Inertial Iterative Algorithms for Split Variational Inclusion and Fixed Point Problems

Abstract

This paper aims to present two inertial iterative algorithms for estimating the solution of split variational inclusion $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and its extended version for estimating the common solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and fixed point problem $\left(\mathrm{FPP}\right)$ of a nonexpansive mapping in the setting of real Hilbert spaces. We establish the weak convergence of the proposed algorithms and strong convergence of the extended version without using the pre-estimated norm of a bounded linear operator. We also exhibit the reliability and behavior of the proposed algorithms using appropriate assumptions in a numerical example.

1. Introduction

The split feasibility problems $\left({\mathrm{S}}_{\mathrm{p}}\mathrm{FP}\right)$, due to Censor and Elfving [1], have ample applications in medical science. Therefore, $\left({\mathrm{S}}_{\mathrm{p}}\mathrm{FP}\right)$ has been widely used over the past twenty years in the design of intensity-modulation therapy treatments and other areas of applied sciences, see, e.g., [2,3,4,5]. Censor et al. [6,7] merged the variational inequality problem $\left({\mathrm{VI}}_{\mathrm{t}}\mathrm{P}\right)$ and $\left({\mathrm{S}}_{\mathrm{p}}\mathrm{FP}\right)$, and a different kind of problem came to existence known as split variational inequality problem $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{t}}\mathrm{P}\right)$ defined as:

$$\begin{array}{c}\hfill \mathrm{Find}\mathrm{out}{l}^{*}\in Q_1\mathrm{such}\mathrm{that}{l}^{*}\in {\mathrm{VI}}_{\mathrm{t}}\mathrm{P}(F_1;Q_1)\mathrm{and}B\left(l^{*}\right)\in {\mathrm{VI}}_{\mathrm{t}}\mathrm{P}(F_2;Q_2),\end{array}$$

(1)

where $Q_1$ and $Q_2$ are subsets of Hilbert spaces $X_1$ and $X_2$, respectively, $B:X_1\to X_2$ is a bounded linear operator, $F_1:X_1\to X_1$ and $F_2:X_2\to X_2$ are two operators, ${\mathrm{VI}}_{\mathrm{t}}\mathrm{P}(F_1;Q_1)=\{q\in C:⟨F_1\left(q\right),p-q⟩\ge 0,\forall p\in Q_1\}$ and ${\mathrm{VI}}_{\mathrm{t}}\mathrm{P}(F_2;Q_2)=\{r\in Q_2:⟨F_2\left(r\right),s-r⟩\ge 0,\forall s\in Q_2\}$.

Moudafi [8] extended ${\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{t}}\mathrm{P}$ into a split monotone variational inclusion problem $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{MVI}}_{\mathrm{s}}\mathrm{P}\right)$ defined as:

$$\begin{array}{c}\hfill \mathrm{Find}\mathrm{out}{l}^{*}\in X_1\mathrm{such}\mathrm{that}{l}^{*}\in {\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(F_1;A_1;X_1)\mathrm{and}B\left(l^{*}\right)\in {\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(A_2;F_2;X_2),\end{array}$$

(2)

where $A_1:X_1\to 2^{X_1}$ and $A_2:X_2\to 2^{X_2}$ are set-valued mappings on Hilbert spaces $X_1$ and $X_2$, respectively, ${\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(F_1,A_1;X_1)=\{p\in X_1:0\in F_1\left(p\right)+A_1\left(p\right)\}$ and ${\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(F_2,A_2;X_2)=\{q\in X_2:0\in F_2\left(q\right)+A_2\left(q\right)\}$. Moudafi [8] proposed the following iterative scheme for $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{MVI}}_{\mathrm{s}}\mathrm{P}\right)$. Let $\mu >0$, choose any starting point $z_0\in X_1$ and compute

$$\begin{array}{c}\hfill z_{n+1}=V[z_n+\lambda B^{*}(W-I)B{z}_n],\end{array}$$

(3)

where $B^{*}$ is an adjoint operator of B, $\lambda \in (0,1/R)$ with R being the spectral radius of the operator $B^{*}B$, $V=R_{\mu }^{A_1}(I-\mu F_1)={(I+\mu A_1)}^{-1}(I-\mu F_1)$ and $W=R_{\mu }^{A_2}(I-\mu F_2)={(I+\mu A_2)}^{-1}(I-\mu F_2).$

If $F_1=F_2=0$, then $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{MVI}}_{\mathrm{s}}\mathrm{P}\right)$ turns into the split inclusion problem (in short, $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ suggested and discussed by Byrne et al. [9]:

$$\begin{array}{c}\hfill \mathrm{Find}\mathrm{out}{l}^{*}\in X_1\mathrm{such}\mathrm{that}{l}^{*}\in {\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(A_1;X_1)\mathrm{and}B\left(l^{*}\right)\in {\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(A_2;X_2),\end{array}$$

(4)

where ${\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(A_1;X_1)=\{p\in X_1:0\in A_1\left(p\right)\}$ and ${\mathrm{VI}}_{\mathrm{s}}\mathrm{P}(A_2;X_2)=\{q\in X_2:0\in A_2\left(q\right)\}$, $A_1,A_2$ are the same as in (2). Moreover, Byrne et al. [9] suggested the following iterative scheme for $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$. Let $\mu >0$ and select a starting point $z_0\in X_1$; then, compute

$$\begin{array}{c}\hfill z_{n+1}=R_{\mu }^{A_1}[z_n+\lambda B^{*}(I-R_{\mu }^{A_2})B{z}_n],\end{array}$$

(5)

where $B^{*}$ is the adjoint operator of B, $R=\parallel B^{*}{B\parallel =\parallel B\parallel }^2$, $\lambda \in (0,2/R)$ and $R_{\mu }^{A_1}$, $R_{\mu }^{A_2}$ are the resolvents of monotone mappings $A_1,A_2$, respectively. It is obvious to see that $l^{*}$ solves $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ if and only if $l^{*}=R_{\mu }^{A_1}[l^{*}+\lambda B^{*}(I-R_{\mu }^{A_2})B{l}^{*}]$. Kazmi and Rizvi [10] studied the following iterative scheme for calculating the common solutions of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and $\left(\mathrm{FPP}\right)$ of a nonexpansive mapping S. For $z_0\in X_1$, compute

$$\begin{array}{cc}\hfill y_n& =R_{\mu }^{A_1}[z_n+\lambda B^{*}(R_{\mu }^{A_2}-I)B{z}_n],\hfill \\ \hfill z_{n+1}& ={\zeta }_{n}f\left(z_n\right)+(1-{\zeta }_n)S{y}_n,\hfill \end{array}$$

(6)

where f is contraction and $\lambda \in (0,\frac{2}{{\parallel B\parallel }^2})$. By extending the work of Kazmi and Rizvi [10], Dilshad et al. [11] discussed the common solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and the fixed point of a finite collection of nonexpansive mappings. Sitthithakerngkiet et al. [12] investigated the common solutions of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and a fixed point of a countably infinite collection of nonexpansive mappings and proposed and discussed the following method. For $z_0\in X_1$, compute

$$\begin{array}{cc}\hfill y_n& =R_{\mu }^{A_1}[z_n+\lambda B^{*}(R_{\mu }^{A_2}-I)B{z}_n],\hfill \\ \hfill z_{n+1}& ={\zeta }_{n}u+{\xi }_n{z}_n+[(1-{\xi }_n)I-{\zeta }_{n}D]W_n{y}_n,\forall n\ge 1,\hfill \end{array}$$

(7)

where $u\in X_1$ is arbitrary, and $W_n$ is W-mapping, which is created by an infinite collection of nonexpansive mappings. Furthermore, Akram et al. [13] modify the method discussed in [10] and investigate the common solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and $\left(\mathrm{FPP}\right)$:

$$\begin{array}{cc}\hfill y_n& =z_n-\lambda \left[(I-R_{{\mu }_1}^{A_1})z_n+B^{*}(I-R_{{\mu }_2}^{A_2})B{z}_n\right],\hfill \\ \hfill z_{n+1}& ={\zeta }_{n}f\left(z_n\right)+(1-{\zeta }_n)S\left(y_n\right),\hfill \end{array}$$

(8)

where $\lambda =\frac{1}{1+{\parallel B\parallel }^2}$, ${\zeta }_n\in (0,1)$ satisfying $\underset{n\to \infty }{lim}{\zeta }_n=0,{\displaystyle \sum _{n=1}^{\infty }}{\zeta }_n=\infty$ and ${\displaystyle \sum _{n=1}^{\infty }}|{\zeta }_n-{\zeta }_{n-1}|<\infty$. Some results related to $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and $\left(\mathrm{FPP}\right)$ can be found in [14,15,16,17,18,19] and the references therein.

