Hybrid Inertial Self-Adaptive Iterative Methods for Split Variational Inclusion Problems

Abstract

Herein, we present two hybrid inertial self-adaptive iterative methods for determining the combined solution of the split variational inclusions and fixed-point problems. Our methods include viscosity approximation, fixed-point iteration, and inertial extrapolation in the initial step of each iteration. We employ two self-adaptive step sizes to compute the iterative sequence, which do not require the pre-calculated norm of a bounded linear operator. We prove strong convergence theorems to approximate the common solution of the split variational inclusions and fixed-point problems. Further, we implement our methods and results to examine split variational inequality and split common fixed-point problems. Finally, we illustrate our methods and compare them with some known methods existing in the literature.

1. Introduction

The fixed-point theory is the coherent and logical framework for indispensable nonlinear interdisciplinary problems including differential equations, control theory, game theory, variational inequality, equilibrium problems, optimization problems, split feasibility problems, etc. Over the last few years, fixed-point theory has become an active research area, which has led to the designing and development of efficient, effective, flexible, and easily implementable approximation methods for approximating the solutions of nonlinear and inverse problems. The fixed-point problem ($\mathrm{FPP}$) of a self-mapping $Z:\mathbb{U}\to \mathbb{U}$ is defined by

$$\begin{array}{c}\hfill \mathrm{find}{v}^{★}\in \mathbb{U}\mathrm{so}\mathrm{that}Z\left(v^{★}\right)=v^{★}.\end{array}$$

(1)

Numerous methods have been used to address fixed-point problems. Among them, the majority of methods used to approximate the fixed points are motivated by Mann’s iterative method [1]. In order to obtain a fast convergence rate, Moudafi [2] introduced the viscosity approximation technique by blending Z with a contraction mapping.

The first split problem, namely, the split feasibility problem, was initially presented by Censor and Elfving [3]. The most recent inverse problem is the split inverse problem studied by Censor et al. [4]. Because of their relevance to mathematical models of real-life problems appearing in cancer therapy [3,5], image restoration [6], computerized tomography, and data compression [7,8], several inverse problems and methods for solving them have been developed and studied in the last few years. Maudafi [9] explored the split monotone variational inclusion problems (S^plitMVIP) in the framework of Hilbert spaces. Byrne et al. [10] introduced the split common null-point problem (S^plitCNPP). A special case of (S^plitCNPP) is the split variational inclusion problem (S^plitVIP), which is defined by

$$\begin{array}{c}\hfill \mathrm{find}{v}^{★}\in \mathbb{U}\mathrm{so}\mathrm{that}0\in M\left(v^{★}\right)\\ \hfill \mathrm{and}{u}^{★}=B{v}^{★}\in \mathbb{U}\mathrm{solves}0\in N\left(u^{★}\right),\end{array}$$

where $M:\mathbb{U}\to 2^{\mathbb{U}}$, $N:\mathbb{U}\to 2^{\mathbb{U}}$ are monotone operators, $B:\mathbb{U}\to \mathbb{U}$ is a bounded linear operator, and $\mathbb{U}$ is a Hilbert space. By using the fact that the zero of the monotone operator M is the fixed point of resolvent of M, that is, $0\in M\left(v^{★}\right)\iff v^{★}=J_{\lambda }^M\left(v^{★}\right)$, Byrne et al. [10] suggested the following method for (S^plitVIP):

$$\begin{array}{c}\hfill v_{n+1}=J_{\mu }^M[v_n-\eta B^{*}(I-J_{\mu }^N)B{v}_n],\end{array}$$

(2)

where $B^{*}$ is the adjoint of B, $\mu >0$, $\eta \in (0,2/Q)$, and $Q=\parallel B^{*}B\parallel$. Based on this iterative method (2), numerous iterative methods have been developed and studied to solve (S^plitVIP). Kazmi and Rizvi [11] extended method (2) to investigate the common solution of (S^plitVIP) and ($\mathrm{FPP}$) as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_n& =J_{\mu }^M[v_n+\eta B^{*}(J_{\mu }^N-I)B{v}_n],\hfill \\ \hfill v_{n+1}& ={\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)S{u}_n,\hfill \end{array}\end{array}$$

(3)

where F is a contraction; $\eta \in (0,{\displaystyle \frac{1}{Q}})$, ${\zeta }_n\in (0,1)$ is a real sequence such that $\underset{n\to \infty }{lim}{\zeta }_n=0$, ${\displaystyle \sum _{n=1}^{\infty }}{\zeta }_n=\infty$, and ${\displaystyle \sum _{n=1}^{\infty }}|{\zeta }_n-{\zeta }_{n-1}|=\infty$. Akram et al. [12] modified the method (3) in the following manner to study the same problem:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_n& =v_n-\eta [(I-J_{{\mu }_1}^M)v_n+B\ast (I-\left(J_{{\mu }_2}^N\right)B{v}_n],\hfill \\ \hfill v_{n+1}& ={\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)S{u}_n,\hfill \end{array}\end{array}$$

(4)

where ${\mu }_1>0,{\mu }_2>0$ and $\eta ={\displaystyle \frac{1}{1+{\parallel B\parallel }^2}}$. Some other iterative methods for solving (S^plitVIP) and ($\mathrm{FPP}$) can also be seen in [13,14,15,16] and references therein.

The common disadvantage of these methods is the calculation of the step size, which depends on the calculation of $\parallel B^{*}B\parallel$ and the calculation of $\parallel B^{*}B\parallel$ is a challenging task. To address this challenge, researchers developed iterative methods that eliminate the estimation of $\parallel B^{*}B\parallel$. Lopez et al. [17] investigated split feasibility problems without knowing the norm of the matrix. Dilshad et al. [18] studied the split common null-point problem without a pre-existing estimation of the operator norm as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_n& =v_n-J_{{\mu }_1}^M{v}_n+B^{*}(I-\left(J_{{\mu }_2}^N\right)B{v}_n,\hfill \\ \hfill v_{n+1}& ={\zeta }_{n}u+(1-{\zeta }_n)(v_n-{\eta }_n{u}_n),\hfill \end{array}\end{array}$$

(5)

for some fixed $u\in \mathbb{U}$ and

$$\begin{array}{c}\hfill {\eta }_n={\displaystyle \frac{\parallel v_n-J_{{\mu }_1}^M{v}_n{\parallel }^2+{\parallel (I-J_{{\mu }_2}^N)B{v}_n\parallel }^2}{\parallel v_n-J_{{\mu }_1}^M{v}_n+B^{*}(I-J_{{\mu }_2}^N)B{v}_n{\parallel }^2}}.\end{array}$$

(6)

In this direction, several research papers have caught the attention of researchers: see [19,20,21,22] and references therein.

To obtain the fast convergence of iterative algorithms, Alvarez and Attouch [23] introduced a new algorithm for estimating the solution of variational inclusions. This algorithm was named as an inertial proximal point algorithm. It is observed that the sequence derived from the inertial proximal point method converges rapidly because of its design. As a result, numerous researchers have applied the inertial term since it plays a crucial role in accelerating the convergence; see [24,25,26,27,28] and references therein.

In continuation to the above study, our aim is to present two hybrid inertial self-adaptive iterative methods to estimate the common solution of (S^plitVIP) and ($\mathrm{FPP}$), which can be summarized as follows:

• Our motive is to introduce fast and traditionally different viscosity methods to estimate the common solution of (S^plitVIP) and ($\mathrm{FPP}$). Unlike method (3) and method (4) [or method (5)], our hybrid algorithms compute the viscosity approximation and fixed-point iteration [or Halpern-type iteration] in the initial step of each iteration.
• To accelerate the convergence, we also add the inertial term in the initial step of the iteration. Therefore, in the first step, we compute the inertial extrapolation, fixed-point iteration, and viscosity approximation all at the same time.
• In method (3) and method (4), the pre-calculated norm of B is essential, which is a tedious task. However, we are using two self-adaptive step-sizes, which do not require the pre-calculated norms of a bounded linear operator B.
• Our methods are efficient and an accelerated version of method (2) of [10], method (3) of [11], method (4) of [12], and method (5) of [18], which is demonstrated by numerical examples.

2. Preliminaries

All through the text, we symbolize the real Hilbert space by $\mathbb{U}$ and by $\mathbb{D}\ne \varnothing$, which is the close and convex subset of $\mathbb{U}$. We denote the strong and weak convergence of the sequence $\left\{v_n\right\}$ to v by $v_n\to v$ and $v_n\rightharpoonup v$, respectively.

