The Split Various Variational Inequalities Problems for Three Hilbert Spaces

Abstract

There are many methods for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. They proved the strong convergence theorem. Many split feasibility problems are generated in real Hillbert spaces. The open problem is proving a strong convergence theorem of three Hilbert spaces with different methods from the lasted method. In this research, a new split variational inequality in three Hilbert spaces is proposed. Important tools which are used to solve classical problems will be developed. The convergence theorem for finding a common element of the set of solution of such problems and the sets of fixed-points of discontinuous mappings has been proved.

1. Introduction

We use the following symbols throughout this paper: let H be a real Hilbert space and C be a nonempty closed convex subset of H. We also use the symbols $“⟶”$ and $“\rightharpoonup ”$, which represent strong and weak convergence, respectively. The variational inequality problem (VIP) is a well known problem. That is to find a point ${\varpi }_{*}$ such that

$$\langle y-{\varpi }_{*},G{\varpi }_{*}\rangle \ge 0,\mathrm{for}\mathrm{all}y\in C,$$

(1)

where $G:C⟶H$ is a mapping. The set of all solutions of (1) is denoted by $Var(C,G)$.

The variational inequality problem has been applied in various fields such as industry, finance, economics, social, ecology, regional, pure and applied sciences; see [1,2,3]. For every $i=1,2$, let $H_i$ be a real Hilbert space and $C,Q$ be nonempty closed convex subset of $H_1$, and $H_2$, respectively. Recently, Censor [4] has introduced a new variational problem called the split inequality problem (SIP). It entails finding a solution of one variational inequality problem (VIP), the image of which, under a given bounded linear transformation, is a solution of another VIP.

The split variational inequality problem is assigned to the following formula; find a point ${\varpi }_{*}\in C$ such that

$$\begin{array}{c}\hfill \langle f\left({\varpi }_{*}\right),x-{\varpi }_{*}\rangle \ge 0,\mathrm{for}\mathrm{all}x\in C\end{array}$$

(2)

and a point $y^{*}=A{\varpi }_{*}$ solves

$$\begin{array}{c}\hfill \langle y-y^{*},g\left(y^{*}\right)\rangle \ge 0,\mathrm{for}\mathrm{all}y\in Q,\end{array}$$

(3)

where $A:H_1\to H_2$ is a bounded linear operator and $f:H_1\to H_1,g:H_2\to H_2$ are mappings. The set of all solutions of (2) and (3) is denoted by

$$\begin{array}{c}\hfill \Omega =\left\{x\in Var\left(C,f\right):\mathrm{for}\mathrm{all}x\in Var\left(Q,g\right)\right\}.\end{array}$$

The split variational inequality problem can be applied to model in intensity-modulated radiation therapy (IMRT) treatment planning.

There are also a lot of authors who have introduced convergence theorem related to the split variational inequality and fixed point problems; see [5,6,7] for an example. In [8], they have studied the Mann implicit iterations for strongly accreative and strongly pseudo-contractive mappings and found that this implicit scheme gives a better convergence rate estimate.

The following definitions are important tools used in this research. A mapping $\vartheta$ of H into itself is called nonexpansive if $\parallel \vartheta x-\vartheta y\parallel \le \parallel x-y\parallel$, for all $x,y\in H.$ We denote by $F\left(\vartheta \right)$ the set of fixed points of $\vartheta$ (i.e., $F\left(\vartheta \right)=\{x\in C:\vartheta x=x\}$). A nonexpansive mapping $\vartheta$ is equivalent to the following inequality;

$$\langle \left(I-\vartheta \right)x-\left(I-\vartheta \right)y,\vartheta y-\vartheta x\rangle \le {\displaystyle \frac{1}{2}}{\parallel \left(I-\vartheta \right)x-\left(I-\vartheta \right)y\parallel }^2,$$

for all $x,y\in H$. From the equation above if $y\in F\left(\vartheta \right)$ and $x\in H,$ we can conclude that;

$$\langle \left(I-\vartheta \right)x,y-\vartheta x\rangle \le {\displaystyle \frac{1}{2}}{∥\left(I-\vartheta \right)x∥}^2.$$

A mapping A of C into H is called inverse strongly monotonic, if there exists $\alpha >0$ such that

$$\langle x-y,Ax-Ay\rangle \ge {\alpha \parallel Ax-Ay\parallel }^2,$$

for all $x,y\in C$. In [9], Kohsaka and Takahashi introduced the nonspreading mapping in Hilbert spaces H which is defined by the following inequality ${2\parallel \vartheta x-\vartheta y\parallel }^2\le {\parallel \vartheta x-y\parallel }^2+{\parallel x-\vartheta y\parallel }^2,$ for all $x,y\in C$.