It is noted that the step size depending upon the norm $\parallel B^{*}B\parallel$ is commonly used in the above-mentioned iterative schemes. To skip this restruction, a new type of iterative method with a self-adaptive step size has been invented. López et al. [20] composed a relaxed iterative method for $\left({\mathrm{S}}_{\mathrm{p}}\mathrm{FP}\right)$ with a self-adaptive step size. Dilshad et al. [21] studied the $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ without using a pre-calculated norm $\parallel B\parallel$. Some useful related work can be found in [22,23,24,25,26] and the references therein.

In recent years, great efforts have been made to speed up various algorithms. The inertia term as one of the speed-up techniques has been studied by many scientists because of its simple form and good speed-up effect. Recall that using the concepts of implicit descritization for the derivatives, Alvarez and Attouch [27] have developed the inertial proximal point method, which can be expressed as

$$z_{n+1}=R_{\mu }^A[z_n+{\varphi }_n(z_n-z_{n-1})],$$

where A is monotone mapping, $R_{\mu }^A$ is the resolvent of A and $\mu >0$. Such types of schemes have a better convergence rate, and hence, this scheme was modified and applied to solve numerous nonlinear problems; see [28,29,30,31,32,33,34] and the references therein.

Following the above-mentioned inertial method, we consider two inertial iterative algorithms for approximating the solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and common solutions of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and $\left(\mathrm{FPP}\right)$ of a nonexpansive mapping.

The next section contains some theory and auxiliary results which are helpful in the proof of the main results. In Section 3, we explain two self-adaptive inertial iterative methods. Section 4 is focused on the proof of the main results discussing the solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and a common solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and $\left(\mathrm{FPP}\right)$. At last, we illustrate a numerical example in favor of the proposed iterative algorithms showing their behavior and reliability.

2. Preliminaries

Assume that $(X,\parallel ·\parallel )$ is a real Hilbert space with the inner product $⟨·,·⟩$. The strong convergence of the real sequence $\left\{z_n\right\}$ to z is indicated by $z_n\to z$ and the weak convergence is indicated by $z_n\rightharpoonup z$. If $\left\{z_n\right\}$ is a sequence in X, ${\omega }_w\left(z_n\right)$ indicates the weak $\omega$-limit set of $\left\{z_n\right\}$, that is

$${\omega }_w\left(z_n\right)=\{z\in H:z_{n_j}\rightharpoonup z\mathrm{as}j\to \infty \mathrm{where}{z}_{n_j}\mathrm{is}\mathrm{a}\mathrm{subsequnce}\mathrm{of}{z}_n\}.$$

We know that for some $z\in X$, there exists a unique nearest point in Q denoted by $P_{Q}z$ such that

$$\begin{array}{c}\hfill \parallel z-P_{Q}z\parallel \le \parallel z-v\parallel ,\forall v\in Q.\end{array}$$

$P_{Q}z$ is called the projection of z onto $Q\subset X$, which satisfies

$$\begin{array}{c}\hfill ⟨z-v,P_{Q}z-P_{C}v⟩\ge {\parallel P_{Q}z-P_{Q}v\parallel }^2,\forall z,v\in X.\end{array}$$

Moreover, $P_{Q}z$ is identified by the fact

$$\begin{array}{c}\hfill P_{Q}z=x\iff ⟨z-v,v-x⟩\ge 0,v\in Q.\end{array}$$

For all $p,q,r$ in Hilbert space X, $\varphi ,\phi ,\psi \in [0,1]$ such that $\varphi +\phi +\psi =1$; then, we have the following equality and inequality

$$\begin{array}{c}\hfill {\parallel \varphi p+\phi q+\psi r\parallel }^2={\varphi \parallel p\parallel }^2+{\phi \parallel q\parallel }^2+{\psi \parallel r\parallel }^2{-\varphi \phi \parallel p-q\parallel }^2{-\phi \psi \parallel q-r-w\parallel }^2-\psi \varphi {\parallel p-r\parallel }^2,\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\parallel p+q\parallel }^2\le {\parallel p\parallel }^2+2⟨q,p+q⟩.\end{array}$$

(10)

Definition 1.

A mapping $F:X\to X$ is called

(i) 

Contraction, if $\parallel F\left(p\right)-F\left(q\right)\parallel \le \kappa \parallel p-q\parallel ,\forall p,q\in X,\kappa \in (0,1);$

(ii) 

Nonexpansive, if $\parallel F\left(p\right)-F\left(q\right)\parallel \le \parallel p-q\parallel ,\forall p,q\in X;$

(iii) 

Firmly nonexpansive, if ${\parallel F\left(p\right)-F\left(q\right)\parallel }^2\le ⟨p-q,F\left(p\right)-F\left(q\right)⟩,\forall p,q\in X;$

(iv) 

τ-inverse strongly monotone, if there exists $\tau >0$ such that

$$⟨F\left(p\right)-F\left(q\right),p-q⟩\ge {\tau \parallel F\left(p\right)-F\left(q\right)\parallel }^2,\forall p,q\in X.$$

Definition 2.

Let $A:X\to 2^X$ be a set valued mapping. Then

(i) 

The mapping A is called monotone if $⟨u-v,p-q⟩\ge 0,\forall u,v\in X,u\in A\left(p\right),v\in A\left(q\right)$;

(ii) 

$Graph\left(A\right)=\left\{\right(u,p)\in X\times X:u\in A(p\left)\right\}$;

(iii) 

The mapping A is called maximal monotone if $Graph\left(A\right)$ is not properly contained in the graph of any other monotone operator.

Lemma 1

([35]). If $\left\{s_n\right\}$ is a sequence of non-negative real numbers such that

$$s_{n+1}\le (1-{\xi }_n)s_n+{\delta }_n,n\ge 0,$$

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

(i) 

${\sum }_{n=1}^{\infty }{\xi }_n=\infty ;$

(ii) 

$lim\underset{n\to \infty }{sup}\frac{{\delta }_n}{{\xi }_n}\le 0$ or ${\displaystyle \sum _{n=1}^{\infty }}\left|{\delta }_n\right|<\infty .$

Then $\underset{n\to \infty }{lim}{s}_n=0.$

Lemma 2

([36]). In a Hilbert space X,

(i) 

a mapping $A:X\to X$ is τ-inverse strongly monotone if and only if $I-\tau A$ is firmly nonexpansive for $\tau >0$.

(ii) 

If $A:X\to 2^X$ is monotone and $R_{\mu }^A$ is the resolvent of A, then $R_{\mu }^A$ and $I-R_{\mu }^A$ are firmly nonexpansive for $\mu >0$.

(iii) 

If $A:X\to X$ is nonexpansive, then $I-A$ is demiclosed at zero and if A is firmly nonexpansive, then $I-A$ is firmly nonexpansive.

Lemma 3

([37]). Let $\left\{{\psi }_n\right\}$ be a bounded sequence in Hilbert space X. Assume there exists a subset $Q\ne \varnothing$ and $Q\subset X$ satisfying the properties

(i) 

${lim}_{n\to \infty }\parallel {\psi }_n-z\parallel$ exists, $\forall z\in Q$,

(ii) 

${\omega }_w\left({\psi }_n\right)\subset Q.$

Then, there exists $z^{*}\in C$ such that ${\psi }_n\rightharpoonup z^{*}$.