For all ${\eta }_1,{\eta }_2,{\eta }_3$ in $\mathbb{U}$, $t_1,t_2,t_3\in [0,1]$ such that $t_1+t_2+t_3=1$, the following equality and inequality hold:

$$\begin{array}{ccc}\hfill \parallel t_1{\eta }_1+t_2{\eta }_2+t_3{\eta }_3{\parallel }^2& =& t_1\parallel {\eta }_1{\parallel }^2+t_2\parallel {\eta }_2{\parallel }^2+t_3{\parallel {\eta }_1\parallel }^2\hfill \\ & & -t_1{t}_2\parallel {\eta }_1-{\eta }_2{\parallel }^2-t_2{t}_3\parallel {\eta }_2-{\eta }_3{\parallel }^2-t_3{t}_1{\parallel {\eta }_3-{\eta }_2\parallel }^2\hfill \end{array}$$

(7)

and

$$\begin{array}{c}\hfill \parallel {\eta }_1\pm {\eta }_2{\parallel }^2=\parallel {\eta }_1{\parallel }^2\pm 2\langle {\eta }_1,{\eta }_2\rangle +\parallel {\eta }_2{\parallel }^2\le {\parallel {\eta }_1\parallel }^2\pm 2\langle {\eta }_2,{\eta }_1+{\eta }_2\rangle .\end{array}$$

(8)

Definition 1.

A mapping $Z:\mathbb{U}\to \mathbb{U}$ is said to be

(1) 

Averaged if there exists a nonexpansive mapping $f:\mathbb{U}\to \mathbb{U}$ and $\alpha \in (0,1)$ such that $Z=(1-\alpha )I+\alpha f$;

(2) 

Lipschitz continuous if there exists $\theta >0$, such that $\parallel Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right)\parallel \le \theta \parallel {\varrho }_1-{\varrho }_2\parallel ,\forall {\varrho }_1$, ${\varrho }_2\in \mathbb{U};$

(3) 

Contraction if $\parallel Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right)\parallel \le \theta \parallel {\varrho }_1-{\varrho }_2\parallel ,\forall {\varrho }_1,{\varrho }_2\in \mathbb{U}$, for some $\theta \in (0,1);$

(4) 

Nonexpansive if $\parallel Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right)\parallel \le \parallel {\varrho }_1-{\varrho }_2\parallel ,\forall {\varrho }_1,{\varrho }_2\in \mathbb{U};$

(5) 

Firmly nonexpansive if $\parallel Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right){\parallel }^2\le \langle {\varrho }_1-{\varrho }_2,Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right)\rangle ,\forall {\varrho }_1,{\varrho }_2\in \mathbb{U};$

(6) 

κ-inverse strongly monotone (κ-ism) if there exists $\kappa >0$ such that

$$\langle Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right),{\varrho }_1-{\varrho }_2\rangle \ge \kappa {\parallel Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right)\parallel }^2,\forall {\varrho }_1,{\varrho }_2\in \mathbb{U};$$

(7) 

Monotone if

$$\langle Z\left({\varrho }_1\right)-Z\left({\varrho }_2\right),{\varrho }_1-{\varrho }_2\rangle \ge 0,\forall {\varrho }_1,{\varrho }_2\in \mathbb{U}.$$

Definition 2

([29]). Let $N:\mathbb{U}\to 2^{\mathbb{U}}$ be a set-valued mapping. Then,

(1) 

N is called monotone if $\langle {\varrho }_1-{\varrho }_2,{\eta }_1-{\eta }_2\rangle \ge 0,\forall {\varrho }_1,{\varrho }_2,\in \mathbb{U},{\eta }_1\in M\left({\varrho }_1\right),{\eta }_2\in N\left({\varrho }_2\right)$;

(2) 

$Graph\left(N\right)=\{({\varrho }_1,{\eta }_1)\in \mathbb{U}\times \mathbb{U}:{\eta }_1\in N\left({\varrho }_1\right)\}$;

(3) 

N is said to be maximal monotone if N is monotone and $(I+\mu N)\left(\mathbb{U}\right)=\mathbb{U}$, for $\mu >0$, where I is an identity mapping on $\mathbb{U}$;

(4) 

The resolvent of N is defined by $J_{\mu }^N={[I+\mu N]}^{-1}$, where I is an identity mapping and $\mu >0$.

Remark 1.

(1) 

It can be easily seen that a κ-inverse strongly monotone mapping is also monotone and ${\displaystyle \frac{1}{\kappa }}$-Lipschitz continuous.

(2) 

Every averaged mapping is nonexpansive, but the converse need not be true in general.

(3) 

The operator Z is firmly nonexpansive if and only if $I-Z$ is firmly nonexpansive.

(4) 

The composition of two averaged operators is also averaged.

Remark 2.

(1) 

The resolvent $J_{\mu }^N$ of the maximal monotone mapping M is single-valued, nonexpansive, as well as firmly nonexpansive for any $\mu >0$.

(2) 

The resolvent $J_{\mu }^N$ is firmly nonexpansive if and only if

$$\begin{array}{c}\hfill \parallel J_{\mu }^N{\varrho }_1-J_{\mu }^N{\varrho }_2{\parallel }^2\le \parallel {\varrho }_1-{\varrho }_2{\parallel }^2-{\parallel (I-J_{\mu }^N){\varrho }_1-(I-J_{\mu }^N){\varrho }_2\parallel }^2,\mathrm{for}\mathrm{all}{\varrho }_1,{\varrho }_2\in \mathbb{U}.\end{array}$$

(3) 

The operator $I-J_{\mu }^N$ is nonexpansive and so it is demiclosed at zero.

(4) 

If $N:\mathbb{U}\to 2^{\mathbb{U}}$ is monotone, then $J_{\mu }^N$ and $I-J_{\mu }^N$ are firmly nonexpansive for $\mu >0$, and $J_{\mu }^N$ is the resolvent of N.

Lemma 1

([30]). Let $\varnothing \ne \mathbb{D}$ be a closed and convex subset of Hilbert space $\mathbb{U}$, and $Z:\mathbb{D}\to \mathbb{D}$ is nonexpansive mapping such that

(1) 

$\mathrm{Fix}\left(Z\right)\ne \varnothing ,$

(2) 

The sequence $\left\{v_n\right\}\rightharpoonup v^{★}$ and $\underset{n\to \infty }{lim}\parallel Z\left(v_n\right)-v_n\parallel =0.$

• Then, $Z\left(v^{★}\right)=v^{★}.$

Lemma 2

([31]). If $\left\{v_n\right\}$ is a sequence of non-negative real numbers satisfying

$$v_{n+1}\le (1-{\psi }_n)v_n+{\psi }_n{\phi }_n,n\ge 0,$$

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

(1) 

$\underset{n\to \infty }{lim}{\psi }_n=0,\mathrm{and}{\displaystyle \sum _{n=1}^{\infty }}{\psi }_n=\infty ,$

(2) 

$lim\underset{n\to \infty }{sup}{\phi }_n\le 0.$

• Then, $\underset{n\to \infty }{lim}{v}_n=0.$

Lemma 3

([32]). Suppose $\mathbb{D}\ne \varnothing$ is a closed and convex subset of $\mathbb{U}.$ If the sequence $\left\{v_n\right\}\in \mathbb{U}$ satisfies the following:

(1) 

$\underset{n\to \infty }{lim}\parallel v_n-v\parallel$ exists for all $v\in \mathbb{D}$,

(2) 

Any weak cluster point of $\left\{v_n\right\}$ belongs to $\mathbb{D}$;

• then, there exists $v^{★}\in \mathbb{D}$ such that $v_n\rightharpoonup v^{★}$.

Lemma 4

([33]). Let $\left\{v_n\right\}$ be a real sequence that does not decrease at infinity in the sense that there exists a subsequence $\left\{v_{n_k}\right\}$ of $\left\{v_n\right\}$ such that $v_{n_k}<v_{n_k+1},$$\forall k\ge 0$. Also consider the sequence of integers ${\left\{\epsilon \left(n\right)\right\}}_{n\ge n_0}$ defined by

$$\begin{array}{c}\hfill \epsilon \left(n\right)=max\{k\le n:v_k\le v_{k+1}\}.\end{array}$$

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

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

3. Main Results

The solution sets of (S^plitVIP) and ($\mathrm{FPP}$) are denoted by ${\Delta }_1$ and ${\Delta }_2$, respectively. To establish the convergence of the suggested methods, we make the following assumptions:

(X₁)

$F:\mathbb{U}\to \mathbb{U}$ is $\theta$-contraction;

(X₂)

$M,N:\mathbb{U}\to 2^{\mathbb{U}}$ are monotone operators and $Z:\mathbb{U}\to \mathbb{U}$ is a nonexpansive mapping;

(X₃)

$\left\{{\zeta }_n\right\}$ is a sequence in $(0,1)$ so that $\underset{n\to \infty }{lim}{\zeta }_n=0$, and ${\displaystyle \sum _{n=1}^{\infty }}{\zeta }_n=\infty$;

(X₄)

$\left\{{\varphi }_n\right\}$ is a positive and bounded sequence such that $\underset{n\to \infty }{lim}{\displaystyle \frac{{\varphi }_n}{{\zeta }_n}}=0$;

(X₅)

The common solution set of (S^plitVIP) and ($\mathrm{FPP}$) is expressed by ${\Delta }_1\cap {\Delta }_2$ and ${\Delta }_1\cap {\Delta }_2\ne \varnothing$.