Following the terminology of Browder and Petryshyn [10], in [11], Osilike and Isiogugu introduced the mapping $\vartheta :C\to C$, which is called $\kappa$—strictly pseudo-nonspreading mapping if there exists $\kappa \in [0,1)$ such that

$${\parallel \vartheta x-\vartheta y\parallel }^2\le {\parallel x-y\parallel }^2+\kappa {\parallel \left(I-\vartheta \right)x-\left(I-\vartheta \right)y\parallel }^2+2\langle x-\vartheta x,y-\vartheta y\rangle ,$$

for all $x,y\in C$. Clearly every nonspreading mapping is $\kappa$-strictly pseudo-nonspreading; see, for example, [11].

In [12], Bnouhachem modified a projection process for finding a common solution of a system of variational inequalities, a split equilibrium problem and a hierarchical fixed-point problem in the setting of real Hilbert spaces and also proved the strong convergence theorem of the sequence $\left\{x_n\right\}$ generated by

$$\left\{\begin{array}{c}\begin{array}{c}{u}_n={\vartheta }_{r_n}^{F_1}\left(x_n+\gamma A^{*}\left(T_{r_n}^{F_2}-I\right)A{x}_n\right);\hfill \\ z_n=P_C\left[P_C\left[u_n-{\alpha }_1{B}_2{u}_n\right]-{\alpha }_1{B}_1{P}_C\left[u_n-{\alpha }_2{B}_2{u}_n\right]\right];\hfill \\ y_n={\beta }_n\varrho x_n+\left(1-{\beta }_n\right)z_n;\hfill \\ x_{n+1}=P_C\left[{\alpha }_n\rho U\left(x_n\right)+\left(I-{\alpha }_n\mu F\right)\vartheta \left(y_n\right))\right],\mathrm{for} \mathrm{all} n\ge 0,\hfill \end{array}\hfill \end{array}\right.$$

where $A:H_1\to H_2$ is a bounded linear operator. Assume that $F_1:C\times C\to \mathbb{R}$ and $F_2:Q\times Q\to \mathbb{R}$ are the bifunctions; $B_i:C\to H$ is a ${\theta }_i$-inverse strongly monotonic mapping for each $i=1,2$ and $S,\vartheta :C\to C$ nonexpansive mapping; $F:C\to C$ is a k-Lipschitzian mapping and is $\eta$-strongly monotonic; $U:C\to C$ is a $\tau$-Lipschitzian mapping; and the positive parameters are $r_n,{\alpha }_n,{\alpha }_1,{\alpha }_2,\rho ,\mu$, for all $n\in \mathbb{N}$.

Let C and Q be nonempty closed convex subsets of the real Hilbert spaces $H_1$ and $H_2$, respectively. The split feasibility problem (SFP) is formulated as:

$$\mathrm{to}\mathrm{find}{x}^{*}\in C\mathrm{such}\mathrm{that}A{x}^{*}\in Q,$$

(4)

where $A:H_1\to H_2$ is a bounded linear operator. The SFP are also applied in [13,14].

Recently, Moudafi [15] introduced the following new split feasibility problem, which is also called general split equality problem:

Let $H_1,H_2,H_3$ be real Hilbert spaces, $C\subset H_1,Q\subset H_2$ be two nonempty closed convex sets and $A:H_1\to H_3,B:H_2\to H_3$ be two bounded linear operators. Moudafi studied the convergence of a relaxed alternating CQ-algorithm for solving the new split feasibility problem, aiming to find

$$\begin{array}{c}\hfill {\varpi }_{*}\in C,y^{*}\in Q\mathrm{such}\mathrm{that}A{\varpi }_{*}=B{y}^{*}.\end{array}$$

(5)

