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

Abstract

In this paper, we propose and study an averaged Halpern-type algorithm for approximating the solution of a common fixed-point problem for a couple of nonexpansive and demicontractive mappings with a variational inequality constraint in the setting of a Hilbert space. The strong convergence of the sequence generated by the algorithm is established under feasible assumptions on the parameters involved. In particular, we also obtain the common solution of the fixed point problem for nonexpansive or demicontractive mappings and of a variational inequality problem. Our results extend and generalize various important related results in the literature that were established for two pairs of mappings: (nonexpansive, nonspreading) and (nonexpansive, strongly quasi-nonexpansive). Numerical tests to illustrate the superiority of our algorithm over the ones existing in the literature are also reported.

1. Introduction

Let $\mathcal{H}$ be a real Hilbert space with norm $\parallel ·\parallel$ and inner product $⟨·,·⟩.$ 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 as follows:

(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 reverse cases are no more true, as illustrated by the next example.

Example 1

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

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

(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 the literature that 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 [2] 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 [3] considered a slightly different Krasnoselsij–Mann iterative procedure for the same problem that he called a “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.

Furthermore, Iemoto and Takahashi [2] 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 they 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.

Ceng and Yuan [4] introduced and investigated composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem and a common fixed-point problem for finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework of infinite-dimensional Hilbert spaces. A modified implicit extragradient iteration has been used by Ceng and Yuan [5] for finding a common solution of the common fixed-point problem of a countable family of nonexpansive mappings, a general system of variational inequalities, and a variational inclusion in a uniformly convex and q-uniformly smooth Banach space with $1<q\le 2$.

To the best of our knowledge, the are no research works regarding the solution of variational inequality problems and common fixed-point problems in the class of demicontractive mappings. 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. [6], Falset et al. [7], Lemoto and Takahashi [8], 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.

Moreover, using Theorem 2, we extend Theorem 1 to the case when the averaged iterative Halpern-type algorithm is used to approximate a solution of the common fixed-point problem for a pair consisting of a k-strictly pseudocontractive mapping and a demicontractive mapping, with a variational inequality constraint.

In order to validate the effectiveness of our general theoretical results, we provide 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 the existing literature.

2. Preliminaries

We recall some important lemmas used in the proofs of our main results. The following two lemmas are taken from Berinde [9].

Lemma 1

(Berinde [9]). Let $\mathcal{H}$ be a real Hilbert space and $D\subset \mathcal{H}$ be 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

(Berinde [9]). Let $\mathcal{H}$ be a real Hilbert space and $D\subset \mathcal{H}$ be 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 [10]). 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

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

Lemma 5

(Xu [11]). Let $\left\{{\alpha }_n\right\}$ be a sequence of non-negative 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

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

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

Lemma 6

(Maingé [12]). 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

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

(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

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

(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 $\overline{u}\in Fix\left(F\right)\cap Fix\left(G\right)$, which is the unique solution of the variational inequality

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

(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

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

(12)

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

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

(13)

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

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

(14)

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 (14) imply that

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

(15)

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

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

(16)

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

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

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

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

Finally,

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

where

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

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

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

(17)

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

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

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. Hence, by (12), (13), and the fact that both sequences $\left\{a_p\right\}$ and $\left\{{\displaystyle \frac{b_p}{a_p}}\right\}$ converging to 0 and $G_{\alpha }$ are strongly quasi-nonexpansive, we obtain

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

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

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

(18)

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 nonincreasing. 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

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

(19)

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 (18) 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

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

(20)

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

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

(21)

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 (20), it follows that

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

Replacing p by $\tau \left(p\right)$ in (17) yields

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

As a consequent, we have

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

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

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

Since $b_p/a_p\to 0,$ by (21), 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 (19), 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

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

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

(22)

Similar to (II), we have two cases.

Case (III) 1. $\{\parallel u_p-\overline{u}\parallel \}$ is eventually nonincreasing. 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

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

(23)

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

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

(24)

Thus,

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

(25)

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

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

(26)

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

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

(27)

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

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

Since both F and $G_{\alpha }$ are demiclosed at 0 and by (26) and (27), 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 (28). Furthermore, from (26) and (27), we have

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

(28)

Finally,

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

Putting

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

the conclusion follows from Lemma 5.

