$ is demiclosed at $0.$ Assume that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$ Let $\left\{a_p\right\}$ and $\left\{b_p\right\}$ be sequences in $[0,1]$ such that $a_p\to 0$ and ${\sum }_{p=1}^{\infty }{a}_p=\infty .$ Let $\left\{u_p\right\}$ be the sequence generated in the following manner:

$$\left\{\begin{array}{cc}x,u_1\in C,\hfill & \\ u_{p+1}=a_{p}x+(1-a_p)[b_{p}F{u}_p+(1-b_p)((1-\alpha )u_p+\alpha G{u}_p)],p\ge 1.\hfill & \end{array}\right.$$

(8)

Then, the following assertions hold.

(I)

If ${\sum }_{p=1}^{\infty }(1-b_p)<\infty$ and ${\sum }_{p=1}^{\infty }|a_p=a_{p+1}|<\infty ,$ then $\left\{u_p\right\}$ strongly converges to $u^{*}\in Fix\left(F\right)$ which is the unique point in $Fix\left(F\right)$ that solves the variational inequality

$$\langle u^{*}-x,u-u^{*}\rangle \ge 0,\forall u\in Fix\left(F\right),$$

(9)

i.e. $u^{*}=P_{Fix\left(F\right)}x.$

(II)

If ${\sum }_{p=1}^{\infty }(1-b_p)<\infty$ and ${\displaystyle \frac{b_p}{a_p}}\to 0,$ then $\left\{u_p\right\}$ converges strongly to $v^{*}\in Fix\left(G\right)$ which is the unique point in $Fix\left(G\right)$ that solves the variational inequality

$$\langle u^{*}-x,u-v^{*}\rangle \ge 0,\forall u\in Fix\left(G\right),$$

(10)

i.e. $v^{*}=P_{Fix\left(G\right)}x.$

(III)

If ${lim\; inf}_{p\to \infty }{b}_p(1-b_p)>0,$ then $\left\{u_p\right\}$ strongly converges to $\overline{u}\in Fix\left(F\right)\cap Fix\left(G\right)$ that is the unique solution of the variational inequality

$$\langle \overline{u}-x,u-\overline{u}\rangle \ge 0,\forall u\in Fix\left(F\right)\cap Fix\left(G\right),$$

(11)

i.e. $\overline{u}=P_{Fix\left(F\right)\cap Fix\left(G\right)}x.$

Proof. 

Since G is a $\beta$-demicontractive mapping, by Lemma 2 it follows that the averaged mapping $G_{\alpha }=(1-\alpha )I+\alpha G$ is quasi-nonexpansive, for $\alpha \in (0,1-\beta ).$ Clearly, $G_{\alpha }-I$ is demiclosed at zero. One can also see that $G_{\alpha }$ is strongly quasi-nonexpansive from the fact that G satisfies Condition A. Now we can write

$$u_{p+1}=a_{p}x+(1-a_p)[b_{p}F{u}_p+(1-b_p)G_{\alpha }{u}_p],p\in \mathbb{N}.$$

Let w be a common fixed point of F and $G_{\alpha }.$ Define

$$T_p=b_{p}F+(1-b_p)G_{\alpha },\forall p\in \mathbb{N}.$$

For all $p\in \mathbb{N},$

$$\begin{array}{cc}\hfill \parallel u_{p+1}-w\parallel & =\parallel a_{p}x+(1-a_p)T_p{u}_p-w\parallel \hfill \\ \hfill & \le (1-a_p)\parallel u_p-w\parallel +a_p\parallel x-w\parallel \hfill \\ \hfill & \le max\{\parallel u_p-w\parallel ,\parallel x-w\parallel \}\hfill \\ \hfill & \le max\{\parallel u_1-w\parallel ,\parallel x-w\parallel \},\hfill \end{array}$$

that is, $\left\{u_p\right\}$ is a bounded sequence.

Furthermore, since $a_p\to 0$ as $p\to \infty ,$ we have

$$u_{p+1}-T_p{u}_p=a_p(x-T_p{u}_p)\to 0,\mathrm{as}p\to \infty .$$

(12)

To prove (I), for all $p\in \mathbb{N}$ compute

$$\begin{array}{cc}\hfill \parallel u_{p+1}-u_p\parallel & =\parallel (1-a_p)(T_p{u}_p-T_{p-1}{u}_{p-1})-(a_p-a_{p-1})T_{p-1}{u}_{p-1}+(a_p-a_{p-1})x\parallel \hfill \\ \hfill & =\parallel (1-a_p)(T_p{u}_p-T_{p-1}{u}_{p-1})+(a_{p-1}-a_p)[T_{p-1}{u}_{p-1}-x]\parallel \hfill \\ \hfill & =\parallel (a_{p-1}-a_p)[T_{p-1}{u}_{p-1}-x]\hfill \\ \hfill & \phantom{=}+(1-a_p)[b_{p}F{u}_p+(1-b_p)G_{\alpha }F{u}_p-b_{p-1}F{u}_{p-1}-(1-b_{p-1})G_{\alpha }{u}_{p-1}]\parallel \hfill \\ \hfill & =\parallel (a_{p-1}-a_p)[T_{p-1}{u}_{p-1}-x]+(1-a_p)[b_p(F{u}_p-F{u}_{p-1})\hfill \\ \hfill & \phantom{=}+(1-b_p)(G_{\alpha }{u}_p-G_{\alpha }{u}_{p-1})+(b_p-b_{p-1}(F{u}_{p-1}-G_{\alpha }{u}_{p-1})]]\parallel \hfill \\ \hfill & \le |a_{p-1}-a_p|\parallel T_{p-1}{u}_{p-1}-x\parallel +(1-a_p)[b_p\parallel u_p-u_{p-1}\parallel \hfill \\ \hfill & \phantom{=}+(1-b_p)\parallel G_{\alpha }{u}_p-G_{\alpha }{u}_{p-1}\parallel +|b_p-b_{p-1}|\parallel F{u}_{p-1}-G_{\alpha }{u}_{p-1}]\parallel \hfill \\ \hfill & =(1-c_p)\parallel u_p-u_{p-1}\parallel +{\mu }_p,\hfill \end{array}$$