In order to prove the weak convergence theory to solve general split equality problem (5), Moudafi defined the following iteration process $\left\{x_k\right\}$:

$$\left\{\begin{array}{c}\begin{array}{c}{x}_{k+1}=P_C\left(x_k-{\gamma }_k{A}^{*}\left(A{x}_k-B{y}_k\right)\right),\hfill \\ y_{k+1}=P_Q\left(y_k+{\gamma }_k{B}^{*}\left(A{x}_{k+1}-B{y}_k\right)\right),\hfill \end{array}\hfill \end{array}\right.$$

where $A^{*},B^{*}$ are adjoint operators of $A,B$ respectively, proper conditions of the positive paramiter ${\gamma }_k$, for all $k\ge 1$. In order to avoid using the projection, Moudafi [16] introduced and studied the following problem: Let $T:H_1\to H_1$ and $S:H_2\to H_2$ be nonlinear operators such that Fix$\left(T\right)$ ≠ Ø and Fix$\left(S\right)$ ≠ Ø, where Fix$\left(T\right)$ and Fix$\left(S\right)$ denote the sets of fixed points of T and S, respectively. If C = Fix$\left(T\right)$ and Q = Fix$\left(S\right)$; then the split equality problem reduces to

$$\mathrm{to}\mathrm{find}x\in \mathrm{Fix}\left(T\right)\mathrm{and}y\in \mathrm{Fix}\left(S\right)\mathrm{such}\mathrm{that}Ax=By,$$

(6)

which is called a split equality fixed point problem (SEFPP).

Denote by $\Gamma$ the solution set of split equality fixed point problem (6). There were recently SEFPP research articles in [17,18].

Question A. Can we prove a strong convergence theorem of three Hilbert spaces by different methods from Moudafi [15]?

For every $i=1,2,3$, let $H_i$ be a real Hilbert space and $C_i$ be a nonempty closed convex subset of $H_i$. Let $B_i:C_i\to H_i$ be a mapping, for all $i=1,2,3$, and let $A_2:H_1\to H_2$ and $A_3:H_2\to H_3$. The split various variational inequality is to find the points

$$\left\{\begin{array}{c}\begin{array}{c}{\varpi }_1^{*}\in C_1,\mathrm{such} \mathrm{that}\langle B_1{\varpi }_1^{*},x_1-{\varpi }_1^{*}\rangle \ge 0,\mathrm{for} \mathrm{all} x_1\in C_1,\mathrm{and}\hfill \\ {\varpi }_2^{*}=A_2{\varpi }_1^{*}\in C_2,\mathrm{such} \mathrm{that}\langle B_2{\varpi }_2^{*},x_2-{\varpi }_2^{*}\rangle \ge 0,\mathrm{for} \mathrm{all} x_2\in C_2,\mathrm{and}\hfill \\ {\varpi }_3^{*}=A_3{\varpi }_2^{*}\in C_3,\mathrm{such} \mathrm{that}\langle B_3{\varpi }_3^{*},x_3-{\varpi }_3^{*}\rangle \ge 0,\mathrm{for} \mathrm{all} x_3\in C_3.\hfill \end{array}\hfill \end{array}\right.$$

(7)

The set of the solutions of (7) is denoted by $\Omega =\{{\varpi }^{*}=\left({\varpi }_1^{*},{\varpi }_2^{*},{\varpi }_3^{*}\right)\in C_1\times C_2\times C_3:{\varpi }_i^{*}\in Var\left(C_i,B_i\right),$ for all $i=1,2,3\}$.

To answer question A, we have created a new tool to prove a strong convergence theorem for three Hilbert spaces to be used for finding the solution of the problem (7) and the fixed points problem of nonspreading and pseudo-nonspreading mappings. Preliminaries In this section, we collect some definitions and lemmas in Hilbert space, which will be needed for proving our main results. More properties of Hilbert space can be found in [19].

Definition 1.

The (nearest point) projection $P_C$ from H onto C assigns to each $x\in H$ the unique point $P_{C}x\in C$ satisfying the property

$$\parallel x-P_{C}x\parallel =\underset{y\in C}{min}\parallel x-y\parallel .$$

Lemmas 1 and 2 are properties of $P_C$.

Lemma 1.

([20]) For a given $x\in H$ and $y\in C$, $P_{C}x=y$ if and only if there holds the inequality $\langle y-x,z-y\rangle \ge 0,$ for all $z\in C.$

Lemma 2.