Case (III) 2. The sequence $\{\parallel u_p-\overline{u}\parallel \}$ is not eventually nonincreasing, 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 (19). 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

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

(29)

Now, using (26) with $\tau \left(p\right)$ instead of $p,$ we have

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

(30)

and (22) 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 (29), and since ${lim\; inf}_{p\to \infty }{b}_p(1-b_p)>0,$

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

(31)

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 (30) and (31), we have that

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

(32)

Similar to (28) changing $\tau \left(p\right)$ with $p,$ we have

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

Following the same proof of (28), replacing p with $\tau \left(p\right),$ we obtain

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

(33)

Now we compute,

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

Consequently,

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

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

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

Taking the limsup and recalling the hypotheses of (32) and (33), 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 (19), we conclude that $u_p\to \overline{u}.$ □

A more general result can be proven 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 $\mathcal{T}:C\to C$ be a k-strictly pseudocontractive mapping and $\mathcal{S}:C\to C$ be a β-demicontractive mapping satisfying Condition A such that $I-\mathcal{S}$ is demiclosed at $0.$ Assume that $Fix\left(\mathcal{T}\right)\cap Fix\left(\mathcal{S}\right)\ne \varnothing .$ 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:

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

(34)

where ${\mathcal{T}}_{\lambda }u=(1-\lambda )u+\lambda \mathcal{T}u$, 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 $u^{*}\in Fix\left(\mathcal{T}\right)$ that is the unique point in $Fix\left(\mathcal{T}\right)$ that solves the variational inequality

$$⟨u^{*}-x,u-u^{*}⟩\ge 0,\forall u\in Fix\left(\mathcal{T}\right),$$

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

(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 $v^{*}\in Fix\left(\mathcal{S}\right)$ that is the unique point in $Fix\left(\mathcal{S}\right)$ that solves the variational inequality

$$⟨v^{*}-x,u-v^{*}⟩\ge 0,\forall u\in Fix\left(\mathcal{S}\right),$$

(35)

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

(III)

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

$$⟨\overline{u}-x,u-\overline{u}⟩\ge 0,\forall u\in Fix\left(\mathcal{T}\right)\cap Fix\left(\mathcal{S}\right),$$

i.e., $\overline{u}=P_{Fix\left(\mathcal{T}\right)\cap Fix\left(\mathcal{S}\right)}x.$

Proof. 

Since $\mathcal{T}$ is k-strictly pseudocontractive, by Lemma 3, we have that the averaged mapping

$${\mathcal{T}}_{\lambda }=(1-\lambda )I+\lambda \mathcal{T}$$

is nonexpansive for any $\lambda \in (0,1-k)$ and that $Fix\left(\mathcal{T}\right)=Fix\left({\mathcal{T}}_{\lambda }\right)$. Hence, ${\mathcal{T}}_{\lambda }$ and $\mathcal{S}$ satisfy the conditions of the operators in Theorem 1. Now, the rest of the proof is similar to that of Theorem 1 by replacing F and $G_{\alpha }$ in Theorem 1 by ${\mathcal{T}}_{\lambda }$ and $\mathcal{S},$ respectively. □

Remark 1.

Most of the results obtained in Takahashi and Tamura [2], Moudafi [3], Cianciaruso et al. [6], Falset et al. [7], and Lemoto and Takahashi [8] 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 Lemoto and Takahashi [8] 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 Lemoto and Takahashi [8];
• 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. [6];
• If F and G are both nonexpansive, then by Theorem 1, we obtain an improvement of the main result in Takahashi and Tamura [2] in the sense that for our averaged Halpern-type algorithm (8) we obtain strong convergence, while for the Krasnoselsij–Mann iterative procedure (5), only weak convergence is obtained by Takahashi and Tamura [2];
• 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. [7];

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:

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

(36)

and

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

(37)

Note that F is nonexpansive. It is easy to check that $Fix\left(F\right)\cap Fix\left(G\right)=\{5/6\}.$

Now, we show that G is ${\displaystyle \frac{1}{2}}$-demicontractive but neither quasi-nonexpansive nor nonexpansive (and hence, G is neither strongly quasi-nonexpansive nor nonspeading).