Now, we are in the position to design our hybrid Algorithm 1. The hybrid Algorithm 1 is constructed in such a way that the initial step iterates the inertial extrapolation term ${\tau }_n(v_n-v_{n-1})$ combined with the viscosity approximation. We implement our hybrid Algorithm 1 to estimate the common solution of (S^plitVIP) and ($\mathrm{FPP}$).

Algorithm 1. Hybrid Algorithm 1

Choose $\lambda >0,\mu >0$, $\tau \in [0,1)$, and  $0<{\varsigma }_n<2$. Select initial points $v_0$ and $v_1$ and fix $n=0$.

Iterative Step: For  $n\ge 1$ iterate $v_n$, $v_{n-1}$ and select  $0<{\tau }_n\le {\overline{\tau }}_n$, where  $$\begin{array}{c}\hfill {\overline{\tau }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{{\varphi }_n}{\parallel v_n-v_{n-1}\parallel }},\tau \right\},\mathrm{if}{v}_n\ne v_{n-1},\hfill \\ \tau ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$ (9)

Compute  $$\begin{array}{ccc}\hfill s_n& =& {\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1}),\hfill \end{array}$$ (10) $$\begin{array}{ccc}\hfill u_n& =& s_n-{\lambda }_n(I-J_{\lambda }^M)\left(s_n\right),\hfill \end{array}$$ (11) $$\begin{array}{ccc}\hfill v_{n+1}& =& u_n-{\mu }_n{B}^{*}(I-J_{\mu }^N)\left(B{u}_n\right),\hfill \end{array}$$ (12)   where ${\lambda }_n$ and ${\mu }_n$ are defined by  $$\begin{array}{c}\hfill {\lambda }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\lambda }^M)s_n\parallel }{\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$ (13)  and  $$\begin{array}{c}\hfill {\mu }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\mu }^N)B{u}_n\parallel }{\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$ (14)

If  $v_{n+1}=u_n=v_n=s_n$, then stop; otherwise, fix  $n=n+1$ and go back to the computation.

Remark 3.

Let $u_n=s_n$ in Algorithm 1; then, if $\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel \ne 0$, we obtain from (11) that ${\lambda }_n\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel =0$, that is, ${\displaystyle \frac{{\varsigma }_n{\parallel (I-J_{\lambda }^M)s_n\parallel }^2}{\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel }}=0$, which implies that ${\varsigma }_n\parallel (I-J_{\lambda }^M)s_n\parallel =0$, which concludes that $0\in M\left(s_n\right)$. If $v_{n+1}=u_n$ and $\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel \ne 0$, we obtain from (12) that ${\mu }_n\parallel B^{*}(I-J_{\lambda }^N)\left(B{u}_n\right)\parallel =0.$ Since B is a bounded linear operator, we obtain $(I-J_{\lambda }^N)\left(B{u}_n\right)=0$, that is, $0\in N\left(B{u}_n\right)$. If $\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel =0$, then there is nothing to show.

Remark 4.

From (9) and Assumption ($X_4$), we have $\underset{n\to \infty }{lim}{\displaystyle \frac{{\tau }_n\parallel v_n-v_{n-1}\parallel }{{\zeta }_n}}\le {\displaystyle \frac{{\varphi }_n}{{\zeta }_n}}=0$. Therefore, there exists a constant $L_1$ such that ${\displaystyle \frac{{\tau }_n\parallel v_n-v_{n-1}\parallel }{{\zeta }_n}}\le L_1$ or ${\tau }_n\parallel v_n-v_{n-1}\parallel \le L_1{\zeta }_n$.

Next, we utilize our hybrid Algorithm 1 to establish the strong convergence theorem, which approximates the common solution of (S^plitVIP) and ($\mathrm{FPP}$). In the presentation of the strong convergence theorem, the implemented method computes two step sizes, which makes us to calculate the norm of the bounded linear operator B.

Theorem 1.

If assumptions $\left(X_1\right)-\left(X_5\right)$ hold, then the sequence $\left\{v_n\right\}$ induced by Algorithm 1 converges strongly to v, where $v=P_{{\Delta }_1\cap {\Delta }_2}F\left(v\right)$.

Proof. 

Let $l^{★}\in {\Delta }_1\cap {\Delta }_2$. By using (8), (11) and Remark 2 (4), we have

$$\begin{array}{cc}\hfill & \parallel u_n-l^{★}{\parallel }^2\hfill \\ =& \parallel s_n-{\lambda }_n(I-J_{\lambda }^M)\left(s_n\right)-l^{★}{\parallel }^2\hfill \\ \le & \parallel s_n-l^{★}{\parallel }^2+{\lambda }_n^2{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^2-2{\lambda }_n\langle (I-J_{\lambda }^M)\left(s_n\right),s_n-l^{★}\rangle \hfill \\ =& \parallel s_n-l^{★}{\parallel }^2+{\lambda }_n^2{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^2-2{\lambda }_n\langle (I-J_{\lambda }^M)\left(s_n\right)-(I-J_{\lambda }^M)\left(l^{★}\right),s_n-l^{★}\rangle \hfill \\ \le & \parallel s_n-l^{★}{\parallel }^2+{\lambda }_n^2\parallel (I-J_{\lambda }^M)\left(s_n\right){\parallel }^2-2{\lambda }_n{\parallel (I-J_{\lambda }^M)\left(s_n\right)-(I-J_{\lambda }^M)\left(l^{★}\right)\parallel }^2\hfill \\ =& \parallel s_n-l^{★}{\parallel }^2+({\lambda }_n^2-2{\lambda }_n){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^2.\hfill \end{array}$$

(15)

Now, using (13), we estimate that

$$\begin{array}{cc}\hfill & ({\lambda }_n^2-2{\lambda }_n){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^2\hfill \\ \hfill =& \parallel (I-J_{\lambda }^M)\left(s_n\right){\parallel }^2[{\displaystyle \frac{{{\varsigma }_n}^2{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^2}{{\left(\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel \right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{2{\varsigma }_n\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}]\hfill \end{array}$$

$$\begin{array}{ccc} & =& \parallel (I-J_{\lambda }^M)\left(s_n\right){\parallel }^3\left[{\displaystyle \frac{{{\varsigma }_n}^2\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel -2{\varsigma }_n(\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel )}{{\left(\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel \right)}^2}}\right]\hfill \\ & \le & \parallel (I-J_{\lambda }^M)\left(v_n\right){\parallel }^3\left[{\displaystyle \frac{({{\varsigma }_n}^2-2{\varsigma }_n)(\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel )}{{\left(\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel \right)}^2}}\right]\hfill \\ & =& {\displaystyle \frac{({{\varsigma }_n}^2-2{\varsigma }_n){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}.\hfill \end{array}$$

(16)

From (15) and (16), we obtain

$$\begin{array}{c}\hfill \parallel u_n-l^{★}{\parallel }^2\le {\parallel s_n-l^{★}\parallel }^2+{\displaystyle \frac{({{\varsigma }_n}^2-2{\varsigma }_n){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}.\end{array}$$

(17)

Since ${\varsigma }_n\in (0,2)$, we obtain

$$\begin{array}{c}\hfill \parallel u_n-l^{★}\parallel \le \parallel s_n-l^{★}\parallel .\end{array}$$

(18)

Applying the same steps as in the calculation of (16) and (17), we can easily obtain the following:

$$\begin{array}{ccc}\hfill \parallel v_{n+1}-l^{★}{\parallel }^2& \le & \parallel u_n-l^{★}{\parallel }^2+({\mu }^2-2{\mu }_n){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^2.\hfill \end{array}$$

(19)

By using (14), we can obtain

$$\begin{array}{ccc}\hfill ({\mu }_n^2-2{\mu }_n){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^2& =& {\displaystyle \frac{({{\varsigma }_n}^2-2{\varsigma }_n){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_n\right)\parallel }}.\hfill \end{array}$$

(20)

It follows from (19) and (20) that

$$\begin{array}{c}\hfill \parallel v_{n+1}-l^{★}{\parallel }^2\le {\parallel u_n-l^{★}\parallel }^2+{\displaystyle \frac{({{\varsigma }_n}^2-2{\varsigma }_n){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_n\right)\parallel }},\end{array}$$

(21)

or

$$\begin{array}{c}\hfill \parallel v_{n+1}-l^{★}\parallel \le \parallel u_n-l^{★}\parallel .\end{array}$$

(22)

Combining (17) and (21), we obtain

$$\begin{array}{ccc}\hfill \parallel v_{n+1}-l^{★}{\parallel }^2\le {\parallel s_n-l^{★}\parallel }^2& +& {\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}\hfill \\ & & +{\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_n\right)\parallel }}.\hfill \end{array}$$

(23)

Since ${\varsigma }_n\in (0,2)$, we conclude that

$$\begin{array}{c}\hfill \parallel v_{n+1}-l^{★}\parallel \le \parallel s_n-l^{★}\parallel .\end{array}$$