Lemma 4

([38]). Let ${\Gamma }_n$ be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence ${\Gamma }_{n_k}$ of ${\Gamma }_n$ such that ${\Gamma }_{n_k}<{\Gamma }_{n_k+1}$ for all $k\ge 0$. In addition, consider the sequence of integers ${\left\{\sigma \left(n\right)\right\}}_{n\ge n_0}$ defined by

$$\begin{array}{c}\hfill \sigma \left(n\right)=max\{k\le n:{\Gamma }_k\le {\Gamma }_{k+1}\}.\end{array}$$

Then, ${\left\{\sigma \left(n\right)\right\}}_{n\ge n_0}$ is a nondecreasing sequence verifying ${lim}_{n\to \infty }\sigma \left(n\right)=\infty$ and $\forall n\ge n_0$,

$$\begin{array}{c}\hfill max\{{\Gamma }_{\sigma \left(n\right)},{\Gamma }_{\left(n\right)}\}\le {\Gamma }_{\sigma \left(n\right)+1}.\end{array}$$

Lemma 5

([38]). Assume that $\left\{s_n\right\}$ is a non-negative sequence of real numbers satisfying

(i) 

$s_{n+1}-s_n\le {\delta }_n(s_n-s_{n-1})+{\theta }_n$;

(ii) 

${\displaystyle \sum _{n=1}^{\infty }}{\theta }_n<\infty$;

(iii) 

${\delta }_n\in [0,\kappa ]$, where $\kappa \in [0,1)$.

Then, $\left\{s_n\right\}$ is convergent and ${\displaystyle \sum _{n=1}^{\infty }}(s_{n+1}-s_n)<\infty$, where ${\left[h\right]}_{+}=max\{h,0\}$ for any $h\in \mathbb{R}$.

3. Inertial Iterative Methods

Suppose that $X_1$ and $X_2$ are real Hilbert spaces and $A_1:X_1\to 2^{X_1},A_2:X_2\to 2^{X_2}$ are monotone mappings; $R_{{\mu }_1}^{A_1}$, $R_{{\mu }_2}^{A_2}$ are the resolvents of $A_1$ and $A_2$, respectively. We assume that $\mathsf{\Lambda }\cap \mathrm{Fix}\left(\mathrm{F}\right)\ne \varnothing$, where $\mathsf{\Lambda }$ denotes the solution set of ${\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}$ and $\mathrm{Fix}\left(\mathrm{F}\right)$ denotes the fixed point set of $\mathrm{FPP}$. First, we suggest the following iterative algorithm for ${\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}$.

Algorithm 1.

Choose $\varphi$ such that $0\le \varphi <1$ and let ${\delta }_n$ be a positive sequence satisfying ${\displaystyle \sum _{n=1}^{\infty }}{\delta }_n<\infty$.

• Iterative Step: Given arbitrary $x_0$, and $x_1$, for $n\ge 1$, choose $0<{\varphi }_n<{\tilde{\varphi }}_n$, where

  $$\begin{array}{c}\hfill {\tilde{\varphi }}_n=\left\{\begin{array}{c}min\left\{\frac{{\delta }_n}{\parallel x_n-x_{n-1}\parallel },\varphi \right\},\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ \varphi ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

  (11)

Compute

$$\begin{array}{ccc}\hfill v_n& =& x_n+{\varphi }_n(x_n-x_{n-1}),\hfill \\ \hfill u_n& =& v_n-{\sigma }_n(I-R_{{\mu }_1}^{A_1})\left(v_n\right),\hfill \\ \hfill x_{n+1}& =& u_n-{\varrho }_n{B}^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right),\hfill \end{array}$$

where ${\sigma }_n$ and ${\varrho }_n$ are defined as

$$\begin{array}{ccc}\hfill {\sigma }_n& =& \left\{\begin{array}{cc}\frac{{\tau }_n{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^2}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2},\hfill & \mathrm{if}\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2\ne 0\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.\hfill \end{array}$$

(12)

and

$$\begin{array}{ccc}\hfill {\varrho }_n& =& \left\{\begin{array}{cc}\frac{{\tau }_n{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2},\hfill & \mathrm{if}\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\ne 0\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.\hfill \end{array}$$

(13)

where ${\mu }_1>0$, ${\mu }_2>0$ and ${\tau }_n\in (0,2).$

Algorithm 2.

Choose $\varphi$ such that $0\le \varphi <1$ and let ${\delta }_n$ be a positive sequence satisfying ${\displaystyle \sum _{n=1}^{\infty }}{\delta }_n<\infty$.

• Iterative Step: Given arbitrary $x_0$, and $x_1$, for $n\ge 1$, choose $0<{\varphi }_n<{\tilde{\varphi }}_n$, where

  $$\begin{array}{c}\hfill {\tilde{\varphi }}_n=\left\{\begin{array}{c}min\left\{\frac{{\delta }_n}{\parallel x_n-x_{n-1}\parallel },\varphi \right\},\mathrm{if}{x}_n\ne x_{n-1},\hfill \\ \varphi ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

  (14)

Compute

$$\begin{array}{ccc}\hfill v_n& =& x_n+{\varphi }_n(x_n-x_{n-1}),\hfill \\ \hfill u_n& =& v_n-{\sigma }_n(I-R_{{\mu }_1}^{A_1})\left(v_n\right),\hfill \\ \hfill w_n& =& u_n-{\varrho }_n{B}^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right),\hfill \\ \hfill x_{n+1}& =& (1-{\zeta }_n-{\xi }_n)u_n+{\zeta }_{n}F\left(w_n\right).\hfill \end{array}$$

where ${\sigma }_n$ and ${\varrho }_n$ are defined as

$$\begin{array}{ccc}\hfill {\sigma }_n& =& \left\{\begin{array}{cc}\frac{{\tau }_n{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^2}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2},\hfill & \mathrm{if}\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2\ne 0\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.\hfill \end{array}$$

(15)

and

$$\begin{array}{ccc}\hfill {\varrho }_n& =& \left\{\begin{array}{cc}\frac{{\tau }_n{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2},\hfill & \mathrm{if}\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\ne 0\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.\hfill \end{array}$$

(16)

where ${\zeta }_n$, ${\xi }_n\in (0,1)$, ${\mu }_1>0$, ${\mu }_2>0$, and ${\tau }_n\in (0,2)$.

Remark 1.

It is not difficult to show that if $\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2=0$ or $\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2=0$ for some $n\ge 0$, then $x_n\in \mathsf{\Lambda }$. In this case, the iteration process ended after a finite number of iterations. We suppose that the proposed algorithms generate infinite sequences which do not end in a finite number of terms.

Remark 2.

From the selection of ${\delta }_n$, such that ${\displaystyle \sum _{n=1}^{\infty }}{\delta }_n<\infty$, we can conclude that $\underset{n\to \infty }{lim}{\varphi }_n\parallel x_n-x_{n-1}\parallel =0.$

Remark 3.

By using the definitions of resolvent of monotone mappings $A_1$ and $A_2$, we can easily obtain that $l^{★}\in \mathsf{\Lambda }$ if and only if $R_{{\mu }_1}^{A_1}\left(l^{★}\right)=l^{*}$ and $R_{{\mu }_2}^{A_2}\left(B{l}^{★}\right)=B\left(l^{★}\right)$.

4. Main Results

Theorem 1.

Let $X_1$, $X_2$ be real Hilbert spaces; $A_1:X_1\to 2^{X_1}$, $A_2:X_2\to 2^{X_2}$ be maximal monotone mappings and $B:X_1\to X_2$ be a bounded linear operator. If ${\tau }_n\in (0,2)$ and ${\zeta }_n,{\xi }_n\in (0,1)$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\xi }_n=0,\sum _{n=0}^{\infty }{\xi }_n=\infty ,\underset{n\to \infty }{lim}(1-{\zeta }_n-{\xi }_n){\zeta }_n>0,\underset{n}{inf}{\tau }_n(2-{\tau }_n)>0.\end{array}$$

(17)

Then, the sequence $\left\{x_n\right\}$ generated from Algorithm 1 converges weakly to a point $z\in \mathsf{\Lambda }$.