([21]) Let H be a Hilbert space, let C be a nonempty closed convex subset of H and let A be a mapping of C into H. Let $u\in C$. Then for $\lambda >0$, $u=P_C(I-\lambda A)u$ if and only if $u\in Var(C,A)$, where $P_C$ is the metric projection of H onto C.

Lemma 3.

([22]) Let $\left\{{{\rm Y}}_n\right\}$ be a sequence of nonnegative real numbers satisfying ${{\rm Y}}_{n+1}=(1-{\alpha }_n){{\rm Y}}_n+{\delta }_n,$ for all $n\ge 0$,

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

(1)

${\displaystyle \sum _{n=1}^{\infty }}{\alpha }_n=+\infty ;$

(2)

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

Then ${lim}_{n\to +\infty }{{\rm Y}}_n=0.$

Lemma 4.

([23]) Let $\left\{{{\rm Y}}_n\right\}$ be a sequence of nonnegative real number satisfying,

$${{\rm Y}}_{n+1}=(1-{\alpha }_n){{\rm Y}}_n+{\alpha }_n{\beta }_n,\mathit{for} \mathit{all} n\ge 0$$

where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}$ satisfy the conditions

(1)

$\left\{{\alpha }_n\right\}\subset [0,1],{\displaystyle \sum _{n=1}^{+\infty }}{\alpha }_n=+\infty$;

(2)

$\underset{n\to +\infty }{lim\; sup}{\beta }_n\le 0or{\displaystyle \sum _{n=1}^{+\infty }}\left|{\alpha }_n{\beta }_n\right|<+\infty .$

Then ${lim}_{n\to +\infty }{{\rm Y}}_n=0.$

Lemma 5.

For every $i=1,2,3$, let $H_i$ be a real Hilbert spaces and $C_i$ be a nonempty closed convex subset of $H_i$. Let $B_i:C_i\to H_i$ be ${\beta }_i$-inverse strongly monotonic mappings with $\eta ={min}_{i=1,2,3}\left\{{\beta }_i\right\}$ and let $A_2:H_1\to H_2,A_3:H_2\to H_3$ be bounded linear operators with the adjoint operator $A_2^{*}$ and $A_3^{*}$, respectively. Assume that ${\overline{x}}_1\in C_1,A_2{\overline{x}}_1={\overline{x}}_2,A_3{\overline{x}}_2={\overline{x}}_3$ and $\Omega \ne Ø$. The following are equivalent:

(i)

$\overline{x}\in \Omega ,where\overline{x}=\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right)\in C_1\times C_2\times C_3$.

(ii)

${\overline{x}}_1=P_{C_1}\left(I_1-{\lambda }_1{B}_1\right)({\overline{x}}_1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3))$,

where $I_i:H_i\to H_i$ is an identity mapping, for all $i=1,2,3$, ${\gamma }_2(1+{\gamma }_3)\le {\displaystyle \frac{1}{L}}$, $L=max\{L_1,L_2\}\le 1$ which $L_1,L_2$ are spectral radii of $A_2{A}_2^{*}$ and $A_3{A}_3^{*}$, respectively, ${\lambda }_i\in \left(0,2\eta \right),$ for all $i=1,2,3$ and ${\gamma }_2,{\gamma }_3\ge 0.$

Proof. 

Let the conditions hold.

$\left(i\right)\Rightarrow \left(ii\right)$ Let $\overline{x}\in \Omega \mathrm{where}\overline{x}=\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right)\in C_1\times C_2\times C_3$; we have

$${\overline{x}}_i\in Var(C_i,B_i),\mathrm{for}\mathrm{all}i=1,2,3.$$

From Lemma 2, we have

$${\overline{x}}_i\in F\left(P_{C_i}(I_i-{\lambda }_i{B}_i)\right),\mathrm{for}\mathrm{all}i=1,2,3.$$

From determining the definition of $\overline{x}$, we have

$${\overline{x}}_1=P_{C_1}\left(I_1-{\lambda }_1{B}_1\right)({\overline{x}}_1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3)).$$

$\left(ii\right)\Rightarrow \left(i\right)$ Let $\overline{x}=\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right)\in C_1\times C_2\times C_3,$ where ${\overline{x}}_2=A_2{\overline{x}}_1,{\overline{x}}_3=A_3{\overline{x}}_2$ and

$${\overline{x}}_1=P_{C_1}\left(I_1-{\lambda }_1{B}_1\right)({\overline{x}}_1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3)).$$

Since $B_i$ is ${\beta }_i$-inverse strongly monotonic with ${\lambda }_i<2\eta ,$ for all $i=1,2,3$, we have $P_{C_i}(I_i-{\lambda }_i{B}_i)$ which is a nonexpansive mapping, for all $i=1,2,3$.