(24)

Since F is $\theta$-contraction, using (10) and Remark 4, we have

$$\begin{array}{ccc}\hfill \parallel s_n-l^{★}\parallel & =& \parallel {\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})-l^{★}\parallel \hfill \\ & =& {\zeta }_n\parallel F\left(v_n\right)-l^{★}\parallel +(1-{\zeta }_n)\parallel Z\left(v_n\right)-l^{★}\parallel +{\tau }_n\parallel v_n-v_{n-1}\parallel \hfill \\ & \le & {\zeta }_n\parallel F\left(v_n\right)-F\left(l^{★}\right)\parallel +{\zeta }_n\parallel F\left(l^{★}\right)-l^{★}\parallel +(1-{\zeta }_n)\parallel Z\left(v_n\right)-l^{★}\parallel +{\tau }_n\parallel v_n-v_{n-1}\parallel \hfill \\ & \le & {\zeta }_n\theta \parallel v_n-l^{★}\parallel +{\zeta }_n\parallel F\left(l^{★}\right)-l^{★}\parallel +(1-{\zeta }_n)\parallel v_n-l^{★}\parallel +{\zeta }_n{L}_1\hfill \\ & =& [1-{\zeta }_n(1-\theta )]\parallel v_n-l^{★}\parallel +{\zeta }_n(1-\theta ){\displaystyle \frac{\parallel F\left(l^{★}\right)-l^{★}\parallel +L_1}{1-\theta }}\hfill \\ & \le & max\left\{\parallel v_n-l^{★}\parallel ,{\displaystyle \frac{\parallel F\left(l^{★}\right)-l^{★}\parallel +L_1}{1-\theta }}\right\}.\hfill \end{array}$$

Taking advantage of (24) and by mathematical induction, we achieve that the sequence $\left\{s_n\right\}$ is bounded, and so are $\left\{v_n\right\}$ and $\left\{u_n\right\}$. Let $r_n={\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)Z\left(v_n\right)$, which is also bounded. By using (8), we obtain

$$\begin{array}{ccc}\hfill & & \parallel r_n-l^{★}{\parallel }^2\hfill \\ & =& \parallel {\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)Z\left(v_n\right)-l^{★}{\parallel }^2\hfill \\ \hfill & =& {\zeta }_n^2\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{(1-{\zeta }_n)}^2{\parallel Z\left(v_n\right)-l^{★}\parallel }^2+2{\zeta }_n(1-{\zeta }_n)\langle F\left(v_n\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle \hfill \\ & =& {\zeta }_n^2\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{(1-{\zeta }_n)}^2{\parallel v_n-l^{★}\parallel }^2+2{\zeta }_n\langle F\left(v_n\right)-F\left(l^{★}\right),Z\left(v_n\right)-l^{★}\rangle \hfill \\ & & +2{\zeta }_n\langle F\left(l^{★}\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle -2{\zeta }_n^2\langle F\left(v_n\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle \hfill \\ & \le & {\zeta }_n^2\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{(1-{\zeta }_n)}^2\parallel v_n-l^{★}{\parallel }^2+2{\zeta }_n\theta \parallel v_n-l^{★}\parallel \parallel v_n-l^{★}\parallel \hfill \\ & & +2{\zeta }_n\langle F\left(l^{★}\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle +2{\zeta }_n^2\parallel F\left(v_n\right)-l^{★}\parallel \parallel v_n-l^{★}\parallel \hfill \\ & =& \left[1-2{\zeta }_n(1-\theta )\right]\parallel v_n-l^{★}{\parallel }^2+{\zeta }_n[{\zeta }_n(\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}{\parallel )}^2\hfill \\ & & +2\langle F\left(l^{★}\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle ].\hfill \end{array}$$

(25)

We also estimate

$$\begin{array}{ccc}\hfill & & \langle r_n-l^{★},{\tau }_n(v_n-v_{n-1})\rangle \hfill \\ & =& \langle {\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)Z\left(v_n\right)-l^{★},{\tau }_n(v_n-v_{n-1})\rangle \hfill \\ & =& {\zeta }_n{\tau }_n\langle F\left(v_n\right)-l^{★},(v_n-v_{n-1})\rangle +(1-{\zeta }_n){\tau }_n\langle Z\left(v_n\right)-l^{★},(v_n-v_{n-1})\rangle \hfill \\ & \le & {\zeta }_n{\tau }_n\parallel F\left(v_n\right)-l^{★}\parallel \parallel v_n-v_{n-1}\parallel +(1-{\zeta }_n){\tau }_n\parallel Z\left(v_n\right)-l^{★}\parallel \parallel v_n-v_{n-1}\parallel \hfill \\ & =& {\tau }_n\parallel v_n-v_{n-1}\parallel \left[{\zeta }_n\parallel F\left(v_n\right)-l^{★}\parallel +(1-{\zeta }_n)\parallel v_n-l^{★}\parallel \right]\hfill \\ & \le & {\varphi }_n\left[\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}\parallel \right],\hfill \end{array}$$

(26)

since ${\tau }_n\parallel v_n-v_{n-1}\parallel \le {\varphi }_n$. By using the above estimated values in (25) and (26), we obtain

$$\begin{array}{ccc}\hfill \parallel s_n-l^{★}{\parallel }^2& =& \parallel r_n+{\tau }_n(v_n-v_{n-1})-l^{★}{\parallel }^2\hfill \\ & =& \parallel r_n-l^{★}{\parallel }^2+2{\tau }_n\langle v_n-v_{n-1},r_n-l^{★}\rangle +{\tau }_n^2{\parallel v_n-v_{n-1}\parallel }^2\hfill \\ & \le & \left[1-{\zeta }_n(1-\theta )\right]{\parallel v_n-l^{★}\parallel }^2+2{\varphi }_n\left[\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}\parallel \right]\hfill \\ & & +{\zeta }_n\left[{\zeta }_n(\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}{\parallel )}^2+2\langle F\left(l^{★}\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle \right]+{\varphi }_n^2\hfill \\ & =& (1-e_n){\parallel v_n-l^{★}\parallel }^2+e_n{d}_n,\hfill \end{array}$$

(27)

where $e_n={\zeta }_n(1-\theta )$ and

$$\begin{array}{c}\hfill d_n={\displaystyle \frac{\frac{2{\varphi }_n}{{\zeta }_n}\left(\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}\parallel \right)+{\zeta }_n{\left(\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}\parallel \right)}^2+2\langle F\left(l^{★}\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle +\frac{{\varphi }_n^2}{{\zeta }_n}}{(1-\theta )}}.\end{array}$$

Combining (23) and (27), we obtain

$$\begin{array}{ccc}\hfill \parallel v_{n+1}-l^{★}{\parallel }^2\le (1-e_n){\parallel v_n-l^{★}\parallel }^2+e_n{d}_n& +& {\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}\hfill \\ & & +{\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_n\right)\parallel }}.\hfill \end{array}$$

(28)

The remaining proof can be split in two possible cases:

• Case I: If $\{\parallel v_n-l^{★}\parallel \}$ is not monotonically increasing, then there exists a number $N_1$ such that $\parallel v_{n+1}-l^{★}\parallel \le \parallel v_n-l^{★}\parallel$ for all $n\ge N_1$. Hence, the boundedness of $\{\parallel v_n-l^{★}\parallel \}$ implies that $\{\parallel v_n-l^{★}\parallel \}$ is convergent. Therefore, using (28), we have

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}+{\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_n\right)\parallel }}\hfill \\ & \le & \parallel v_n-l^{★}{\parallel }^2-\parallel v_{n+1}-l^{★}{\parallel }^2-e_n{\parallel v_n-l^{★}\parallel }^2+e_n{d}_n\hfill \end{array}$$

Since ${\varsigma }_n\in (0,2)$ and $e_n\to 0$ as $n\to \infty$, by taking limit $n\to \infty$, we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\displaystyle \frac{\parallel (I-J_{\lambda }^M)\left(s_n\right){\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}=0,\end{array}$$

$$\begin{array}{c}\hfill \mathrm{and}\underset{n\to \infty }{lim}{\displaystyle \frac{\parallel (I-J_{\mu }^N)\left(B{u}_n\right){\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_n\right){\parallel }^2+\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_n\right)\parallel }}=0,\end{array}$$

which implies that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel =\underset{n\to \infty }{lim}\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel =0.\end{array}$$

(29)

From (11) and (13), we infer that

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

(30)

Using (12) and (14), we obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_{n+1}-u_n\parallel =0.\end{array}$$

(31)

Taking together (30) and (31), we have

$$\begin{array}{c}\hfill \parallel v_{n+1}-s_n\parallel \le \parallel v_{n+1}-u_n\parallel +\parallel u_n-s_n\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(32)

It is not difficult to obtain that

$$\begin{array}{c}\hfill \parallel v_{n+1}-v_n\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(33)