Proof. 

Let $l\in \mathsf{\Lambda }$, then $(I-R_{{\mu }_1}^{A_1})\left(l\right)=0$. Since resolvent operator $R_{{\mu }_1}^{A_1}$ is firmly nonexpansive, hence, so is $(I-R_{{\mu }_1}^{A_1})$ for ${\mu }_1>0$, then by Algorithm 1 and (10), we have

$$\begin{array}{ccc}\hfill \parallel u_n{-l\parallel }^2& =& \parallel v_n-{\sigma }_n(I-R_{{\mu }_1}^{A_1})\left(v_n\right){-l\parallel }^2\hfill \\ & \le & \parallel v_n{-l\parallel }^2+{\sigma }_n^2{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^2-2{\sigma }_n⟨(I-R_{{\mu }_1}^{A_1})\left(v_n\right),v_n-l⟩\hfill \\ & =& \parallel v_n{-l\parallel }^2+{\sigma }_n^2{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^2-2{\sigma }_n⟨(I-R_{{\mu }_1}^{A_1})\left(v_n\right)-(I-R_{{\mu }_1}^{A_1})\left(l\right),v_n-l⟩\hfill \\ & \le & \parallel v_n{-l\parallel }^2+{\sigma }_n^2\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2-2{\sigma }_n{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)-(I-R_{{\mu }_1}^{A_1})\left(l\right)\parallel }^2\hfill \\ & =& \parallel v_n{-l\parallel }^2+({\sigma }_n^2-2{\sigma }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^2.\hfill \end{array}$$

(18)

Now, using (12), we estimate that

$$\begin{array}{ccc}\hfill & & ({\sigma }_n^2-2{\sigma }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^2\hfill \\ & =& \parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2[\frac{{{\tau }_n}^2{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{{\left(\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2\right)}^2}\hfill \\ & & -\frac{2{\tau }_n{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^2}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}]\hfill \end{array}$$

$$\begin{array}{ccc}& =& \parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^4\left[\frac{{{\tau }_n}^2\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2-2{\tau }_n(\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right){\parallel }^2)}{{\left(\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2\right)}^2}\right]\hfill \\ & \le & \parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^4\left[\frac{({{\tau }_n}^2-2{\tau }_n)(\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right){\parallel }^2)}{{\left(\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2\right)}^2}\right]\hfill \\ & =& \frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}.\hfill \end{array}$$

(19)

From (18) and (19), we obtain

$$\begin{array}{c}\hfill \parallel u_n{-l\parallel }^2\le {\parallel v_n-l\parallel }^2+\frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}.\end{array}$$

(20)

Since $(I-R_{{\mu }_2}^{A_2})$ is firmly nonexpasive and using $(I-R_{{\mu }_2}^{A_2})\left(Bl\right)=0$ and (10), we estimate

$$\begin{array}{ccc}\hfill \parallel x_{n+1}{-l\parallel }^2& =& \parallel u_n-{\varrho }_n(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){-l\parallel }^2\hfill \\ & \le & \parallel u_n{-l\parallel }^2+{\varrho }_n^2{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2-2{\varrho }_n⟨(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right),u_n-l⟩\hfill \\ & =& \parallel u_n{-l\parallel }^2+{\varrho }_n^2{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\hfill \\ & & -2{\varrho }_n⟨(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)-(I-R_{{\mu }_2}^{A_2})\left(Bl\right),u_n-l⟩\hfill \\ & =& \parallel u_n{-l\parallel }^2+{\varrho }_n^2\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^2-2{\varrho }_n{\parallel (I-R_{{\mu }_1}^{A_1})\left(B{u}_n\right)\parallel }^2\hfill \\ & =& \parallel u_n{-l\parallel }^2+({\varrho }_n^2-2{\varrho }_n){\parallel J_{{\lambda }_1}^{A_2}\left(B{u}_n\right)\parallel }^2.\hfill \end{array}$$

(21)

By (13), it turns out that

$$\begin{array}{ccc}\hfill & & ({\varrho }_n^2-2{\varrho }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\hfill \\ & =& \parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^2[\frac{{{\tau }_n}^2{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{{\left(\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\right)}^2}\hfill \\ & & -\frac{2{\tau }_n{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}]\hfill \\ & =& \parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^4\times \hfill \\ & & \left[\frac{{{\tau }_n}^2\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^2-2{\tau }_n(\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^2)}{{\left(\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\right)}^2}\right]\hfill \\ & \le & \parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^4\left[\frac{({{\tau }_n}^2-2{\tau }_n)\left(\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\right)}{{\left(\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2\right)}^2}\right]\hfill \\ & =& \frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}.\hfill \end{array}$$

(22)

It follows from (21) and (22) that

$$\begin{array}{c}\hfill \parallel x_{n+1}{-l\parallel }^2\le {\parallel u_n-l\parallel }^2+\frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}.\end{array}$$

(23)

Combining (20) and (23), we obtain

$$\begin{array}{ccc}\hfill \parallel x_{n+1}{-l\parallel }^2& \le & \parallel v_n{-l\parallel }^2+\frac{{\tau }_n\left({\tau }_n-2\right){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\hfill \\ & & +\frac{{\tau }_n\left({\tau }_n-2\right){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}\hfill \end{array}$$

(24)

By using the Cauchy–Schwartz inequality, we observe that

$$\begin{array}{ccc}\hfill \parallel v_n{-l\parallel }^2& =& \parallel x_n-l+{\varphi }_n(x_n-x_{n-1}){\parallel }^2\hfill \\ & =& \parallel x_n{-l\parallel }^2+2{\varphi }_n⟨x_n-x_{n-1},x_n-l⟩+{\varphi }_n^2{\parallel x_n-x_{n-1}\parallel }^2.\hfill \\ & \le & \parallel x_n{-l\parallel }^2+2{\varphi }_n\parallel x_n-x_{n-1}\parallel \parallel x_n-l\parallel +{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2.\hfill \end{array}$$

Since $2\parallel x_n-x_{n-1}\parallel \parallel x_n-l\parallel =\parallel x_n-x_{n-1}{\parallel }^2+\parallel x_n{-l\parallel }^2-{\parallel (x_n-x_{n-1})-(x_n-l)\parallel }^2$, we get

$$\begin{array}{ccc}\hfill \parallel v_n{-l\parallel }^2& \le & \parallel x_n{-l\parallel }^2+2{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2+{\varphi }_n\left\{\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right\}.\hfill \end{array}$$

(25)

From (25) and (24), we obtain

$$\begin{array}{ccc}\hfill \parallel x_{n+1}{-l\parallel }^2& \le & \parallel x_n{-l\parallel }^2+2{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2+{\varphi }_n\left\{\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right\}\hfill \\ & +& \frac{{\tau }_n({\tau }_n-2){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\hfill \\ & & +\frac{{\tau }_n\left({\tau }_n-2\right){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}.\hfill \end{array}$$

(26)

Since, ${\tau }_n\in (0,2)$, that is ${\tau }_n-2<0$, we obtain

$$\begin{array}{ccc}\hfill (\parallel x_{n+1}-l{\parallel }^2-\parallel x_n-l{\parallel }^2)& \le & {\varphi }_n\left\{\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right\}+2{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2\hfill \end{array}$$

Applying Lemma 5, we deduce that the limit $\parallel x_n-l\parallel$ exists, which guarantees the boundednesss of sequence $\left\{x_n\right\}$ and hence $\left\{u_n\right\}$ and $\left\{v_n\right\}$. From (26), it follows that ${\displaystyle \sum _{n=1}^{\infty }}{\varphi }_n\left(\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right)<\infty$ and