Let $w\in \Omega$ where $w=\left(w_1,w_2,w_3\right)\in C_1\times C_2\times C_3$ where $w_2=A_2{w}_1,w_3=A_3{w}_2$. From $\left(i\right)$ implies $\left(ii\right)$, we have

$$w_1=P_{C_1}(I-{\lambda }_1{B}_1)(w_1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I_2-{\lambda }_2{B}_2))w_2+{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I_3-{\lambda }_3{B}_3))w_3)).$$

Put $M=(I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3$ and $N=(I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right))w_2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right)w_3.$

From determining the definition of $\overline{x}$ and w, we have

$$\begin{array}{ccc}\hfill \parallel {\overline{x}}_1-w_1{\parallel }^2& \le & {\parallel {\overline{x}}_1-w_1-{\gamma }_2{A}_2^{*}(M-N)\parallel }^2\hfill \\ & =& {\parallel {\overline{x}}_1-w_1\parallel }^2-2{\gamma }_2\langle {\overline{x}}_1-w_1,A_2^{*}(M-N)\rangle +{\gamma }_2^2{\parallel A_2^{*}(M-N)\parallel }^2\hfill \\ & \le & {\parallel {\overline{x}}_1-w_1\parallel }^2-2{\gamma }_2\langle {\overline{x}}_2-w_2,M-N\rangle +{\gamma }_2^{2}L{\parallel M-N\parallel }^2\hfill \\ & \le & {\parallel {\overline{x}}_1-w_1\parallel }^2-2{\gamma }_2\langle {\overline{x}}_2-w_2,(I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3\rangle \hfill \\ & & +{\gamma }_2^{2}L{\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3\parallel }^2\hfill \\ & =& {\parallel {\overline{x}}_1-w_1\parallel }^2-2{\gamma }_2(\langle {\overline{x}}_2-w_2,(I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2\rangle +{\gamma }_3\langle {\overline{x}}_3-w_3,\hfill \\ & & \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3\rangle )+{\gamma }_2^{2}L\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2\hfill \\ & & +{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{\parallel }^2\hfill \\ & =& \parallel {\overline{x}}_1-w_1{\parallel }^2+2{\gamma }_2\langle w_2-{\overline{x}}_2,(I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2\rangle +2{\gamma }_2{\gamma }_3\langle w_3-{\overline{x}}_3,\hfill \\ & & \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3\rangle +{\gamma }_2^{2}L(\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2{\parallel }^2\hfill \\ & & +{\gamma }_3^{2}L\parallel \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{\parallel }^2+2{\gamma }_3\langle (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2,\hfill \\ & & A_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3\rangle )\hfill \\ & \le & \parallel {\overline{x}}_1-w_1{\parallel }^2+2{\gamma }_2\langle w_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right){\overline{x}}_2+P_{C_2}\left(I_2-{\lambda }_2{B}_2\right){\overline{x}}_2-{\overline{x}}_2,\hfill \\ & & (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2\rangle +2{\gamma }_2{\gamma }_3\langle w_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right){\overline{x}}_3+P_{C_3}\left(I_3-{\lambda }_3{B}_3\right){\overline{x}}_3-{\overline{x}}_3,\hfill \\ & & \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3\rangle +{\gamma }_2^{2}L(\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2{\parallel }^2\hfill \\ & & +{\gamma }_3^{2}L{\parallel \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3\parallel }^2\hfill \\ & & +{\gamma }_3\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2{\parallel }^2+{\gamma }_3\parallel A_3^{*}\left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{\parallel }^2)\hfill \\ & \le & {\parallel {\overline{x}}_1-w_1\parallel }^2+2{\gamma }_2\left({\displaystyle \frac{1}{2}}\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2{\parallel }^2-\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2{\parallel }^2\right)\hfill \\ & & +2{\gamma }_2{\gamma }_3\left({\displaystyle \frac{1}{2}}\parallel \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{\parallel }^2-\parallel \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{\parallel }^2\right)\hfill \\ & & +{\gamma }_2^{2}L(\parallel ((I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2{\parallel }^2+{\gamma }_3^{2}L\parallel \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{)\parallel }^2\hfill \\ & & +{\gamma }_3\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2{\parallel }^2+{\gamma }_{3}L\parallel \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{)\parallel }^2)\hfill \\ & =& {\parallel {\overline{x}}_1-w_1\parallel }^2-{\gamma }_2(1-{\gamma }_{2}L(1+{\gamma }_3)){\parallel (I_2-P_{C_2}\left(I_2-{\lambda }_2{B}_2\right)){\overline{x}}_2\parallel }^2\hfill \\ & & -{\gamma }_2{\gamma }_3(1-{\gamma }_2{L}^2(1+{\gamma }_3))\parallel \left(I_3-P_{C_3}\left(I_3-{\lambda }_3{B}_3\right)\right){\overline{x}}_3{)\parallel }^2.\hfill \end{array}$$