where $c_p=1-b_p+a_p{b}_p$ and

$${\mu }_p=|a_{p-1}-a_p|\parallel T_{p-1}{u}_{p-1}-x\parallel +(1-a_p)[(1-b_p)+|b_p-b_{p-1}|\parallel F{u}_{p-1}-G_{\alpha }{u}_{p-1}]\parallel .$$

We see that $c_p\to 0,$${\sum }_{p=1}^{\infty }{c}_p=\infty$ and ${\sum }_{p=1}^{\infty }{\mu }_p<\infty .$ Hence, by Lemma 5 we conclude that $u_{p+1}-u_p\to 0.$ This and (12) imply that

$$u_p-T_p{u}_p\to 0.$$

(13)

Moreover,

$$\begin{array}{cc}\hfill \parallel u_p-T_p{u}_p\parallel & =\parallel u_p-b_{p}F{u}_p-(1-b_p)\parallel \hfill \\ \hfill & \ge \parallel u_p-b_{p}F{u}_p\parallel -(1-b_p)\parallel G_{\alpha }{u}_p\parallel ,\hfill \\ \hfill \parallel u_p-b_{p}F{u}_p\parallel & \le \parallel u_p-T_p{u}_p\parallel +(1-b_p)\parallel G_{\alpha }{u}_p\parallel ,\hfill \end{array}$$

and thus, from the hypothesis that ${\sum }_{p=1}^{\infty }(1-b_p)<\infty ,$ we also have

$$u_p-b_{p}F{u}_p\to 0.$$

We can conclude that $u_p-F{u}_p\to 0.$ Hence, any weak limit of $\left\{u_p\right\}$ is in $Fix\left(F\right).$

Let $\left\{u_{p_k}\right\}$ be a subsequence of $\left\{u_p\right\}$ such that

$$\underset{p\to \infty }{lim\; sup}\langle u_p-u^{*},x-u^{*}\rangle =\underset{k\to \infty }{lim}\langle u_{p_k}-u^{*},x-u^{*}\rangle$$

(14)

and $u_{p_k}\to y.$ Thus, $y\in Fix\left(F\right)$ and

$$\underset{p\to \infty }{lim\; sup}\langle u_p-u^{*},x-u^{*}\rangle =\langle y-u^{*},x-u^{*}\rangle ,$$

which is nonpositive by the definition of $u^{*}.$ We obtain

$$\begin{array}{cc}\hfill \underset{p\to \infty }{lim\; sup}\langle T_p{u}_p-u^{*},x-u^{*}\rangle & =\underset{p\to \infty }{lim\; sup}[\langle u_p-u^{*},x-u^{*}\rangle +\langle T_p{u}_p-u_p,x-u^{*}\rangle ]\hfill \\ \hfill & =\underset{p\to \infty }{lim\; sup}\langle u_p-u^{*},x-u^{*}\rangle \le 0.\hfill \end{array}$$

Finally,

$$\begin{array}{cc}\hfill \parallel u_{p+1}-u^{*}{\parallel }^2& =\parallel (1-a_p)(T_p{u}_p-u^{*})+a_p(x-u^{*}){\parallel }^2\hfill \\ \hfill & ={(1-a_p)}^2\parallel T_p{u}_p-u^{*}{\parallel }^2+a_p^2{\parallel x-u^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_p\langle (1-a_p)(T_p{u}_p-u^{*}),x-u^{*}\rangle \hfill \\ \hfill & ={(1-a_p)}^2\parallel b_p(F{u}_p-u^{*})+(1-b_n)(G_{\alpha }{u}_p-u^{*}){\parallel }^2+a_p^2{\parallel x-u^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_p\langle T_p{u}_p-u^{*},x-u^{*}\rangle -2a_p^2\langle T_p{u}_p-u^{*},x-u^{*}\rangle \hfill \\ \hfill & \le {(1-a_p)}^2\left[b_p\parallel u_p-u^{*}\parallel +(1-b_p){\parallel G_{\alpha }{u}_p-u^{*}\parallel }^2\right]+a_p^2{\parallel x-u^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_p\langle T_p{u}_p-u^{*},x-u^{*}\rangle \hfill \\ \hfill & \le {(1-a_p)}^2\parallel u_p-u^{*}{\parallel }^2+(1-b_p)\parallel G_{\alpha }{u}_p-u^{*}{)\parallel }^2+a_p{\parallel x-u^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_p\langle T_p{u}_p-u^{*},x-u^{*}\rangle \hfill \\ \hfill & =(1-t_p){\parallel u_p-u^{*}\parallel }^2+t_p{r}_p+s_p,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill & t_p=2a_p-a_p^2,r_p=a_p\left[\parallel x-u^{*}{\parallel }^2+2\langle T_p{u}_p-u^{*},x-u^{*}\rangle \right],\hfill \\ \hfill & s_p=(1-b_p){\parallel G_{\alpha }{u}_p-u^{*}\parallel }^2.\hfill \end{array}$$