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty }\left[\frac{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}+\frac{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}\right]<\infty ,\end{array}$$

which concludes

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\left[\frac{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}+\frac{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}\right]=0,\end{array}$$

hence

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\frac{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}=0,\\ \hfill \underset{n\to \infty }{lim}\frac{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}=0,\end{array}$$

which concludes that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel =\underset{n\to \infty }{lim}\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel =0.\end{array}$$

(27)

It remains to show that ${\omega }_w\left(x_n\right)\in \mathsf{\Lambda }$. Let $l^{★}\in {\omega }_w\left(x_n\right)$ and $\left\{x_{n_k}\right\}$ be a subsequence of $\left\{x_n\right\}$ so that $x_{n_k}\rightharpoonup l^{★}$, as $k\to \infty$. Applying (27) and Remark 2, in Algorithm 1, it follows that

$$\begin{array}{c}\hfill \parallel x_n-v_n\parallel ={\varphi }_n\parallel x_n-x_{n-1}\parallel \to 0,n\to \infty \end{array}$$

$$\begin{array}{ccc}\hfill \parallel x_n-u_n\parallel & =& \parallel x_n-[v_n-{\sigma }_n(I-R_{{\mu }_1}^{A_1})\left(v_n\right)]\parallel \hfill \\ & \le & \parallel x_n-v_n\parallel +{\sigma }_n\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel \hfill \\ & \le & \parallel x_n-v_n\parallel +\frac{{\tau }_n{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^3}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\to 0,n\to \infty \hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \parallel v_n-u_n\parallel & \le & \parallel x_n-u_n\parallel +\parallel v_n-x_n\parallel \to 0,n\to \infty .\hfill \end{array}$$

Hence, there exist subsequences $\left\{u_{n_k}\right\}$ and $\left\{v_{n_k}\right\}$ of $\left\{u_n\right\}$ and $\left\{v_n\right\}$, respectively, which converge to $l^{★}$. From (27), we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel (I-R_{{\mu }_1}^{A_1})\left(v_{n_k}\right)\parallel =\underset{k\to \infty }{lim}\parallel (I-R_{{\mu }_1}^{A_1})\left(l^{★}\right)\parallel =0,\\ \hfill \underset{k\to \infty }{lim}\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_{n_k}\right)\parallel =\underset{k\to \infty }{lim}\parallel (I-R_{{\mu }_2}^{A_2})\left(B{l}^{★}\right)\parallel =0,\end{array}$$

which imply that $l^{★}\in A_1^{-1}\left(0\right)$ and $B\left(l^{★}\right)\in A_2^{-1}\left(0\right)$. □

Theorem 2.

Let $X_1$, $X_2$ be real Hilbert spaces; and let $A_1:X_1\to 2^{X_1}$, $A_2:X_2\to 2^{X_2}$ be set-valued maximal monotone mappings. If $\left\{{\zeta }_n\right\}$, $\left\{{\xi }_n\right\}$ are real sequences in $(0,1)$, ${\tau }_n\in (0,2)$ and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\xi }_n=0,\sum _{n=0}^{\infty }{\xi }_n=\infty ,\underset{n\to \infty }{lim}(1-{\zeta }_n-{\xi }_n){\zeta }_n>0,\underset{n}{inf}{\tau }_n(2-{\tau }_n)>0.\end{array}$$

(28)

Then, the sequence $\left\{x_n\right\}$ obtained from Algorithm 2 converges strongly to $l=P_{\mathsf{\Lambda }\cap \mathrm{Fix}\left(F\right)}\left(0\right)$.

Proof. 

Let $l=P_{\mathsf{\Lambda }\cap \mathrm{Fix}\left(F\right)}\left(0\right)$. From Algorithm 2, we have

$$\begin{array}{ccc}\hfill \parallel v_n-l\parallel & =& \parallel x_n+{\varphi }_n(x_n-x_{n-1})-l\parallel \hfill \\ & \le & (1-{\varphi }_n)\parallel x_n-l\parallel +{\varphi }_n\parallel x_{n-1}-l\parallel \hfill \\ & \le & max\{\parallel x_n-l\parallel ,\parallel x_{n-1}-l\parallel \},\hfill \end{array}$$

(29)

and

$$\begin{array}{ccc}\hfill \parallel x_{n+1}-l\parallel & =& \parallel (1-{\zeta }_n-{\xi }_n)u_n+{\zeta }_{n}F\left(w_n\right)-l\parallel \hfill \\ & \le & (1-{\zeta }_n-{\xi }_n)\parallel u_n-l\parallel +\parallel +{\zeta }_n\parallel F\left(w_n\right)-l\parallel +{\xi }_n\parallel -l\parallel \hfill \\ & \le & (1-{\zeta }_n-{\xi }_n)\parallel u_n-l\parallel +\parallel +{\zeta }_n\parallel w_n-l\parallel +{\xi }_n\parallel -l\parallel \hfill \\ & \le & (1-{\xi }_n)\parallel v_n-l\parallel +{\xi }_n\parallel l\parallel \hfill \\ & \le & (1-{\xi }_n)\left[(1-{\varphi }_n)\parallel x_n-l\parallel +{\varphi }_n\parallel x_{n-1}-l\parallel \right]+{\xi }_n\parallel l\parallel \hfill \\ & \le & max\{\parallel x_n-l\parallel ,\parallel x_{n-1}-l\parallel ,\parallel l\parallel \}\hfill \\ & \le & ⋮\hfill \\ & \le & max\{\parallel x_0-l\parallel ,\parallel x_1-l\parallel ,\parallel l\parallel \},\hfill \end{array}$$

which shows that $\left\{x_n\right\}$ is bounded and hence the $\left\{v_n\right\}$, $\left\{u_n\right\}$, and $\left\{w_n\right\}$ are bounded. From (20) and (23) of the proof of Theorem 1, we have

$$\begin{array}{c}\hfill \parallel u_n{-l\parallel }^2\le \parallel v_n{-l\parallel }^2+\frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\le {\parallel v_n-l\parallel }^2.\end{array}$$

(30)

$$\begin{array}{c}\hfill \parallel w_n{-l\parallel }^2\le \parallel u_n{-l\parallel }^2+\frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}\le {\parallel u_n-l\parallel }^2.\end{array}$$

(31)

Now,

$$\begin{array}{ccc}\hfill \parallel x_{n+1}{-l\parallel }^2& =& \parallel (1-{\zeta }_n-{\xi }_n)u_n+{\zeta }_{n}F\left(w_n\right){-l\parallel }^2\hfill \\ & =& \parallel (1-{\zeta }_n-{\xi }_n)(u_n-l)+{\zeta }_n(F\left(w_n\right)-l)+{\xi }_n(-l){\parallel }^2\hfill \\ & \le & (1-{\zeta }_n-{\xi }_n)\parallel u_n{-l\parallel }^2+{\zeta }_n\parallel F\left(w_n\right){-l\parallel }^2+{\xi }_n{\parallel l\parallel }^2\hfill \\ & & -{\zeta }_n(1-{\zeta }_n-{\xi }_n){\parallel u_n-F\left(w_n\right)\parallel }^2\hfill \\ & =& (1-{\zeta }_n-{\xi }_n)\parallel u_n{-l\parallel }^2+{\zeta }_n\parallel w_n{-l\parallel }^2+{\xi }_n{\parallel l\parallel }^2\hfill \\ & & -{\zeta }_n(1-{\zeta }_n-{\xi }_n){\parallel u_n-F\left(w_n\right)\parallel }^2.\hfill \end{array}$$

(32)