(8)

By applying above equation and Lemma 2, we have

$${\overline{x}}_2\in F\left(P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)=Var(C_2,B_2)\mathrm{and}{\overline{x}}_3\in F\left(P_{C_3}\left(I-{\lambda }_3{B}_3\right)\right)=Var(C_3,B_3).$$

(9)

From determining the definitions of $\overline{x}$ and (9), we have

$${\overline{x}}_1\in F\left(P_{C_1}\left(I-{\lambda }_1{B}_1\right)\right)=Var(C_1,B_1).$$

Hence $\overline{x}\in \Omega$. □

Lemma 6.

Let C be a nonempty closed convex subset of Hilbert space H. Let $\vartheta :C\to C$ be a nonspreading mapping and $\varrho :C\to C$ be κ-pseudo-nonspreding mapping with $F\left(\vartheta \right)\cap F\left(\varrho \right)\ne Ø$. Then $F\left(P_C(I-\gamma (a\left(I-\vartheta \right)+\left(1-a\right)\left(I-\varrho \right)))\right)=F\left(\vartheta \right)\cap F\left(\varrho \right)$ for all $a\in \left(0,1\right)$ and $\gamma >0$. Moreover, if $\gamma <1-\kappa$, then

$$\begin{array}{c}\hfill ∥I-\gamma \left(a\left(I-\vartheta \right)+\left(1-a\right)\left(I-\varrho \right))\right)x-{\varpi }^{*}∥\le ∥{\varpi }^{*}-x∥,\end{array}$$

for all $x\in C$ and ${\varpi }^{*}\in F\left(\varrho \right)\cap F\left(\vartheta \right)$.

Proof. 

Let ${\varpi }_0\in F\left(\varrho \right)\cap F\left(\vartheta \right)$, we have

$$\begin{array}{c}\hfill P_C(I-\gamma \left(a(I-\vartheta )+(1+a)(I-\varrho ))\right){\varpi }_0={\varpi }_0.\end{array}$$

It follows that ${\varpi }_0\in F\left(P_C(I-\gamma \left(a(I-\vartheta )+(1+a)(I-\varrho ))\right)\right)$. Therefore

$$\begin{array}{c}\hfill F\left(\vartheta \right)\cap F\left(\varrho \right)\subseteq F(P_C(I-\gamma \left(a(I-\vartheta )+(1+a)(I-\varrho )\right)))\end{array}$$

Let ${\varpi }_0\in F\left(P_C(I-\gamma (a(I-\vartheta )+(1+a)(I-\varrho )))\right)$ and ${\varpi }^{*}\in F\left(\vartheta \right)\cap F\left(\varrho \right)$. From Lemma 2, we have

$$\langle y-{\varpi }_0,a(I-\vartheta ){\varpi }_0+(1-a)(I-\varrho ){\varpi }_0\rangle \ge 0,$$

for all $y\in C.$

From determining the definition of $\varrho$, we have

$$\begin{array}{ccc}{\parallel {\varpi }_0-{\varpi }^{*}\parallel }^2+\kappa {\parallel (I-\varrho ){\varpi }_0\parallel }^2\hfill & \ge \hfill & {\parallel \varrho {\varpi }_0-{\varpi }^{*}\parallel }^2\hfill \\ & =\hfill & {\parallel (I-\varrho ){\varpi }_0-({\varpi }_0-{\varpi }^{*})\parallel }^2\hfill \\ \hfill & =\hfill & {\parallel \left(I-\varrho \right){\varpi }_0\parallel }^2-2\langle (I-\varrho ){\varpi }_0,{\varpi }_0-{\varpi }^{*}\rangle +{\parallel {\varpi }_0-{\varpi }^{*}\parallel }^2\hfill \end{array}$$