Now, Lemma 5 implies that $u_p\to u^{*}.$

To prove (II), let $v^{*}$ be the unique solution of the variational inequality (10) and compute

$$\begin{array}{cc}\hfill \parallel u_{p+1}-v^{*}{\parallel }^2& =\parallel a_{n}x+(1-a_n)T_p{u}_p-v^{*}+a_p{v}^{*}-a_p{v}^{*}{\parallel }^2\hfill \\ \hfill & =\parallel a_p(x-v^{*})+(1-a_n)(T_p{u}_p-v^{*}){\parallel }^2\hfill \\ \hfill & \le {(1-a_n)}^2{\parallel (T_p{u}_p-v^{*})\parallel }^2+2a_p\langle x-v^{*},u_{p+1}-v^{*}\rangle \hfill \\ \hfill & ={(1-a_n)}^2{\parallel b_p(F{u}_p-v^{*})+(1-b_p)(G_{\alpha }{u}_p-v^{*})\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_p\langle x-v^{*},u_{p+1}-v^{*}\rangle \hfill \\ \hfill & \le {(1-a_p)}^2\parallel u_p-v^{*}{\parallel }^2+b_p{\parallel F{u}_p-v^{*}\parallel }^2\hfill \\ \hfill & \phantom{==}+2a_p\langle x-v^{*},u_{p+1}-v^{*}\rangle \hfill \end{array}$$

(15)

We have two cases, namely, the sequence $\{\parallel u_p-v^{*}\parallel \}$ is eventually not increasing or not eventually not increasing.

Case (II) 1. There exists $p_0\in \mathbb{N}$ such that $\parallel u_{p+1}-v^{*}\parallel \le \parallel u_p-v^{*}\parallel$ for all $p\ge p_0.$ Put

$$\begin{array}{cc}\hfill {\phi }_p& =2\langle u_{p+1}-v^{*},x-v^{*}\rangle ,\mathrm{and}\hfill \\ \hfill {\mu }_p& =(1-b_p){\parallel (G_{\alpha }{u}_p-v^{*})\parallel }^2.\hfill \end{array}$$

Since ${(1-a_p)}^2\le (1-a_p),$ we have

$$\parallel u_{p+1}-v^{*}{\parallel }^2\le (1-a_p){\parallel u_p-v^{*}\parallel }^2+a_p{\phi }_p+{\mu }_p.$$

Since $\{\parallel u_p-v^{*}\parallel \}$ is not eventually increasing, ${lim}_{p\to \infty }\parallel u_p-v^{*}\parallel$ exists. Thus,

$$\begin{array}{cc}\hfill 0& =\underset{p\to \infty }{lim}(\parallel u_{p+1}-v^{*}\parallel -\parallel u_p-v^{*}\parallel )\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; inf}(a_p\parallel x-v^{*}\parallel +(1-a_n)\parallel T_p{u}_p-v^{*}\parallel -\parallel u_p-v^{*}\parallel )\hfill \\ \hfill & =\underset{p\to \infty }{lim\; inf}(\parallel T_p{u}_p-v^{*}\parallel -\parallel u_p-v^{*}\parallel )\hfill \\ \hfill & =\underset{p\to \infty }{lim\; inf}(\parallel b_p(F{u}_p-v^{*})+(1-b_p)(G_{\alpha }{u}_p-v^{*})\parallel -\parallel u_p-v^{*}\parallel )\hfill \\ \hfill & =\underset{p\to \infty }{lim\; inf}(\parallel G_{\alpha }{u}_p-v^{*}\parallel -\parallel u_p-v^{*}\parallel )\hfill \\ \hfill & =\underset{p\to \infty }{lim\; sup}(\parallel u_p-v^{*}\parallel -\parallel u_p-v^{*}\parallel )=0\hfill \end{array}$$

Hence,

$$\underset{p\to \infty }{lim}(\parallel G_{\alpha }{u}_p-v^{*}\parallel -\parallel u_p-v^{*}\parallel )=0.$$

From the strong quasi-nonexpansiveness of $G_{\alpha }$, we conclude that

$$G_{\alpha }{u}_p-u_p\to 0.$$

(16)

The rest of the proof is similar to the proof of (I).