Combining (30)–(32), we obtain

$$\begin{array}{ccc}\hfill \parallel x_{n+1}{-l\parallel }^2& \le & (1-{\zeta }_n-{\xi }_n)\left[\parallel v_n{-l\parallel }^2+\frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\right]\hfill \\ & & +{\zeta }_n\left[\parallel u_n{-l\parallel }^2+\frac{({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}\right]+{\xi }_n{\parallel l\parallel }^2\hfill \\ & & -{\zeta }_n(1-{\zeta }_n-{\xi }_n){\parallel u_n-F\left(w_n\right)\parallel }^2\hfill \\ & \le & (1-{\zeta }_n-{\xi }_n)\parallel v_n{-l\parallel }^2+{\zeta }_n\parallel u_n{-l\parallel }^2-{\zeta }_n(1-{\zeta }_n-{\xi }_n){\parallel u_{)}-F\left(w_n\right)\parallel }^2\hfill \\ & & -\frac{(1-{\zeta }_n-{\xi }_n)({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\hfill \\ & & -\frac{{\zeta }_n({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}+{\xi }_n{\parallel l\parallel }^2\hfill \end{array}$$

$$\begin{array}{ccc}& \le & (1-{\xi }_n)\left[\parallel x_n{-l\parallel }^2+2{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2+{\varphi }_n\left\{\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right\}\right]\hfill \\ & & -{\zeta }_n(1-{\zeta }_n-{\xi }_n){\parallel u_n-F\left(w_n\right)\parallel }^2+\frac{(1-{\zeta }_n-{\xi }_n)({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\hfill \\ & & +\frac{{\zeta }_n({{\tau }_n}^2-2{\tau }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}+{\xi }_n{\parallel l\parallel }^2\hfill \\ & \le & \parallel x_n{-l\parallel }^2+2{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2+{\varphi }_n\left\{\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right\}\hfill \\ & & -{\zeta }_n(1-{\zeta }_n-{\xi }_n){\parallel u_n-F\left(w_n\right)\parallel }^2+\frac{(1-{\zeta }_n-{\xi }_n){\tau }_n(2-{\tau }_n){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\hfill \\ & & +\frac{{\zeta }_n{\tau }_n(2-{\tau }_n){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}+{\xi }_n{\parallel l\parallel }^2.\hfill \end{array}$$

(33)

Two possible cases occur.

Case I. Suppose the sequence $\{\parallel x_n-l\parallel \}$ is nonincreasing; then, there exists $m\ge 0$ such that $\parallel x_{n+1}-l\parallel \le \parallel x_n-l\parallel$, for each $n\ge m$. Then, $\underset{n\to \infty }{lim}\parallel x_n-l\parallel$ exists and hence $\underset{n\to \infty }{lim}(\parallel x_{n+1}-l\parallel -\parallel x_n-l\parallel )=0.$ Since ${\xi }_n\to 0$, ${\tau }_n\in (0,2)$, and $inf{\zeta }_n(1-{\zeta }_n-{\xi }_n)>0$, hence from (33), we have

$$\begin{array}{ccc}& & \frac{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel }^2}\to 0,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}& & \frac{\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right){\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right)\parallel }^2}\to 0,\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & \parallel u_n-F\left(w_n\right)\parallel \to 0.\hfill \end{array}$$

(36)

From (34) and (35), we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel (I-R_{{\mu }_1}^{A_1})\left(u_n\right)\parallel =0\mathrm{and}\underset{n\to \infty }{lim}\parallel (I-R_{{\mu }_2}^{A_2})\left(B{v}_n\right)\parallel =0.\end{array}$$

(37)

From Algorithm 2, using Remark 2, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_n-x_n\parallel =0.\end{array}$$

(38)

From Algorithm 2, using (34) and (35), we obtain as $n\to \infty$

$$\begin{array}{ccc}& & \parallel u_n-v_n\parallel \to 0,\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & \parallel w_n-u_n\parallel \to 0.\hfill \end{array}$$

(40)

By using (38)–(40), we obtain

$$\begin{array}{ccc}\hfill \parallel u_n-x_n\parallel & \le & \parallel u_n-v_n\parallel +\parallel v_n-x_n\parallel \to 0,\mathrm{as}n\to \infty \hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill \parallel w_n-x_n\parallel & \le & \parallel w_n-u_n\parallel +\parallel u_n-x_n\parallel \to 0,\mathrm{as}n\to \infty .\hfill \end{array}$$

(42)

Thus, since ${\xi }_n\to 0$, and using (36), (40) and (41), we obtain

$$\begin{array}{ccc}\hfill \parallel x_{n+1}-x_n\parallel & =& \parallel (1-{\zeta }_n-{\xi }_n)u_n+{\zeta }_{n}F\left(w_n\right)-x_n\parallel \hfill \\ & \le & \parallel u_n-x_n\parallel +{\zeta }_n\parallel F\left(w_n\right)-u_n\parallel +{\xi }_n\parallel -u_n\parallel \to 0asn\to \infty ,\hfill \end{array}$$

(43)

and

$$\begin{array}{ccc}\hfill \parallel F\left(w_n\right)-w_n\parallel & \le & \parallel F\left(w_n\right)-u_n\parallel +\parallel u_n-w_n\parallel \to 0asn\to \infty .\hfill \\ \hfill \parallel F\left(u_n\right)-u_n\parallel & \le & \parallel F\left(u_n\right)-F\left(w_n\right)\parallel +\parallel F\left(w_n\right)-u_n\parallel \to 0asn\to \infty .\hfill \\ & \le & \parallel u_n-w_n\parallel +\parallel F\left(w_n\right)-u_n\parallel \to 0asn\to \infty .\hfill \end{array}$$

(44)

Hence, there exists a subsequence $\left\{u_{nk}\right\}$ of $\left\{u_n\right\}$ which converges weakly to l. By using Lemma 3, we conclude that $l\in \mathrm{Fix}\left(F\right)$. By Theorem 1, we have that ${\omega }_w\left(x_n\right)\subset \mathsf{\Lambda }$. So, we obtain $l\in \mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }$. Setting $s_n=(1-{\zeta }_n)u_n+{\zeta }_{n}F\left(w_n\right)$ and rewrite $x_{n+1}=(1-{\xi }_n)s_n+{\zeta }_n{\xi }_n(F\left(w_n\right)-u_n)$, we have

$$\begin{array}{ccc}\hfill \parallel s_n-l\parallel & =& \parallel (1-{\zeta }_n)u_n+{\zeta }_{n}F\left(w_n\right)-l\parallel \hfill \\ & \le & (1-{\zeta }_n)\parallel u_n-l\parallel +{\zeta }_n\parallel F\left(w_n\right)-l\parallel \hfill \\ & \le & (1-{\zeta }_n)\parallel v_n-l\parallel +{\zeta }_n\parallel w_n-l\parallel \hfill \\ & \le & \parallel v_n-l\parallel .\hfill \end{array}$$

From (45) and Algorithm 2, we obtain

$$\begin{array}{ccc}\hfill \parallel x_{n+1}{-l\parallel }^2& =& \parallel (1-{\xi }_n)(s_n-l)+{\xi }_n{\zeta }_n(F\left(w_n\right)-u_n){-l\parallel }^2\hfill \\ & \le & (1-{\xi }_n){\parallel s_n-l\parallel }^2+2{\xi }_n⟨{\zeta }_n(F\left(w_n\right)-u_n)-l,x_{n+1}-l⟩\hfill \\ & \le & (1-{\xi }_n)\left[\parallel x_n{-l\parallel }^2+2{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2+{\varphi }_n\left\{\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right\}\right]\hfill \\ & & +2{\xi }_n\{{\zeta }_n⟨F\left(w_n\right)-u_n,x_{n+1}-l⟩+⟨-l,x_{n+1}-l⟩\},\hfill \\ & \le & (1-{\xi }_n)\parallel x_n{-l\parallel }^2+2{\varphi }_n{\parallel x_n-x_{n-1}\parallel }^2+{\varphi }_n\left\{\parallel x_n{-l\parallel }^2-{\parallel x_{n-1}-l\parallel }^2\right\}\hfill \\ & & +2{\xi }_n\{{\zeta }_n⟨F\left(w_n\right)-u_n,x_{n+1}-l⟩+⟨-l,x_{n+1}-l⟩\}.\hfill \end{array}$$

(45)

Since ${\omega }_w\left(x_n\right)\subset \mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }$ and $l=P_{\mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }}\left(0\right)$, then using (35), we obtain

$$\begin{array}{ccc}\hfill lim\underset{n\to \infty }{sup}\frac{b_n}{{\xi }_n}& =& lim\underset{n\to \infty }{sup}\{2{\zeta }_n⟨F\left(w_n\right)-u_n,x_{n+1}-l⟩+⟨-l,x_{n+1}-l⟩\}\hfill \\ & =& lim\underset{n\to \infty }{sup}⟨-l,x_{n+1}-l⟩\le 0.\hfill \end{array}$$

Thus, by Lemma 1 in (45), we obtain $x_n\to l$.