(10)

From the result of the calculation from the inequality (10), we get

$$\langle (I-\varrho ){\varpi }_0,{\varpi }_0-{\varpi }^{*}\rangle \ge \left({\displaystyle \frac{1-\kappa }{2}}\right){\parallel (I-\varrho ){\varpi }_0\parallel }^2$$

(11)

Assume that ${\varpi }_0\ne \vartheta {\varpi }_0$; then we have $\parallel (I-\vartheta ){\varpi }_0\parallel >0.$ Using the same method as (11) and definitions of $\vartheta$, we get

$$\langle (I-\vartheta ){\varpi }_0,{\varpi }_0-{\varpi }^{*}\rangle \ge {\displaystyle \frac{1}{2}}{\parallel (I-\vartheta ){\varpi }_0\parallel }^2$$

(12)

From (11) and $a\in (0,1)$, we obtain

$$\begin{array}{ccc}\langle {\varpi }^{*}-{\varpi }_0,a(I-\vartheta ){\varpi }_0\rangle & =& \langle {\varpi }^{*}-{\varpi }_0,a(I-\vartheta ){\varpi }_0+(1-a)(I-\varrho ){\varpi }_0\rangle \hfill \\ & & -(1-a)\langle {\varpi }^{*}-{\varpi }_0,(I-\varrho ){\varpi }_0\rangle \hfill \\ & \ge & (1-a)\langle {\varpi }_0-{\varpi }^{*},(I-\varrho ){\varpi }_0\rangle .\hfill \end{array}$$

(13)

From (13), we have

$$\begin{array}{c}\hfill \langle {\varpi }^{*}-{\varpi }_0,(I-\vartheta ){\varpi }_0\rangle \ge 0.\end{array}$$

From above and (12), we have

$$\begin{array}{c}\hfill 0\le \langle {\varpi }^{*}-{\varpi }_0,(I-\vartheta ){\varpi }_0\rangle \le -{\displaystyle \frac{1}{2}}{\parallel (I-\vartheta ){\varpi }_0\parallel }^2.\end{array}$$

Thus, $\parallel (I-\vartheta ){\varpi }_0\parallel \le 0.$ This is a contradiction.

Thus, we have ${\varpi }_0$ = $\vartheta {\varpi }_0$ and it implies that

$$\begin{array}{c}\hfill {\varpi }_0\in F\left(\vartheta \right).\end{array}$$

(14)

Similarly, by using the same technique as (14), we have

$$\begin{array}{c}\hfill {\varpi }_0\in F\left(\varrho \right).\end{array}$$

(15)

From (14) and (15), we have

$$\begin{array}{c}\hfill F\left(P_C(I-\gamma (a\left(I-\vartheta \right)+(1+a)\left(I-\varrho \right)))\right)\subseteq F\left(\varrho \right)\cap F\left(\vartheta \right).\end{array}$$

Let ${\varpi }^{*}\in F\left(\varrho \right)\cap F\left(\vartheta \right)$ and $x\in C$; we have