Indeed, for any $x\in [0,1)$ and $y={\displaystyle \frac{5}{6}}$ (the fixed point of G), we have

$${|Gx-y|}^2=0\le {|x-y|}^2+k{|x-Gx|}^2$$

for any $k\ge 0.$ For $x=1,$ we have

$${\left|G1-{\displaystyle \frac{5}{6}}\right|}^2={\left|{\displaystyle \frac{1}{3}}-{\displaystyle \frac{5}{6}}\right|}^2\le {\left|1-{\displaystyle \frac{5}{6}}\right|}^2+k{\left|1-{\displaystyle \frac{1}{3}}\right|}^2.$$

This holds when $k\ge {\displaystyle \frac{1}{2}}.$ Thus, G is ${\displaystyle \frac{1}{2}}$-demicontractive. If we take $x=1$ and $y={\displaystyle \frac{5}{6}},$ then we obtain

$$|Tx-Ty|=|Tx-y|={\displaystyle \frac{1}{2}}>|x-y|={\displaystyle \frac{1}{6}}.$$

Therefore, G is not quasi-nonexpansive nor nonexpansive.

Therefore, we cannot apply any of the results in Takahashi and Tamura [2], Moudafi [3], Cianciaruso et al. [6], Falset et al. [7], Lemoto and Takahashi [8], 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 $5/6$, which is the unique common fixed point of F and G.

Several numerical experiments were conducted in MATLAB R2022a using the algorithm (8) with different values of the parameters.

The numerical results for three initial values, taken arbitrarily from the set D and with $r=1000$, are presented in Table 1.

Table 1. Numerical results for $x=0.2,r=1000$, with initial values $u_0=1,u_0=0.7$, and $u_0=0.1$.

Iteration (p)	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$
0	1.000000	0.700000	0.100000
1	0.786549	0.822542	0.774552
2	0.837948	0.834349	0.839148
3	0.832452	0.833106	0.832234
4	0.833503	0.833354	0.833553
5	0.833264	0.833303	0.833251
⋯	⋯	⋯	⋯
15	0.833327	0.833327	0.833327
⋯	⋯	⋯	⋯
20	0.833329	0.833329	0.833329
⋯	⋯	⋯	⋯
50	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
51	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
111	0.833333	0.833333	0.833333
112	0.833333	0.833333	0.833333

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

Table 2. Numerical results for $x=0.7,r=5000$, with initial values $u_0=1,u_0=0.7$, and $u_0=0.1$.

Iteration (p)	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$
0	1.000000	0.700000	0.100000
1	0.786649	0.822642	0.774652
2	0.837988	0.834389	0.839188
3	0.832478	0.833133	0.832260
4	0.833522	0.833373	0.833572
5	0.833279	0.833318	0.833266
⋯	⋯	⋯	⋯
15	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
20	0.833332	0.833332	0.833332
⋯	⋯	⋯	⋯
26	0.833333	0.833333	0.833333
27	0.833333	0.833333	0.833333

We note that for the initial value $\overline{u}=0.7$, one obtains the solution of the split common fixed point after 57 iterations, while for the initial value $\overline{u}=0.2$, one obtains the same solution after 63 iterations.

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 (34). Moreover, Theorem 2 extends Theorem 1 to the case when the averaged iterative Halpern-type algorithm is used to approximate a solution of the common fixed-point problem for the pair consisting of a k-strictly pseudocontractive mapping and a demicontractive mapping, with a variational inequality constraint.
• We validated the effectiveness of our general theoretical results through 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 the existing literature.
• For other related results, we refer the reader to Agwu et al. [13], Araveeporn et al. [14], Ceng and Yao [15], Ceng and Yuan [4], Jaipranop and Saejung [16], Kraikaew and Saejung [17], Mebawondu et al. [18], Nakajo et al. [19], Petruşel and Yao [20], Rizvi [21], Sahu et al. [22], Thuy [23], Uba et al. [24], Xu [25], Yao et al. [26,27], Yotkaew et al. [28] etc.
• Our future scope is to apply the obtained results in order to solve some relevant real-world problems.