Case II. If the sequence $\{\parallel x_n-l\parallel \}$ is increasing, we can construct a subsequence $\{\parallel x_{n_k}-l\parallel \}$ of $\{\parallel x_n-l\parallel \}$ such that $\parallel x_{n_k}-l\parallel \le \parallel x_n-l\parallel$ for all $k\in \mathbb{N}$. In this case, we define a subsequence of positive integers $\gamma \left(n\right)$

$$\gamma \left(n\right)=max\{k\le n:\parallel x_k-l\parallel \le \parallel x_{k+1}-l\parallel \},$$

then $\gamma \left(n\right)\to \infty$ and $n\to \infty$ and $\parallel x_{\gamma \left(n\right)}-l\parallel \le \parallel x_{\gamma \left(n\right)+1}-l\parallel$, it follows from (33) that

$$\begin{array}{ccc}\hfill \parallel x_{\gamma \left(n\right)}{-l\parallel }^2& \le & (1-{\xi }_{\gamma \left(n\right)})\parallel x_{\gamma \left(n\right)}{-l\parallel }^2+2{\varphi }_{\gamma \left(n\right)}\parallel x_n-x_{\gamma \left(n\right)-1}{\parallel }^2+{\varphi }_{\gamma \left(n\right)}\{{\parallel x_{\gamma \left(n\right)}-l\parallel }^2\hfill \\ & & -\parallel x_{\gamma \left(n\right)-1}{-l\parallel }^2\}-{\zeta }_{\gamma \left(n\right)}(1-{\zeta }_{\gamma \left(n\right)}-{\xi }_{\gamma \left(n\right)}){\parallel u_{\gamma \left(n\right)}-F\left(w_{\gamma \left(n\right)}\right)\parallel }^2\hfill \\ & & -\frac{(1-{\zeta }_{\gamma \left(n\right)}-{\xi }_{\gamma \left(n\right)}){\tau }_{\gamma \left(n\right)}({\tau }_{\gamma \left(n\right)}-2){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_{\gamma \left(n\right)}\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_{\gamma \left(n\right)}\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_{\gamma \left(n\right)}\right)\parallel }^2}\hfill \\ & & -\frac{{\zeta }_{\gamma \left(n\right)}{\tau }_{\gamma \left(n\right)}({\tau }_{\gamma \left(n\right)}-2){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_{\gamma \left(n\right)}\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_{\gamma \left(n\right)}\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_{\gamma \left(n\right)}\right)\parallel }^2}+{\xi }_{\gamma \left(n\right)}{\parallel l\parallel }^2\hfill \end{array}$$

that is

$$\begin{array}{ccc}\hfill & & {\xi }_{\gamma \left(n\right)}{(\parallel l\parallel }^2-\parallel x_{\gamma \left(n\right)}{-l\parallel }^2)+2{\varphi }_{\gamma \left(n\right)}{\parallel x_{\gamma \left(n\right)}-x_{\gamma \left(n\right)-1}\parallel }^2+{\varphi }_n\left\{\parallel x_{\gamma \left(n\right)}{-l\parallel }^2-{\parallel x_{\gamma \left(n\right)-1}-l\parallel }^2\right\}\hfill \\ & & +\frac{(1-{\zeta }_{\gamma \left(n\right)}-{\xi }_{\gamma \left(n\right)}){\tau }_{\gamma \left(n\right)}({\tau }_{\gamma \left(n\right)}-2){\parallel (I-R_{{\mu }_1}^{A_1})\left(v_{\gamma \left(n\right)}\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(v_{\gamma \left(n\right)}\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{v}_{\gamma \left(n\right)}\right)\parallel }^2}\hfill \\ & & +\frac{{\zeta }_{\gamma \left(n\right)}{\tau }_{\gamma \left(n\right)}({\tau }_{\gamma \left(n\right)}-2){\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_{\gamma \left(n\right)}\right)\parallel }^4}{\parallel (I-R_{{\mu }_1}^{A_1})\left(u_{\gamma \left(n\right)}\right){\parallel }^2+{\parallel B^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_{\gamma \left(n\right)}\right)\parallel }^2}\hfill \\ & & \ge {\zeta }_{\gamma \left(n\right)}(1-{\zeta }_{\gamma \left(n\right)}-{\xi }_{\gamma \left(n\right)}){\parallel u_{\gamma \left(n\right)}-F\left(w_{\gamma \left(n\right)}\right)\parallel }^2.\hfill \end{array}$$

Since ${\xi }_{\gamma \left(n\right)}\to 0$ and ${\varphi }_{\gamma \left(n\right)}\to 0$ and $\gamma \left(n\right)\to 0$, then for subsequences $\left\{x_{\gamma \left(n\right)}\right\}$, $\left\{u_{\gamma \left(n\right)}\right\}$ and $\left\{w_{\sigma \left(n\right)}\right\}$, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel u_{\gamma \left(n\right)}-F\left(w_{\gamma \left(n\right)}\right)\parallel =0,\underset{n\to \infty }{lim}\parallel (I-R_{{\mu }_1}^{A_1})\left(v_{\gamma \left(n\right)}\right)\parallel =0\mathrm{and}\underset{n\to \infty }{lim}\parallel (I-R_{{\mu }_2}^{A_2})\left(B{u}_{\gamma \left(n\right)}\right)\parallel =0.\end{array}$$

Similarly, we can show that $\parallel x_{\gamma (n+1)}-x_{\gamma \left(n\right)}\parallel \to 0,$ as $n\to \infty$ and ${\omega }_w\left(x_{\gamma \left(n\right)}\right)\subset \mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }$. It is remaining to show that $x_n\to l$.

By using $\parallel x_{\gamma \left(n\right)}-l\parallel <\parallel x_{\gamma \left(n\right)+1}-l\parallel$ and the boundedness of $\parallel x_n-l\parallel$, we have

$$\begin{array}{ccc}\hfill \parallel x_{\gamma \left(n\right)}{-l\parallel }^2& \le & 2{\alpha }_{\gamma \left(n\right)}⟨F\left(w_{\gamma \left(n\right)}\right)-u_{\gamma \left(n\right)},x_{\gamma (n+1)}-l⟩+2⟨-l,x_{\gamma (n+1)}-l⟩,\hfill \\ & \le & M\parallel F\left(w_{\gamma \left(n\right)}\right)-u_{\gamma \left(n\right)}\parallel -2⟨l,x_{\gamma \left(n\right)+1}-l⟩.\hfill \end{array}$$

Since $\parallel x_{\gamma \left(n\right)+1}-x_{\gamma \left(n\right)}\parallel \to 0$, we obtain

$$\begin{array}{ccc}\hfill lim\underset{n\to \infty }{sup}⟨-l,x_{\gamma \left(n\right)+1}-l⟩& =& -lim\underset{n\to \infty }{sup}⟨l,x_{\gamma \left(n\right)}-l⟩\hfill \\ & =& -\underset{r\in {\omega }_w\left(x_{\gamma \left(n\right)}\right)}{max}⟨l,r-l⟩\le 0,\hfill \end{array}$$

(46)

due to $l=P_{\mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }}\left(0\right)$, $\omega \left(x_{\gamma \left(n\right)}\right)\subset \mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }$ and using $\parallel F\left(w_{\gamma \left(n\right)}\right)-u_{\gamma \left(n\right)}\parallel \to 0$, using Lemma 1 in (46), we obtain that $x_{\gamma \left(n\right)}\to l$, and

$$\begin{array}{c}\hfill \parallel x_n-l\parallel \le \parallel x_{\gamma \left(n\right)+1}-l\parallel \le \parallel x_{\gamma \left(n\right)+1}-x_{\gamma \left(n\right)}\parallel +\parallel x_{\gamma \left(n\right)}-l\parallel \to 0,\end{array}$$

that is, $x_n\to l$. Hence, the theorem is proved. □

For ${\tau }_n=1$, we obtain the following corollary of Theorem 2.