Case (II) 2. The sequence $\{\parallel u_p-v^{*}\parallel \}$ is not eventually not increasing. There exists a subsequence $\{\parallel u_{p_k}-v^{*}\parallel \}$ such that $\parallel u_{p_k}-v^{*}\parallel <\parallel u_{p_k+1}-v^{*}\parallel$ for all $k\in \mathbb{N}.$ By Lemma 6, there exists an increasing sequence of integers $\left\{\tau \right(p\left)\right\}$ satisfying

$$\begin{array}{cc}\hfill & \underset{p\to \infty }{lim}\tau \left(p\right)=\infty ,\parallel u_{\tau \left(p\right)}-v^{*}\parallel \le \parallel u_{\tau \left(p\right)+1}-v^{*}\parallel \hfill \\ \hfill & \parallel u_p-v^{*}\parallel \le \parallel u_{\tau \left(p\right)+1}-v^{*}\parallel ,\forall p\ge p_0.\hfill \end{array}$$

(17)

Thus,

$$0\le \underset{p\to \infty }{lim\; inf}(\parallel u_{\tau \left(p\right)+1}-v^{*}\parallel -\parallel u_{\tau \left(p\right)}-v^{*}\parallel ).$$

Using (16) with $\tau \left(p\right)$ instead of $p,$ we obtain

$$\underset{p\to \infty }{lim}\parallel G_{\alpha }{u}_{\tau \left(p\right)}-v^{*}\parallel -\parallel u_{\tau \left(p\right)}-v^{*}\parallel =0.$$

The strong quasi-nonexpasiveness of $G_{\alpha }$ implies

$$G_{\alpha }{u}_{\tau \left(p\right)}-u_{\tau \left(p\right)}\to 0,$$

(18)

Since $I-G_{\alpha }$ is demiclosed at $0,$ we conclude that

$$\underset{p\to \infty }{lim\; sup}\langle u_{\tau \left(p\right)+1}-v^{*},x-v^{*}\rangle \le 0.$$

(19)

One may observe that

$$\begin{array}{cc}\hfill \parallel u_{\tau \left(p\right)+1}-u_{\tau \left(p\right)}\parallel & =a_{\tau \left(p\right)}\parallel x-u_{\tau \left(p\right)}\parallel +(1-a_{\tau \left(p\right)})b_{\tau \left(p\right)}O\left(1\right)\hfill \\ \hfill & \phantom{=}+(1-a_{\tau \left(p\right)})\parallel G_{\alpha }{u}_{\tau \left(p\right)}{-}_{\tau \left(p\right)}\parallel ,\hfill \end{array}$$

where $O\left(1\right)$ represents a bounded sequence. Thus, from (18) it follows that

$$u_{\tau \left(p\right)+1}-u_{\tau \left(p\right)}\to 0.$$

Replacing p by $\tau \left(p\right)$ in (15) yields

$$\begin{array}{cc}\hfill \parallel u_{\tau \left(p\right)+1}-v^{*}\parallel & \le {(1-a_{\tau \left(p\right)})}^2{\parallel u_{\tau \left(p\right)}-v^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle u_{\tau \left(p\right)+1}-v^{*},x-v^{*}\rangle +b_{\tau \left(p\right)}{\parallel F{u}_{\tau \left(p\right)}-v^{*}\parallel }^2\hfill \\ \hfill & \le {(1-a_{\tau \left(p\right)})}^2{\parallel u_{\tau \left(p\right)+1}-v^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle u_{\tau \left(p\right)+1}-v^{*},x-v^{*}\rangle +b_{\tau \left(p\right)}{\parallel F{u}_{\tau \left(p\right)}-v^{*}\parallel }^2.\hfill \end{array}$$

As a consequent, we have

$$\begin{array}{cc}\hfill 2a_{\tau \left(p\right)}{\parallel u_{\tau \left(p\right)+1}-v^{*}\parallel }^2& \le {\left(a_{\tau \left(p\right)}\right)}^2{\parallel u_{\tau \left(p\right)+1}-v^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle u_{\tau \left(p\right)+1}-v^{*},x-v^{*}\rangle +b_{\tau \left(p\right)}{\parallel F{u}_{\tau \left(p\right)}-v^{*}\parallel }^2.\hfill \end{array}$$

Dividing by $a_{\tau \left(p\right)}$ gives

$$\begin{array}{cc}\hfill 0\le 2\parallel u_{\tau \left(p\right)+1}-v^{*}{\parallel }^2& \le a_{\tau \left(p\right)}{\parallel u_{\tau \left(p\right)+1}-v^{*}\parallel }^2\hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle u_{\tau \left(p\right)+1}-v^{*},x-v^{*}\rangle +{\displaystyle \frac{b_{\tau \left(p\right)}}{a_{\tau \left(p\right)}}}{\parallel F{u}_{\tau \left(p\right)}-v^{*}\parallel }^2.\hfill \end{array}$$

Since $b_p/a_p\to 0,$ by (19) we obtain

$$\underset{p\to \infty }{lim}\parallel u_{\tau \left(p\right)}-v^{*}\parallel \le \underset{p\to \infty }{lim}\parallel u_{\tau \left(p\right)+1}-v^{*}\parallel =0.$$

From (17), we obtain that $u_p\to v^{*}$.