By using $s_n-v_n=r_n+{\tau }_n(v_n-v_{n-1})-v_n$, and using (32), (33), and Remark 4, we immediately see that

$$\begin{array}{c}\hfill \parallel r_n-v_n\parallel \le {\tau }_n\parallel v_n-v_{n-1}\parallel +\parallel v_{n+1}-v_n\parallel +\parallel v_{n+1}-s_n\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(34)

Hence, we can obtain

$$\begin{array}{c}\hfill \parallel s_n-v_n\parallel \le {\tau }_n\parallel v_n-v_{n-1}\parallel +\parallel r_n-v_n\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(35)

From (10), we can easily write that

$$\begin{array}{c}\hfill s_n-Z\left(v_n\right)={\zeta }_n(F\left(v_n\right)-l^{★})+{\zeta }_n(Z\left(l^{★}\right)-Z\left(v_n\right))+{\tau }_n(v_n-v_{n-1}).\end{array}$$

(36)

Using the boundedness of $\left\{v_n\right\}$, Condition $\left(X_3\right)$, the nonexpansive property of Z, and Remark 4, we achieve

$$\begin{array}{c}\hfill \parallel s_n-Z\left(v_n\right)\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(37)

Similarly, we can show that

$$\begin{array}{c}\hfill \parallel u_n-Z\left(v_n\right)\parallel \to 0,\parallel v_n-Z\left(v_n\right)\parallel \to 0\mathrm{as}n\to \infty .\end{array}$$

(38)

Since $\left\{v_n\right\}$ is bounded, which implies the existence of subsequence $\left\{v_{n_k}\right\}$ converging weakly to $\overline{l}$, the subsequences $\left\{s_{n_k}\right\}$, $\left\{u_{n_k}\right\}$ of $\left\{s_n\right\}$ and $\left\{u_n\right\}$, respectively, also converge weakly to $\overline{l}$. It follows from (29) and (38) that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\parallel (I-J_{\lambda }^M)\left(s_{n_k}\right)\parallel =0,\underset{k\to \infty }{lim}\parallel (I-J_{\mu }^N)\left(B{u}_{n_k}\right)\parallel =0\mathrm{and}\underset{k\to \infty }{lim}\parallel v_{n_k}-Z\left(v_{n_k}\right)\parallel =0.\end{array}$$

(39)

Keeping in mind (30), (31), and (39), we infer that $\overline{l}\in {\Delta }_n\cap {\Delta }_2$.

Finally, we prove that the sequence $\left\{v_n\right\}$ strongly converges. From (28), we have

$$\begin{array}{c}\hfill \parallel v_{n+1}-l^{★}{\parallel }^2\le (1-e_n){\parallel v_n-l^{★}\parallel }^2+e_n{d}_n.\end{array}$$

(40)

Furthermore,

$$\begin{array}{ccc}\hfill & & lim\underset{n\to \infty }{sup}{d}_n\hfill \\ & & =lim\underset{n\to \infty }{sup}{\displaystyle \frac{\frac{2{\varphi }_n}{{\varsigma }_n}\left(\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}\parallel \right)+{\varsigma }_n{\left(\parallel F\left(v_n\right)-l^{★}\parallel +\parallel v_n-l^{★}\parallel \right)}^2+2\langle F\left(l^{★}\right)-l^{★},Z\left(v_n\right)-l^{★}\rangle +\frac{{\varphi }_n^2}{{\zeta }_n}}{(1-\theta )}}\hfill \\ & & =lim\underset{k\to \infty }{sup}{\displaystyle \frac{\frac{2{\varphi }_{n_k}}{{\varsigma }_{n_k}}\left(\parallel F\left(v_{n_k}\right)-l^{★}\parallel +\parallel v_{n_k}-l^{★}\parallel \right)+{\zeta }_{n_k}{\left(\parallel F\left(v_{n_k}\right)-l^{★}\parallel +\parallel v_{n_k}-l^{★}\parallel \right)}^2+2\langle F\left(l^{★}\right)-l^{★},Z\left(v_{n_k}\right)-l^{★}\rangle +\frac{{\varphi }_{n_k}^2}{{\zeta }_{n_k}}}{(1-\theta )}}\hfill \\ & & =lim\underset{k\to \infty }{sup}2\langle F\left(l^{★}\right)-l^{★},Z\left(v_{n_k}\right)-v_{n_k}\rangle +2\langle F\left(l^{★}\right)-l^{★},v_{n_k}-l^{★}\rangle \hfill \\ & & =0+\langle F\left(l^{★}\right)-l^{★},\overline{l}-l^{★}\rangle \hfill \\ & & \le 0.\hfill \end{array}$$

Now we are in position to apply Lemma 2 in (40) and conclude that $\left\{v_n\right\}$ converges strongly to $l^{★}$. Hence, the result is proved.

• Case II: If $\{\parallel v_n-l^{★}\parallel \}$ is monotonically increasing, then the sequence $ϵ:\mathbb{N}:\to \mathbb{N}$ for all $n\ge n_0$ defined by $ϵ\left(n\right)=max\{k\in \mathbb{N}:n\ge k:\parallel v_k-l^{★}\parallel \le \parallel v_{k+1}-l^{★}\parallel \}$ is increasing such that $ϵ\left(n\right)\to \infty$ as $n\to \infty$ and

$$\begin{array}{c}\hfill 0\le \parallel v_{ϵ\left(n\right)}-l^{★}\parallel \le \parallel v_{ϵ\left(n\right)+1}-l^{★}\parallel ,\forall n\ge n_0.\end{array}$$

(41)

By using (28), we have

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\varsigma }_{ϵ\left(n\right)}\left({\varsigma }_{ϵ\left(n\right)}-2\right){\parallel (I-J_{\lambda }^M)\left(s_{ϵ\left(n\right)}\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_{ϵ\left(n\right)}\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_{ϵ\left(n\right)}\right)\parallel }}+{\displaystyle \frac{{\varsigma }_n\left({\varsigma }_{ϵ\left(n\right)}-2\right){\parallel (I-J_{\mu }^N)\left(B{u}_{ϵ\left(n\right)}\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_{ϵ\left(n\right)}\right){\parallel }^2+\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_{ϵ\left(n\right)}\right)\parallel }}\hfill \\ & \le & \parallel v_{ϵ\left(n\right)}-l^{★}{\parallel }^2-\parallel v_{ϵ\left(n\right)+1}-l^{★}{\parallel }^2+e_{ϵ\left(n\right)}{d}_{ϵ\left(n\right)}-e_{ϵ\left(n\right)}{\parallel v_{ϵ\left(n\right)}-l^{★}\parallel }^2+e_{ϵ\left(n\right)}{d}_{ϵ\left(n\right)}\hfill \end{array}$$

By passing the limit $n\to \infty$, we obtain

$$\begin{array}{c}\hfill \parallel (I-J_{\lambda }^M)\left(s_{ϵ\left(n\right)}\right)\parallel \to 0,\parallel (I-J_{\mu }^N)\left(B{u}_{ϵ\left(n\right)}\right)\parallel \to 0.\end{array}$$

Using the same techniques as in the proof of Case I, we obtain $\parallel v_{ϵ\left(n\right)+1}-s_{ϵ\left(n\right)}\parallel \to 0$, $\parallel v_{ϵ\left(n\right)+1}-u_{ϵ\left(n\right)}\parallel \to 0$, $\parallel v_{ϵ\left(n\right)+1}-v_{ϵ\left(n\right)}\parallel \to 0$, and $\parallel Z\left(v_{ϵ\left(n\right)}\right)-v_{ϵ\left(n\right)}\parallel \to 0$ as $n\to \infty$. From (40) and (41), we obtain

$$\begin{array}{c}\hfill 0\le \parallel v_{ϵ\left(n\right)}-l^{★}{\parallel }^2\le d_{ϵ\left(n\right)},\end{array}$$

Thus, $lim{sup}_{n\to \infty }\parallel v_{ϵ\left(n\right)}-l^{★}\parallel \le 0$. By passing limit $n\to \infty$ and using Lemma 4,

$$\begin{array}{c}\hfill 0\le \parallel v_n-l^{★}\parallel \le max\{\parallel v_n-l^{★}\parallel ,\parallel v_{ϵ\left(n\right)}-l^{★}\parallel \}\le \parallel v_{ϵ\left(n\right)+1}-l^{★}\parallel .\end{array}$$

It follows that $\parallel v_n-l^{★}\parallel \to 0$, that is, $v_n\to l^{★}$ as $n\to \infty$. This completes the proof. □

Further, we construct the hybrid Algorithm 2, which is a slightly modified version of hybrid Algorithm 1. In hybrid Algorithm 2, the initial step iterates the viscosity approximation, which is the convex combination of $F\left(v_n\right)$ and $Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})$, where the inertial extrapolation term ${\tau }_n(v_n-v_{n-1})$ is added to accelerate the convergence.