Corollary 1.

Let $X_1$, $X_2$, $A_1,A_2$, B, $B^{*}$ and ${\varphi }_n$ be identical as in Theorem 2. Let $\left\{{\zeta }_n\right\}$, $\left\{{\xi }_n\right\}$ be sequences in $(0,1)$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\xi }_n=0,\sum _{n=0}^{\infty }{\xi }_n=\infty ,\underset{n\to \infty }{lim}(1-{\zeta }_n-{\xi }_n){\zeta }_n>0,\end{array}$$

hold. Then, the sequence $\left\{x_n\right\}$ obtained by Algorithm 2 (with ${\tau }_n=1$), converges strongly to $l=P_{\mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }}\left(0\right)$.

For ${\xi }_n=0$, we obtain the following corollary of Theorem 2.

Corollary 2.

Let $X_1$, $X_2$, $A_1,A_2$, B, $B^{*}$ and ${\varphi }_n$ be identical as in Theorem 2. If $\left\{{\zeta }_n\right\}$ is a sequence in $(0,1)$ so that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}(1-{\zeta }_n){\zeta }_n>0,\underset{n}{inf}{\tau }_n(2-{\tau }_n)>0\end{array}$$

holds, then the sequence $\left\{x_n\right\}$ obtained by the following scheme

$$\begin{array}{ccc}\hfill v_n& =& x_n+{\varphi }_n(x_n-x_{n-1}),\hfill \\ \hfill u_n& =& v_n-{\sigma }_n(I-R_{{\mu }_1}^{A_1})\left(v_n\right),\hfill \\ \hfill w_n& =& u_n-{\varrho }_n{B}^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right),\hfill \\ \hfill x_{n+1}& =& (1-{\zeta }_n)w_n+{\zeta }_{n}F\left(w_n\right),\hfill \end{array}$$

where ${\sigma }_n$ and ${\varrho }_n$ are defined by (15) and (16), respectively, converges strongly to $l\in \mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }$.

For ${\tau }_n=1$ and ${\xi }_n=0$, we obtain the following corollary of Theorem 2.

Corollary 3.

Let $X_1$, $X_2$, $A_1,A_2$ and B, $B^{*}$ be identical as in Algorithm 2. Let $\left\{{\zeta }_n\right\}$ be a sequence in $(0,1)$ so that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}(1-{\zeta }_n){\zeta }_n>0.\end{array}$$

Then, the sequence $\left\{x_n\right\}$ obtained by the following scheme

$$\begin{array}{ccc}\hfill v_n& =& x_n+{\varphi }_n(x_n-x_{n-1}),\hfill \\ \hfill u_n& =& v_n-{\sigma }_n(I-R_{{\mu }_1}^{A_1})\left(v_n\right),\hfill \\ \hfill w_n& =& u_n-{\varrho }_n{B}^{*}(I-R_{{\mu }_2}^{A_2})\left(B{u}_n\right),\hfill \\ \hfill x_{n+1}& =& (1-{\zeta }_n)w_n+{\zeta }_{n}F\left(w_n\right),\hfill \end{array}$$

where ${\sigma }_n$ and ${\varrho }_n$ are defined by (15) and (16), respectively (with ${\tau }_n=1$), converges strongly to $l\in \mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }$.

5. Numerical Experiments

Suppose $X_1=X_2=\mathbb{R}$. Let us consider the monotone mappings $A_1$ and $A_2$ defined as $A_1\left(x\right)=\frac{x}{2}+2\mathrm{and}{A}_2\left(x\right)=x+2$. The nonexpansive mapping $F:X_1\to X_1$ is defined as $F\left(x\right)=\frac{x-4}{2}$ and bounded linear operator $B:X_1\to X_2$ is defined as $B\left(x\right)=\frac{x}{2}$. It is not a difficult task to show that $A_1$ and $A_2$ are monotone mappings and B is nonexpansive mapping and $\mathrm{Fix}\left(F\right)\cap \mathsf{\Lambda }=\{-4\}$. The resolvents of $A_1$ and $A_2$ with parameter ${\mu }_1>0,{\mu }_2>0$ are

$$\begin{array}{ccc}\hfill R_{{\mu }_1}^{A_1}\left(x\right)={[I+{\mu }_1{A}_1]}^{-1}\left(x\right)=\frac{2x-4{\mu }_1}{2+{\mu }_1},& & R_{{\mu }_2}^{A_2}\left(x\right)={[I+{\mu }_2{A}_2]}^{-1}\left(x\right)=\frac{x-2{\mu }_2}{1+{\mu }_2}.\hfill \end{array}$$

We choose ${\tau }_n=1-\frac{1}{n+1}$, ${\mathsf{\Lambda }}_n=\frac{1}{n^2}$, $\xi =\frac{1}{n}$ and ${\zeta }_n=(1-\frac{e^{\frac{1}{n}}}{3})$ satisfying the condition (28) in Algorithm 2. We fixed the maximum number of iterations 50 as a stopping criterion. The parameter ${\varphi }_n$ is randomly generated in $(0,{\tilde{\varphi }}_n)$, where ${\tilde{\varphi }}_n$ is calculated by using (14). The behavior of the sequences $\left\{x_n\right\},\left\{v_n\right\}$ and $\left\{u_n\right\}$ are plotted in Figure 1 by applying three distinct cases of parameters which are mentioned below:

Figure 1. Numerical behavior of $\parallel w_n-u_n\parallel ,\parallel x_n-u_n\parallel ,\parallel v_n-u_n\parallel ,\parallel w_n-u_n\parallel ,\parallel x_{n+1}-x_n\parallel$ and $\parallel x_n\parallel$ choosing three cases of parameters.

• Case (I): $x_0=0$, $x_1=5$, $\varphi =0.1$, ${\mu }_1=0.5$, ${\mu }_2=0.9$.
• Case (II): $x_0=-3$, $x_1=4$, $\varphi =0.5$, ${\mu }_1=5$, ${\mu }_2=8$.
• Case (III): $x_0=5$, $x_1=-5$, $\varphi =0.75$, ${\mu }_1=10$, ${\mu }_2=20$.

Observations:

• In Figure 1a–d, we observed that the behavior of $\left\{w_n\right\},\left\{v_n\right\}$ and $\left\{u_n\right\}$ is uniform irrespective of the selection of parameters.
• From Figure 1e–f, we notice that the sequence obtained from Algorithm 2 converges to the same limit with a suitable selection of parameters.
• It is worthwhile to mention that the estimation of $\parallel B{B}^{*}\parallel$ is not required to implement the algorithm, which is not so handy to calculate in general.

6. Conclusions

We have suggested and analyzed inertial methods to estimate the solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and common solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and $\left(\mathrm{FPP}\right)$. We proved the weak and strong convergence of algorithms to estimate the solution of $\left({\mathrm{S}}_{\mathrm{p}}{\mathrm{VI}}_{\mathrm{s}}\mathrm{P}\right)$ and $\left(\mathrm{FPP}\right)$ with suitable assumptions in such a way that the estimation of the step size does not require a pre-estimated norm $\parallel B{B}^{*}\parallel$. Finally, we perform a numerical example to exhibit the behavior of the proposed algorithms using different cases of parameters.