To prove (III), let $\overline{u}$ be the unique point in $Fix\left(F\right)\cap Fix\left(G\right)$ that satisfies the variational inequality (11). We have

$$\begin{array}{cc}\hfill \parallel T_p{u}_p-\overline{u}{\parallel }^2& =\parallel b_p(F{u}_p-\overline{u})+(1-b_p){(G{u}_p-\overline{u}\parallel }^2\hfill \\ \hfill & =b_p\parallel F{u}_p-\overline{u}{\parallel }^2+(1-b_p)\parallel G_{\alpha }{u}_p-\overline{u}{\parallel }^2-b_p(1-b_p){\parallel F{u}_p-G_{\alpha }{u}_p\parallel }^2\hfill \\ \hfill & \le \parallel u_p-\overline{u}{\parallel }^2-b_p(1-b_p){\parallel F{u}_p-G_{\alpha }{u}_p\parallel }^2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel u_{p+1}-\overline{u}{\parallel }^2& =\parallel a_p(x-\overline{u})+(1-a_p){(T_p{u}_p-\overline{u}\parallel }^2\hfill \\ \hfill & \le {(1-a_p)}^2\parallel T_p{u}_p-\overline{u}{\parallel }^2+a_p^2\parallel x-\overline{u}{\parallel }^2+a_p\parallel (x-\overline{u})(T_p{u}_p-\overline{u})\parallel \hfill \\ \hfill & \le \parallel u_p-\overline{u}{\parallel }^2-b_p(1-b_p){\parallel F{u}_p-G_{\alpha }{u}_p\parallel }^2\hfill \\ \hfill & \phantom{==}+a_p^2\parallel x-\overline{u}{\parallel }^2+a_p\parallel (x-\overline{u})(T_p{u}_p-\overline{u})\parallel .\hfill \end{array}$$

(20)

Similar to (II), we have two cases.

Case (III) 1. $\{\parallel u_p-\overline{u}\parallel \}$ is eventually not increasing. There exists $p_0\in \mathbb{N}$ such that $\parallel u_{p+1}-\overline{u}\parallel \le \parallel u_p-\overline{u}\parallel ,$ for all $n\ge n_0.$ Thus, ${lim}_{p\to \infty }\parallel u_p-\overline{u}\parallel$ exists. We have

$$\begin{array}{cc}\hfill b_p(1-b_p)\parallel F{u}_p-G{u}_p\parallel \le & \parallel u_p-\overline{u}{\parallel }^2-\parallel u_{p+1}-\overline{u}{\parallel }^2+a_p^2{\parallel x-\overline{u}\parallel }^2\hfill \\ \hfill & +a_p\parallel (x-\overline{u})(T_p{u}_p-\overline{u})\parallel .\hfill \end{array}$$

Since ${lim\; inf}_{p\to \infty }{b}_p(1-b_p)>0,$ we have

$$F{u}_p-G_{\alpha }{u}_p\to 0.$$

(21)

Moreover,

$$\begin{array}{cc}\hfill 0& =\underset{p\to \infty }{lim}(\parallel u_{p+1}-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; inf}(a_p\parallel x-\overline{u}\parallel +(1-a_p)\parallel T_p{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )\hfill \\ \hfill & =\underset{p\to \infty }{lim\; inf}(\parallel T_p{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )\hfill \\ \hfill & =\underset{p\to \infty }{lim\; inf}(\parallel b_p(F{u}_p-\overline{u}+(1-b_p)(G_{\alpha }{u}_p-\overline{u})\parallel -\parallel u_p-\overline{u}\parallel )\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; inf}(b_p(\parallel F{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )+(1-b_p)(\parallel G_{\alpha }{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel \left)\right)\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; sup}(b_p(\parallel F{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )+(1-b_p)(\parallel G_{\alpha }{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel \left)\right)\hfill \\ \hfill & \le 0\hfill \end{array}$$

Therefore

$$\underset{p\to \infty }{lim}(b_p(\parallel F{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )+(1-b_p)(\parallel G_{\alpha }{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel \left)\right)=0.$$

It follows that

$$\underset{p\to \infty }{lim}(b_p(\parallel F{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )=\underset{p\to \infty }{lim}(1-b_p)(\parallel G_{\alpha }{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel \left)\right)=0.$$

(22)

Thus,

$$\underset{p\to \infty }{lim}\left(\right(\parallel F{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )=\underset{p\to \infty }{lim}(\parallel G_{\alpha }{u}_p-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel \left)\right)=0.$$

(23)

From strong quasi-nonexpansiveness of $G_{\alpha }$ it follows

$$S{u}_p-u_p\to 0.$$

(24)

Since $u_p-F{u}_p=u_p-G_{\alpha }{u}_p+G_{\alpha }{u}_p-F{u}_p,$ we obtain

$$u_p-F{u}_p\to 0.$$

(25)

Choose a subsequence $u_{p_k}\to z$ such that

$$\underset{p\to \infty }{lim\; sup}\langle u_p-\overline{u},x-\overline{u}\rangle =\underset{p\to \infty }{lim}\langle u_{p_k}-\overline{u},x-\overline{u}\rangle =\langle z-\overline{u},x-\overline{u}\rangle .$$

Since both F and $G_{\alpha }$ are demiclosed at 0 and by (24), (25), one can conclude that $z\in Fix\left(F\right)\cap Fix\left(G_{\alpha }\right).$ Hence, by the definition of $\overline{u},$ we obtain (26). Furthermore, from (24) and (25) we have

$$T_p{u}_p-u_p\to 0.$$

(26)