Algorithm 2. Hybrid Algorithm 2

Choose $\lambda >0,\mu >0$, $\tau \in [0,1)$, and $0<{\varsigma }_n<2$. Select initial points $v_0$ and $v_1$ and fix $n=0$.

Iterative Step: For $n\ge 1$, iterate $v_n$,  $v_{n-1}$ and select $0<{\tau }_n\le {\overline{\tau }}_n$, where  $$\begin{array}{c}\hfill {\overline{\tau }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{{\varphi }_n}{\parallel v_n-v_{n-1}\parallel }},\tau \right\},\mathrm{if}{v}_n\ne v_{n-1},\hfill \\ \tau ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$ (42)

Compute  $$\begin{array}{ccc}\hfill s_n& =& {\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)[Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})],\hfill \end{array}$$ (43) $$\begin{array}{ccc}\hfill u_n& =& s_n-{\lambda }_n(I-J_{\lambda }^M)\left(s_n\right),\hfill \end{array}$$ (44) $$\begin{array}{ccc}\hfill v_{n+1}& =& u_n-{\mu }_n{B}^{*}(I-J_{\mu }^M)\left(B{u}_n\right),\hfill \end{array}$$ (45) where ${\mu }_n$ and ${\lambda }_n$ are defined by $$\begin{array}{c}\hfill {\lambda }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\lambda }^M)s_n\parallel }{\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$ (46)   and  $$\begin{array}{c}\hfill {\mu }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\mu }^N)B{u}_n\parallel }{\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$ (47)

If $v_{n+1}=u_n=v_n=s_n$, then stop; otherwise, fix $n=n+1$ and go back to the computation.

The following is the convergence analysis of hybrid Algorithm 2, which is similar to that of the proof of Theorem 1.

Theorem 2.

If assumptions $\left(X_1\right)-\left(X_5\right)$ hold, then the sequence $\left\{v_n\right\}$ generated by Algorithm 2 converges strongly to v, where $v=P_{{\Delta }_1\cap {\Delta }_2}F\left(v\right)$.

Proof. 

Take $l^{★}\in {\Delta }_1\cap {\Delta }_2$; then, from (43) and using Remark 4, we see that

$$\begin{array}{ccc}\hfill \parallel s_n-l^{★}\parallel & =& \parallel {\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)[Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})]-l^{★}\parallel \hfill \\ & =& {\zeta }_n\parallel F\left(v_n\right)-l^{★}\parallel +(1-{\zeta }_n)[\parallel Z\left(v_n\right)-l^{★}\parallel +{\tau }_n\parallel v_n-v_{n-1}\parallel ]\hfill \\ & \le & {\zeta }_n\parallel F\left(v_n\right)-F\left(l^{★}\right)\parallel +{\zeta }_n\parallel F\left(l^{★}\right)-l^{★}\parallel +(1-{\zeta }_n)\parallel Z\left(v_n\right)-l^{★}\parallel +{\tau }_n\parallel v_n-v_{n-1}\parallel \hfill \\ & \le & {\zeta }_n\theta \parallel v_n-l^{★}\parallel +{\zeta }_n\parallel F\left(l^{★}\right)-l^{★}\parallel +(1-{\zeta }_n)\parallel v_n-l^{★}\parallel +{\zeta }_n{L}_1\hfill \\ & =& [1-{\zeta }_n(1-\theta )]\parallel v_n-l^{★}\parallel +{\zeta }_n(1-\theta ){\displaystyle \frac{\parallel F\left(l^{★}\right)-l^{★}\parallel +L_1}{1-\theta }}\hfill \\ & \le & max\left\{\parallel v_n-l^{★}\parallel ,{\displaystyle \frac{\parallel F\left(l^{★}\right)-l^{★}\parallel +L_1}{1-\theta }}\right\}.\hfill \end{array}$$

Keeping in mind (24) and using mathematical induction, we obtain that the sequence $\left\{s_n\right\}$ is bounded, and so is $\left\{v_n\right\}$ and $\left\{u_n\right\}$. Denote $y_n=Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})$; then, by using (8) and denoting $t_n={\tau }_n\parallel v_n-v_{n-1}\parallel \le {\varphi }_n$, we establish that

$$\begin{array}{ccc}\hfill \parallel y_n-l^{★}{\parallel }^2& =& \parallel Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})-l^{★}\parallel \hfill \\ & \le & \parallel Z\left(v_n\right)-l^{★}{\parallel }^2+2{\tau }_n\langle v_n-v_{n-1},y_n-l^{★}\rangle \hfill \\ & \le & \parallel v_n-l^{★}{\parallel }^2+2t_n\parallel y_n-l^{★}\parallel ,\hfill \end{array}$$

(48)

and

$$\begin{array}{ccc}\hfill \langle F\left(v_n\right)-l^{★},y_n-l^{★}\rangle & =& \langle F\left(v_n\right)-F\left(l^{★}\right),y_n-l^{★}\rangle +\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle \hfill \\ & \le & \parallel F\left(v_n\right)-F\left(l^{★}\right)\parallel \parallel y_n-l^{★}\parallel +\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle \hfill \\ & \le & \theta \parallel v_n-l^{★}\parallel \parallel y_n-l^{★}\parallel +\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle \hfill \\ & =& {\displaystyle \frac{1}{2}}\left({\theta }^2\parallel v_n-l^{★}{\parallel }^2+{\parallel y_n-l^{★}\parallel }^2\right)+\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle .\hfill \end{array}$$

(49)

Using (48) and (49), we obtain

$$\begin{array}{ccc}\hfill & & \parallel s_n-l^{★}\parallel \hfill \\ & =& \parallel {\zeta }_{n}F\left(v_n\right)+(1-{\zeta }_n)y_n-l^{★}\parallel \hfill \\ & =& {\zeta }^2\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{(1-{\zeta }_n)}^2\parallel y_n-l^{★}{\parallel }^2+2{\zeta }_n(1-{\zeta }_n)F\left(v_n\right)-l^{★},y_n-l^{★}\rangle \hfill \\ & \le & {\zeta }_n^2\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{\zeta }_n(1-{\zeta }_n)\{{\theta }^2\parallel v_n-l^{★}{\parallel }^2+{\parallel y_n-l^{★}\parallel }^2\hfill \\ & & +2\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle \}+{(1-{\zeta }_n)}^2{\parallel y_n=l^{★}\parallel }^2\hfill \\ & \le & {\zeta }_n^2\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{\zeta }_n{\theta }^2\parallel v_n-l^{★}{\parallel }^2+[{(1-{\zeta }_n)}^2+{\zeta }_n(1-{\zeta }_n)]{\parallel y_n-l^{★}\parallel }^2\hfill \\ & & +2{\zeta }_n(1-{\zeta }_n\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle \hfill \\ & \le & {\zeta }_n^2\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{\zeta }_n{\theta }^2\parallel v_n-l^{★}{\parallel }^2+(1-{\zeta }_n)[\parallel v_n-l^{★}{\parallel }^2+2t_n\parallel y_n-l^{★}\parallel ]\hfill \\ & & +2{\zeta }_n(1-{\zeta }_n\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle \hfill \\ & \le & [1-{\zeta }_n(1-{\theta }^2)]{\parallel v_n-l^{★}\parallel }^2+{\zeta }_n\{2(1-{\zeta }_n)\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle \hfill \\ & & +{\zeta }_n\parallel F\left(v_n\right)-l^{★}{\parallel }^2+{\displaystyle \frac{2t_n}{{\zeta }_n}}\parallel y_n-l^{★}\parallel \}.\hfill \end{array}$$

(50)

Combining (23) and (50), we obtain

$$\begin{array}{ccc}\hfill \parallel v_{n+1}-l^{★}{\parallel }^2& \le & (1-c_n){\parallel v_n-l^{★}\parallel }^2+c_n{g}_n+{\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{s}_n\right)\parallel }}\hfill \\ & & +{\displaystyle \frac{{\varsigma }_n\left({\varsigma }_n-2\right){\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel }^3}{\parallel (I-J_{\lambda }^M)\left(u_n\right)\parallel +\parallel B^{*}(I-J_{\mu }^N)\left(B{u}_n\right)\parallel }},\hfill \end{array}$$

where $c_n={\zeta }_n(1-{\theta }^2)$ and $g_n={\displaystyle \frac{2(1-{\zeta }_n)\langle F\left(l^{★}\right)-l^{★},y_n-l^{★}\rangle +{\zeta }_n\parallel f\left(v_n\right)-l^{★}{\parallel }^2+\frac{2{\varphi }_n}{{\zeta }_n}\parallel y_n-l^{★}\parallel }{(1-{\theta }^2)}}$.