Finally

$$\begin{array}{cc}\hfill \parallel u_{p+1}-\overline{u}{\parallel }^2& =\parallel (1-a_p)(T_p{u}_p-\overline{u})+a_p(x-\overline{u})\parallel \hfill \\ \hfill & ={(1-a_p)}^2\parallel T_p{u}_p-\overline{u}{\parallel }^2+a_p^2{\parallel x-\overline{u}\parallel }^2+2a_p\langle (1-a_p)(T_p{u}_p-\overline{u}),x-\overline{u}\rangle \hfill \\ \hfill & \le {(1-a_p)}^2\parallel u_p-\overline{u}{\parallel }^2+a_p^2[\parallel x-\overline{u}{\parallel }^2-2\langle T_p{u}_p-\overline{u},x-\overline{u}\rangle ]\hfill \\ \hfill & \phantom{=}+2a_p\langle T_p{u}_p-u_p,x-\overline{u}\rangle +2a_p\langle u_p-\overline{u},x-\overline{u}\rangle \hfill \\ \hfill & \le (1-a_p)\parallel u_p-\overline{u}{\parallel }^2+a_p[\parallel x-\overline{u}{\parallel }^2-2\langle T_p{u}_p-\overline{u},x-\overline{u}\rangle ]\hfill \\ \hfill & \phantom{=}+2a_p\langle T_p{u}_p-u_p,x-\overline{u}\rangle +2a_p\langle u_p-\overline{u},x-\overline{u}\rangle .\hfill \end{array}$$

Putting

$$\begin{array}{cc}\hfill s_p& =1-{(1-a_p)}^2,\hfill \\ \hfill {\sigma }_p& =2\langle T_p{u}_p-u_p,x-p_0\rangle +2\langle u_p-\overline{u},x-\overline{u}\rangle +a_p[\parallel x-\overline{u}{\parallel }^2-2\langle T_p{u}_p-\overline{u},x-\overline{u}\rangle ],\hfill \end{array}$$

the conclusion follows from Lemma 5.

Case (III) 2. The sequence $\{\parallel u_p-\overline{u}\parallel \}$ is not eventually not increasing, i.e., there exists a subsequence $\{\parallel u_{p_k}-\overline{u}\parallel \}$ such that $\parallel u_{p_k}-\overline{u}\parallel <\parallel u_{p_{k+1}}-\overline{u}\parallel ,\forall j\in \mathbb{N}.$

Lemma 6 implies that there exists an increasing sequence of integers ${\left\{\tau \left(p\right)\right\}}_{p\in \mathbb{N}}$ satisfying (17). Therefore

$$\begin{array}{cc}\hfill o& \le \underset{p\to \infty }{lim\; inf}(\parallel u_{\tau \left(p\right)+1}-\overline{u}\parallel -\parallel u_{\tau \left(p\right)}-\overline{u}\parallel )\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; sup}(\parallel u_{\tau \left(p\right)+1}-\overline{u}\parallel -\parallel u_{\tau \left(p\right)}-\overline{u}\parallel )\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; sup}(\parallel u_{p+1}-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; sup}(\parallel (1-a_p)(T_p{u}_p-\overline{u})+a_p(x-\overline{u})\parallel -\parallel u_p-\overline{u}\parallel )\hfill \\ \hfill & \le \underset{p\to \infty }{lim\; sup}((1-a_p)\parallel u_p-\overline{u}\parallel +a_p\parallel x-\overline{u}\parallel -\parallel u_p-\overline{u}\parallel )\hfill \\ \hfill & =0\hfill \end{array}$$

We obtain

$$\underset{p\to \infty }{lim}(\parallel u_{\tau \left(p\right)+1}-\overline{u}\parallel -\parallel u_{\tau \left(p\right)}-\overline{u}\parallel )=0$$

(27)

Now, using (24) with $\tau \left(p\right)$ instead of $p,$ we have

$$G_{\alpha }{u}_{\tau \left(p\right)}-u_{\tau \left(p\right)}\to 0,$$

(28)

and (20) can be written as

$$\begin{array}{cc}\hfill 0& \le b_{\tau \left(p\right)}(1-b_{\tau \left(p\right)}){\parallel F{u}_{\tau \left(p\right)}-G_{\alpha }{u}_{\tau \left(p\right)}\parallel }^2\hfill \\ \hfill & \le \parallel u_{\tau \left(p\right)}-\overline{u}{\parallel }^2-\parallel u_{\tau \left(p\right)+1}-\overline{u}{\parallel }^2+a_p\parallel x-\overline{u}\parallel \left(1+\parallel (T_p{u}_p-\overline{u})\parallel \right).\hfill \end{array}$$

From (27) and since ${lim\; inf}_{p\to \infty }{b}_p(1-b_p)>0,$

$$F{u}_{\tau \left(p\right)}-G_{\alpha }{u}_{\tau \left(p\right)}\to 0.$$

(29)

We also obtain that

$$u_{\tau \left(p\right)}-F{u}_{\tau \left(p\right)}=u_{\tau \left(p\right)}-G_{\alpha }{u}_{\tau \left(p\right)}+G_{\alpha }{u}_{\tau \left(p\right)}-F{u}_{\tau \left(p\right)}.$$

By (28) and (29), we have that

$$u_{\tau \left(p\right)}-F{u}_{\tau \left(p\right)}\to 0\mathrm{and}{u}_{\tau \left(p\right)}-T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}\to 0.$$

(30)

Similar to (26) changing $\tau \left(p\right)$ with $p,$ we have

$$\underset{p\to \infty }{lim\; sup}\langle u_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle \le 0.$$

Following the same proof of (26), replacing p with $\tau \left(p\right),$ we obtain

$$\underset{p\to \infty }{lim\; sup}\langle u_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle \le 0.$$

(31)

Now we compute,

$$\begin{array}{cc}\hfill \parallel u_{\tau \left(p\right)+1}-\overline{u}{\parallel }^2& \le {(1-a_{\tau \left(p\right)})}^2{\parallel u_{\tau \left(p\right)}-\overline{u}\parallel }^2\hfill \\ \hfill & \phantom{=}+a_{\tau \left(p\right)}^2[\parallel x-\overline{u}{\parallel }^2-2\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle ]\hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-u_{\tau \left(p\right)},x-\overline{u}\rangle +2a_{\tau \left(p\right)}\langle u_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle \hfill \\ \hfill & \le {(1-a_{\tau \left(p\right)})}^2{\parallel u_{\tau \left(p\right)+1}-\overline{u}\parallel }^2\hfill \\ \hfill & \phantom{=}+a_{\tau \left(p\right)}^2[\parallel x-\overline{u}{\parallel }^2-2\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle ]\hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-u_{\tau \left(p\right)},x-\overline{u}\rangle +2a_{\tau \left(p\right)}\langle u_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle \hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill 2a_{\tau \left(p\right)}{\parallel u_{\tau \left(p\right)+1}-\overline{u}\parallel }^2& \le a_{\tau \left(p\right)}^2[\parallel x-\overline{u}{\parallel }^2-2\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle ]\hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle u_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle \hfill \\ \hfill & \phantom{=}+2a_{\tau \left(p\right)}\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-u_{\tau \left(p\right)},x-\overline{u}\rangle \hfill \end{array}$$

dividing by $a_{\tau \left(p\right)},$ we get

$$\begin{array}{cc}\hfill 0& \le 2\parallel u_{\tau \left(p\right)+1}-\overline{u}{\parallel }^2\hfill \\ \hfill & \le a_{\tau \left(p\right)}[\parallel x-\overline{u}{\parallel }^2-2\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle ]+2\langle u_{\tau \left(p\right)}-\overline{u},x-\overline{u}\rangle \hfill \\ \hfill & \phantom{=}+2\langle T_{\tau \left(p\right)}{u}_{\tau \left(p\right)}-u_{\tau \left(p\right)},x-\overline{u}\rangle \hfill \end{array}$$

Taking the limsup and recalling the hypothesis (30) and (31), we obtain

$$\underset{p\to \infty }{lim}\parallel u_{\tau \left(p\right)}-\overline{u}\parallel \le \underset{p\to \infty }{lim}\parallel u_{\tau \left(p\right)+1}-\overline{u}\parallel =0$$

Now by (17), we conclude that $u_p\to \overline{u}.$ □

A more general result could be proved similarly to the proof of Theorem 1.

Theorem 2.

Let $\mathcal{H}$ be a Hilbert space and C be a closed convex subset of $\mathcal{H}.$ Let $F:C\to C$ be a k-strictly pseudocontractive mapping and $G:C\to C$ be a β-demicontractive mapping satisfying Condition A such that $I-G$ is demiclosed at $0.$ Assume that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$ Let $\left\{a_p\right\}$ and $\left\{b_p\right\}$ be sequences in $[0,1]$ such that $a_p\to 0$ and ${\sum }_{p=1}^{\infty }{a}_p=\infty .$ Let $\left\{u_p\right\}$ be a sequence generated in the following manner:

$$\left\{\begin{array}{cc}x,u_1\in C,\hfill & \\ u_{p+1}=a_{p}x+(1-a_p)[b_p{F}_{\lambda }{u}_p+(1-b_p)((1-\alpha )u_p+\alpha G{u}_p)],p\ge 1.\hfill & \end{array}\right.$$

(32)

where $F_{\lambda }u=(1-\lambda )u+\lambda Fu$, with $\lambda \in (0,1-k)$.

Then, the following assertions hold.

(I)

If ${\sum }_{p=1}^{\infty }(1-b_p)<\infty ,{\sum }_{p=1}^{\infty }|a_p=a_{p+1}|<\infty ,$ then $\left\{u_p\right\}$ strongly converges to $u^{*}\in Fix\left(F\right)$ that is the unique point in $Fix\left(F\right)$ that solves the variational inequality

$$\langle u^{*}-x,u-u^{*}\rangle \ge 0,\forall u\in Fix\left(F\right),$$

i.e. $u^{*}=P_{Fix\left(F\right)}x.$

(II)

If ${\sum }_{p=1}^{\infty }(1-b_p)<\infty ,{\displaystyle \frac{b_p}{a_p}}\to 0,$ then $\left\{u_p\right\}$ converges strongly to $v^{*}\in Fix\left(G\right)$ that is the unique point in $Fix\left(G\right)$ that solves the variational inequality

$$\langle v^{*}-x,u-v^{*}\rangle \ge 0,\forall u\in Fix\left(G\right),$$

(33)

i.e. $v^{*}=P_{Fix\left(G\right)}x$.