Considering Case I of Theorem 1, we can easily obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel (I-J_{\lambda }^M)\left(s_n\right)\parallel =\underset{n\to \infty }{lim}\parallel (I-J_{\mu }^N)\left(B{u}_n\right)\parallel =0,\end{array}$$

(51)

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel u_n-s_n\parallel =0,\underset{n\to \infty }{lim}\parallel v_{n+1}-u_n\parallel =0,\end{array}$$

(52)

and, hence,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel v_{n+1}-s_n\parallel =0,\underset{n\to \infty }{lim}\parallel v_{n+1}-v_n\parallel =0.\end{array}$$

(53)

Next, we show that $\parallel Z{v}_n-v_n\parallel \to 0$ as $n\to \infty$. As $y_n=Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})$, then

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\parallel y_n-Z\left(v_n\right)\parallel =0=0,\end{array}$$

and

$$\begin{array}{ccc}\hfill s_n-Z\left(v_n\right)& =& {\zeta }_{n}F\left(v_n\right)-(1-{\zeta }_n)y_n-Z\left(v_n\right)\hfill \\ & =& {\zeta }_n(F\left(v_n\right)-l^{★})+{\zeta }_n(l^{★}-y_n)+(y_n-Z\left(v_n\right)),\hfill \end{array}$$

The assumption on ${\zeta }_n$ and the boundedness of $v_n$ imply that

$$\begin{array}{c}\hfill \parallel s_n-Z\left(v_n\right)\parallel \le {\zeta }_n\parallel F\left(v_n\right)-l^{★}\parallel +{\zeta }_n\parallel l^{★}-y_n\parallel +\parallel y_n-Z\left(v_n\right)\parallel \to \mathrm{as}n\to \infty .\end{array}$$

Also,

$$\begin{array}{c}\hfill \parallel Z\left(v_n\right)-u_n\parallel \le \parallel Z\left(v_n\right)-s_n\parallel +\parallel u_n-s_n\parallel \to \mathrm{as}n\to \infty .\end{array}$$

(54)

Together with (52)–(54), we obtain by taking $n\to \infty$

$$\begin{array}{c}\hfill \parallel Z\left(v_n\right)-v_n\parallel \le \parallel Z\left(v_n\right)-u_n\parallel +\parallel u_n-s_n\parallel +\parallel v_{n+1}-s_n\parallel +\parallel v_{n+1}-v_n\parallel \to 0.\end{array}$$

(55)

The boundedness of $\left\{v_n\right\}$, $\left\{u_n\right\}$, and $\left\{s_n\right\}$ imply the existence of subsequence $\left\{v_{n_k}\right\}$, $\left\{u_{n_k}\right\}$, and $\left\{s_{n_k}\right\}$, which converge to some point $\widehat{v}$; and from (51)–(53) and (55), we conclude that $\widehat{v}\in {\Delta }_1\cap {\Delta }_2$. The remaining proof can be obtained easily by using similar steps as in the proof of Theorem 1. □

Let $q\in \mathbb{U}$ be arbitrary. Then, by replacing $F\left(v\right)$ with q in hybrid Algorithm 1 and hybrid Algorithm 2, we define the following Halpern-type iterative methods, which can be seen as the particular cases of our hybrid methods:

Corollary 1.

If assumptions $\left({\mathrm{X}}_2\right)$–$\left({\mathrm{X}}_5\right)$ hold good, then the sequence $\left\{v_n\right\}$ induced by Algorithm 3 converges strongly to $q^{★}={\mathrm{P}}_{{\Delta }_1\cap {\Delta }_2}\left(q\right)$.

Algorithm 3. A Particular Case of Hybrid Algorithm 1

Choose $\lambda >0,\mu >0$, $\tau \in [0,1)$, and $0<{\varsigma }_n<2$. Select initial points $v_0$ and $v_1$, any $q\in \mathbb{U}$, and fix $n=0$.

Iterative Step: For $n\ge 1$, iterate $v_n$, $v_{n-1}$, and select $0<{\tau }_n\le {\overline{\tau }}_n$, where $$\begin{array}{c}\hfill {\overline{\tau }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{{\varphi }_n}{\parallel v_n-v_{n-1}\parallel }},\tau \right\},\mathrm{if}{v}_n\ne v_{n-1},\hfill \\ \tau ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

Compute  $$\begin{array}{ccc}\hfill s_n& =& {\zeta }_{n}q+(1-{\zeta }_n)Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1}),\hfill \\ \hfill u_n& =& s_n-{\lambda }_n(I-J_{\lambda }^M)\left(s_n\right),\hfill \\ \hfill v_{n+1}& =& u_n-{\mu }_n{B}^{*}(I-J_{\mu }^N)\left(B{u}_n\right),\hfill \end{array}$$ where${\mu }_n$ and ${\lambda }_n$ are defined by $$\begin{array}{c}\hfill {\lambda }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\lambda }^M)s_n\parallel }{\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$ and $$\begin{array}{c}\hfill {\mu }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\mu }^N)B{u}_n\parallel }{\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

If $v_{n+1}=u_n=v_n=s_n$, then stop; otherwise, fix $n=n+1$ and go back to the computation.

Proof. 

Replacing $F\left(v\right)=q$ in Algorithm 1 as well as in the proof of Theorem 1, we obtain the required result. □

Corollary 2.

If assumptions $\left({\mathrm{X}}_2\right)$–$\left({\mathrm{X}}_5\right)$ hold good, then the sequence $\left\{v_n\right\}$ induced by Algorithm 4 converges strongly to $q^{★}={\mathrm{P}}_{{\Delta }_1\cap {\Delta }_2}\left(q\right)$.

Algorithm 4. A Particular Case of Hybrid Algorithm 2

Let $\lambda >0,\mu >0$, $\tau \in [0,1)$, and $0<{\varsigma }_n<2$ be given. Select initial points $v_0$ and $v_1$, any $q\in \mathbb{U}$, and fix $n=0$.

Iterative Step: For $n\ge 1$, iterate $v_n$, $v_{n-1}$, and select $0<{\tau }_n\le {\overline{\tau }}_n$, where $$\begin{array}{c}\hfill {\overline{\tau }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{{\varphi }_n}{\parallel v_n-v_{n-1}\parallel }},\tau \right\},\mathrm{if}{v}_n\ne v_{n-1},\hfill \\ \tau ,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

Compute $$\begin{array}{ccc}\hfill s_n& =& {\zeta }_{n}q+(1-{\zeta }_n)[Z\left(v_n\right)+{\tau }_n(v_n-v_{n-1})],\hfill \\ \hfill u_n& =& s_n-{\lambda }_n(I-J_{\lambda }^M)\left(s_n\right),\hfill \\ \hfill v_{n+1}& =& u_n-{\mu }_n{B}^{*}(I-J_{\mu }^N)\left(B{u}_n\right),\hfill \end{array}$$ where ${\mu }_n$ and ${\lambda }_n$ are defined by $$\begin{array}{c}\hfill {\lambda }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\lambda }^M)s_n\parallel }{\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)s_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{s}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$   and  $$\begin{array}{c}\hfill {\mu }_n=\left\{\begin{array}{c}{\displaystyle \frac{{\varsigma }_n\parallel (I-J_{\mu }^N)B{u}_n\parallel }{\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel }},\mathrm{if}\parallel (I-J_{\lambda }^M)u_n\parallel +\parallel B^{*}(I-J_{\mu }^N)B{u}_n\parallel \ne 0,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

If $v_{n+1}=u_n=v_n=s_n$, then stop; otherwise, fix $n=n+1$ and go back to the computation.

Proof. 

By replacing q in place of $F\left(v\right)$ in Algorithm 2 as well as in the proof of Theorem 2, we obtain the desired proof. □

4. Some Advantages

Some applications of the suggested methods for solving split variational inequality and split common fixed-point problems are discussed below.

4.1. Split Variational Inequality Problem

Let $\mathbb{D}$ be a nonempty, closed, and convex subset of $\mathbb{U}$ and $P_{\mathbb{D}}$ be the projection on $\mathbb{D}$, $G_1:\mathbb{U}\to \mathbb{U}$, and $G_2:\mathbb{U}\to \mathbb{U}$, which are monotone operators, and $B:\mathbb{U}\to \mathbb{U}$ is a bounded linear operator, then the split variational inequality problem (S^plitVI_tP) is defined by

$$\begin{array}{c}\hfill \mathrm{find}{v}^{★}\in \mathbb{U}\mathrm{so}\mathrm{that}\langle G_1\left(v^{★}\right),v-v^{★},\rangle \ge 0,\forall v\in \mathbb{D},\\ \hfill \mathrm{and}{u}^{★}=B{l}^{★}\in \mathbb{U}\mathrm{solves}\langle G_2\left(u^{★}\right),v-u^{★},\rangle \ge 0,\forall u\in \mathbb{D}.\end{array}$$

Then, by replacing $J_{\lambda }^M=J_{\mu }^N=P_{\mathbb{D}}$ in Algorithms 1–4, we can obtain the hybrid algorithms and their convergence results for (S^plitVI_tP) and (FPP).