(III)

If ${lim\; inf}_{p\to \infty }{b}_p(1-b_p)>0,$ then $\left\{u_p\right\}$ strongly converges to $\overline{u}\in Fix\left(F\right)\cap Fix\left(G\right)$ that is the unique solution of the variational inequality

$$\langle \overline{u}-x,u-\overline{u}\rangle \ge 0,\forall u\in Fix\left(F\right)\cap Fix\left(G\right),$$

i.e. $\overline{u}=P_{Fix\left(F\right)\cap Fix\left(G\right)}x.$

Proof. 

As F is k-strictly pseudocontractive, by Lemma 3 we have that the averaged mapping

$$F_{\lambda }u=(1-\lambda )u+\lambda Fu$$

is nonexpansive, for any $\lambda \in (0,1-k)$ and that $Fix\left(F\right)=Fix\left(F_{\lambda }\right)$. We apply Theorem 1 for $F_{\lambda }$ and G and get the conclusion. □

Remark 1.

Most of the results obtained in Takahashi and Tamura [21], Moudafi [15], Cianciaruso et al. [7], Falset et al. [8], Iemoto and Takahashi [9] could be obtained as corollaries of our main results or could be slightly improved by considering our averaged Halpern type algorithm (8).

We illustrate this fact in the following for four different instances.

• If F is nonexpansive and G is nonspreading, then by Theorem 1 we obtain an improvement of Theorem 4.1 in Iemoto and Takahashi [9], in the sense that for our averaged Halpern type algorithm (8) we have strong convergence, while for the Krasnoselsij-Mann iterative procedure (5) only weak convergence was obtained by Iemoto and Takahashi [9];
• If F is nonexpansive and G is nonspreading, then by Theorem 1 we obtain the main result (i.e., Theorem 14) in Cianciaruso et al. [7];
• If F and G are both nonexpansive, then by Theorem 1 we obtain an improvement of the main result in Takahashi and Tamura [21], in the sense that for our averaged Halpern type algorithm (8) we get strong convergence, while for the Krasnoselsij-Mann iterative procedure (5) only weak convergence is obtained by Takahashi and Tamura [21];
• If F is nonexpansive and G is strongly quasi-nonexpansive, then by Theorem 1 we obtain the main result (i.e., Theorem 3) in Falset et al. [8];
• ...

4. Numerical Illustrations

In this section, we consider some numerical examples to illustrate the numerical behaviour of Algorithm (8), for approximating a common fixed point for a nonexpansive mapping and a $\beta$-demicontractive mapping.

Example 2.

Let $\mathcal{H}$ be the real line with the usual norm and $D=[0,1].$ Define F and G on D as follows as

$$F\left(u\right)=5/3-u,u\in D$$

(34)

and

$$G\left(u\right)=\left\{\begin{array}{cc}5/6,\hfill & if0\le u<1\hfill \\ 1/3,\hfill & \mathrm{if}u=1.\hfill \end{array}\right.$$

(35)

Note that F is nonexpansive, and G is $1/2$-demicontractive. It is easy to check that $Fix\left(F\right)\cap Fix\left(G\right)=\{5/6\}.$ The mapping G is neither quasi-nonexpasive nor nonexpansive (and hence it is neither strongly quasi-nonexpansive nor nonspeading).

Therefore, we cannot apply any of the results in Takahashi and Tamura [21], Moudafi [15], Cianciaruso et al. [7], Falset et al. [8], Iemoto and Takahashi [9] etc. to solve the common fixed point problem for F and G.

If we put

$$a_p={\displaystyle \frac{1}{rp}},b_p={\displaystyle \frac{2p}{1+3p}},p\in \mathbb{N},r\in \mathbb{R}\mathit{and}r\ge 1,$$

then all assumptions in Theorem 1 part (iii) are satisfied. This implies that the sequence $\left\{u_p\right\}$ generated by the algorithm (8) converges to $5/6$, the unique common fixed point of F and G.

Several numerical experiments were conducted in MATLAB using the algorithm(8)with different values of the parameters.

The numerical results for three initial values with $r=1000$ are presented in Table 1.

Table 2 shows numerical results for three initial values with $r=5000$ and $x=0.7.$ One can see that for x near the common fixed point and r large, the iterations converge faster.

5. Conclusions

• We have introduced an averaged iterative Halpern type algorithm intended to find a common fixed point for a pair consisting of a nonexpansive mapping and a demicontractive mapping which also solves a certain variational inequality problem;
• We established a strong convergence theorem (Theorem 1) for the sequence generated by our algorithm;
• We extended Theorem 1 to the more general case of a pair of mappings consisting of a k-strictly pseudocontractive mapping F and a $\beta$-demicontractive mapping G (Theorem 2), by considering the double averaged Halpern type algorithm (32).
• We validated the effectiveness of our general theoretical results by some appropriate numerical experiments, corresponding to part (iii) of Theorem 1, which are reported in Section 4. These results clearly illustrate the progress of our convergence results over existing literature.
• For other related results we refer the reader to Agwu et al. [1], Araveeporn et al. [2], Ceng and Yao [5], Ceng and Yuan [6], Jaipranop and Saejung [10], Kraikaew and Saejung [12], Mebawondu et al. [14], Nakajo et al. [17], Petruşel and Yao [18], Rizvi [19], Sahu et al. [20], Thuy [22], Uba et al. [23], Xu [25], Yao et al. [26,27], Yotkaew et al. [28],...