4.2. Split Common Fixed-Point Problem

Let $T_1:\mathbb{U}\to \mathbb{U}$ and $T_2:\mathbb{U}\to \mathbb{U}$ be self-nonexpansive mappings and $B:\mathbb{U}\to \mathbb{U}$ be a bounded linear operator; then, the split common fixed-point problem (S^plitCFPP) is defined as follows:

$$\begin{array}{c}\hfill \mathrm{find}{v}^{★}\in \mathbb{U}\mathrm{so}\mathrm{that}{v}^{★}\in Fix\left(T_1\right),\\ \hfill \mathrm{and}{u}^{★}=B{v}^{★}\in \mathbb{U}\mathrm{solves}{u}^{★}\in Fix\left(T_2\right).\end{array}$$

Then, by replacing $J_{\lambda }^M=T_1$, $J_{\mu }^N=T_2$, and $Z=I$, identity mapping in Algorithms 1–4, we can obtain the hybrid algorithms and their convergence results for (S^plitCFPP).

Next, we present numerical examples in finite and infinite dimensional Hilbert spaces, showing the efficiency of our hybrid methods and their comparison with the work studied in [10,11,12,18].

5. Numerical Examples

Example 1

(Finite dimensional). Let $\mathbb{U}={\mathbb{R}}^4$, equipped with the inner product $〈t,q〉=q_1{t}_1+q_2{t}_2+q_3{t}_3+q_4{t}_4$ for $t=(t_1,t_2,t_3,t_4)$ and $q=(q_1,q_2,q_3,q_4)$ and the norm ${\parallel q\parallel }^2=|q_1{|}^2+|q_2{|}^2+|q_3{|}^2+{\left|q_4\right|}^2$. The operators M, N, and B are defined by

$$\begin{array}{ccc}\hfill M(q_1,q_2,q_3,q_4)& =& \left(q_1,{\displaystyle \frac{q_2}{2}},{\displaystyle \frac{q_3}{3}},{\displaystyle \frac{q_4}{4}}\right),\hfill \\ \hfill N(q_1,q_2,q_3,q_4)& =& \left(q_1,{\displaystyle \frac{q_2}{3}},q_3,{\displaystyle \frac{q_4}{3}}\right),\hfill \\ \hfill B(q_1,q_2,q_3,q_4)& =& \left(q_1,q_2,q_3,q_4\right),\hfill \end{array}$$

• such that M is ${\displaystyle \frac{1}{4}}$-inverse strongly monotone and N is ${\displaystyle \frac{1}{3}}$-inverse strongly monotone (hence monotone); B is a bounded and linear operator. The nonexpansive mapping Z is defined by $Z(q_1,q_2,q_3,q_4)=\left(q_1,0,q_3,0\right)$, and $F(q_1,q_2,q_3,q_4)=\left({\displaystyle \frac{q_1}{2}},{\displaystyle \frac{q_2}{2}},{\displaystyle \frac{q_3}{2}},{\displaystyle \frac{q_4}{2}}\right)$ is a θ-contraction with $\theta ={\displaystyle \frac{1}{2}}$.

To run our algorithms, we select ${\varphi }_n={\displaystyle \frac{1}{n^2+100}}$, ${\varsigma }_n=2-{\displaystyle \frac{1}{n+1}}$, and $\tau =0.75$, and ${\tau }_n$ is selected randomly from $(0,\overline{{\tau }_n})$, where

$$\begin{array}{c}\hfill {\overline{\tau }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{1}{(n^2+100)\parallel v_n-v_{n-1}\parallel }},0.75\right\},\mathrm{if}{v}_n\ne v_{n-1},\hfill \\ 0.75,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

We compare our algorithms and method (2), method (3), method (4), and method (5) by using the following common parameters: $\lambda =\mu ={\mu }_1={\mu }_2=0.25$, ${\zeta }_n={\displaystyle \frac{1}{\sqrt{n+100}}}$ for all the methods; $\eta =0.5$ for method (2), method (3), and method (4); $u=(0,0,0,1)$ and ${\eta }_n$ given by (6) are used in method (5). The stopping condition is $\parallel v_{n+1}-v_n\parallel <{10}^{-20}$, and we consider the two cases of initial values:

• Case (a): $v_0=(-130,-10,54,95),v_1=(14,30,-105,100)$
• Case (b): $v_0=(0,-100,100,0),v_1=(1/2,0,-1/100,5)$

It can be seen that our algorithms are efficient and effective and can be implemented easily without calculating $\parallel B\parallel$. The convergence of $\left\{v_n\right\}$ to $\left\{0\right\}={\Delta }_1\cap {\Delta }_2$ is shown in Figure 1 and Figure 2 using different initial values. It is found that our algorithms approach the solution in fewer numbers of steps in comparison to the method (2) of [10], method (3) of [11], method (4) of [12], and method (5) of [18].

Example 2. (Infinite dimensional) Let $\mathbb{U}=l_2:=\left\{t:=(t_1,t_2,t_3,\dots ,t_n,\dots ),t_n\in \mathbb{R}:{\displaystyle \sum _{n=1}^{\infty }}{\left|t_n\right|}^2<\infty \right\}$, the space of all square summable sequences with inner product $\langle t,q\rangle ={\displaystyle \sum _{n=1}^{\infty }}{t}_n{q}_n$, and the norm is $\parallel t\parallel ={\left({\displaystyle \sum _{n=1}^{\infty }}{\left|t_n\right|}^2\right)}^{1/2}$. The mappings M, N, and S are defined by

$$\begin{array}{c}\hfill M\left(t\right):={\displaystyle \frac{t}{5}}N\left(t\right):={\displaystyle \frac{t}{3}}Z\left(t\right):=t,F\left(t\right):={\displaystyle \frac{t}{2}},B\left(t\right):=t,\forall t\in l_2.\end{array}$$

Clearly, M and N are monotone, Z is nonexpansive, F is a contraction, and B is a bounded linear operator. We choose ${\varsigma }_n=2-{\displaystyle \frac{2}{n+10}}$, ${\varphi }_n={\displaystyle \frac{1}{n^2+1}}$, and $\tau =0.85$, and ${\tau }_n$ is selected randomly from $(0,\overline{{\tau }_n})$, where

$$\begin{array}{c}\hfill {\overline{\tau }}_n=\left\{\begin{array}{c}min\left\{{\displaystyle \frac{1}{(n^2+100)\parallel v_n-v_{n-1}\parallel }},0.85\right\},\mathrm{if}{v}_n\ne v_{n-1},\hfill \\ 0.85,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

The common parameters for our algorithms and method (2), method (3), method (4), and method (5) are as follows: $\lambda =\mu ={\mu }_1={\mu }_2={\displaystyle \frac{1}{3}}$, ${\zeta }_n={\displaystyle \frac{1}{\sqrt{n+1}}}$ for all the methods; $\eta =0.5$ for method (2), method (3), and method (4); $u=(0,1,0,0,0,\dots )$ and ${\eta }_n$ given by (6) is used in method (5). We plot the convergence of the sequences induced by Algorithms 1–4. The stopping condition is $\parallel v_{n+1}-v_n\parallel <{10}^{-25}$ for the following two initial values:

• Case (a’): $v_0={\left\{{\displaystyle \frac{1}{3^n}}\right\}}_{n=1}^{\infty },v_1={\left\{{\displaystyle \frac{1}{n}}\right\}}_{n=1}^{\infty };$
• Case (b’): $v_0={\left\{{\displaystyle \frac{1}{n^2}}\right\}}_{n=1}^{\infty },v_1={\left\{{\displaystyle \frac{{(-1)}^n}{n^2}}\right\}}_{n=1}^{\infty };$

Our algorithms are effective and efficient in the sense that they are implemented easily without calculating $\parallel B\parallel$. It can be seen in Figure 3 and Figure 4 that the sequences obtained from our methods estimate the solution in fewer numbers of steps as compared to method (2) of [10], method (3) of [11], method (4) of [12], and method (5) of [18].

Remark 5.

We do not present any result on the rate of convergence of the proposed methods. In the future, it will be interesting to study and compare the convergence rate of our proposed methods and other techniques.

6. Conclusions

We present two hybrid inertial self-adaptive iterative methods for estimating the common solution of ($\mathrm{FPP}$) and (S^plitVIP). Two strong convergence theorems are established. Some special cases of the proposed methods are noted. We also implement our hybrid methods to explore the solution of split variational inequality problems and split common fixed-point problems. Our algorithms are simple and different in the sense that they estimate the viscosity approximation, fixed-point iteration, and inertial extrapolation in the initial steps of each iteration. Our methods are also efficient; they involve two self-adaptive step sizes and do not require the pre-estimated norm of a bounded linear operator in the iteration process. The effectiveness and efficiency of the proposed methods are illustrated by numerical examples, Examples 1 and 2. In the study of the numerical examples, it is observed that the presented methods are effective and easily implemented without any hurdle. The iterative sequence obtained by our methods estimates the common solution of (S^plitVIP) and ($\mathrm{FPP}$) in fewer numbers of steps in comparison to method (2) of [10], method (3) of [11], method (4) of [12], and method (5) of [18].