$$\begin{array}{ccc}\hfill {∥(I-\gamma \left(a\left(I-\vartheta \right)+\left(1-a\right)\left(I-\varrho \right))\right)x-{\varpi }^{*}∥}^2& =& \parallel (I-\gamma \left(a\left(I-\vartheta \right)+\left(1-a\right)\left(I-\varrho \right))\right)x\hfill \\ & & -(I-\gamma \left(a\left(I-\vartheta \right)+\left(1-a\right)\left(I-\varrho \right))\right){\varpi }^{*}{\parallel }^2\hfill \\ & =& \parallel x-{\varpi }^{*}-\gamma (a(\left(I-\vartheta \right)x-\left(I-\vartheta \right){\varpi }^{*})\hfill \\ & & +(1-a)(\left(I-\varrho \right)x-\left(I-\varrho \right){\varpi }^{*}){)\parallel }^2\hfill \\ & =& \parallel x-{\varpi }^{*}{\parallel }^2-2\gamma \langle a(\left(I-\vartheta \right)x-\left(I-\vartheta \right){\varpi }^{*})\hfill \\ & & +(1-a)(\left(I-\varrho \right)x-\left(I-\varrho \right){\varpi }^{*}),x-{\varpi }^{*}\rangle \hfill \\ & & +{\gamma }^2\parallel a(\left(I-\vartheta \right)x-\left(I-\vartheta \right){\varpi }^{*})\hfill \\ & & +(1-a)(\left(I-\varrho \right)x-\left(I-\varrho \right){\varpi }^{*}){\parallel }^2\hfill \\ & \le & \parallel x-{\varpi }^{*}{\parallel }^2-2\gamma a\langle \left(I-\vartheta \right)x-\left(I-\vartheta \right){\varpi }^{*},x-{\varpi }^{*}\rangle \hfill \\ & & -2\gamma (1-a)\langle \left(I-\varrho \right)x-\left(I-\varrho \right){\varpi }^{*},x-{\varpi }^{*}\rangle \hfill \\ & & +{\gamma }^{2}a{\parallel \left(I-\vartheta \right)x-\left(I-\vartheta \right){\varpi }^{*}\parallel }^2\hfill \\ & & +(1-a){\gamma }^2{\parallel \left(I-\varrho \right)x-\left(I-\varrho \right){\varpi }^{*}\parallel }^2\hfill \\ & \le & \parallel x-{\varpi }^{*}{\parallel }^2-2\gamma a{\displaystyle \frac{{\parallel (I-T)x\parallel }^2}{2}}\hfill \\ & & -2\gamma (1-a)(1-\kappa ){\displaystyle \frac{{\parallel \left(I-\varrho \right)x\parallel }^2}{2}}\hfill \\ & & +{\gamma }^2{a\parallel \left(I-\vartheta \right)x\parallel }^2+(1-a){\gamma }^2{\parallel \left(I-\varrho \right)x\parallel }^2\hfill \\ & \le & \parallel x-{\varpi }^{*}{\parallel }^2.\hfill \end{array}$$

 □

2. The Split Various Variational Inequality Theorem

Theorem 7.

For every $i=1,2,3$, let $C_i$ be a closed convex subset of a real Hilbert space $H_i$. Let $B_i:C_i\to H_i$ be ${\beta }_i$-inverse strongly monotonic mappings with $\eta ={min}_{i=1,2,3}\left\{{\beta }_i\right\}$ and let $A_2:H_1\to H_2,A_3:H_2\to H_3$ be a bounded linear operator with the adjoint operator $A_2^{*}$ and $A_3^{*}$, respectively. Assume that ${\overline{x}}_1\in H_1,{\overline{x}}_2=A_2{\overline{x}}_1,{\overline{x}}_3=A_3{\overline{x}}_2$ and $\Omega \ne Ø$. Let $\vartheta ,\varrho :C\to C$ be nonspreading and κ-strictly pseudo-nonspreading mappings, respectively. Assume that $\Omega \cap F\left(\vartheta \right)\cap F\left(\varrho \right)\ne Ø$ and let the sequence $\left\{x_n\right\}$ generated by $u,x_1\in C$, and

$$\begin{array}{ccc}\hfill x_{n+1}& =& {\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{P}_{C_1}\left(I_1-{\lambda }_n\left(a\left(I_1-\vartheta \right)+\left(1-a\right)\left(I_1-\varrho \right)\right)\right)x_n\hfill \\ & & +{\delta }_n{P}_{C_1}(I_1-{\lambda }_1{B}_1)(x_n^1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2+{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I-{\lambda }_3{B}_3))x_n^3)),\hfill \end{array}$$

for all $n\ge 1$ and $a\in (0,1)$, $I_i:H_i\to H_i$ are identity mappings, for all $i=1,2,3$, where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\}\subseteq \left[0,1\right]$ and ${\alpha }_n+{\beta }_n+{\gamma }_n+{\delta }_n=1$ and $x_n=x_n^1,x_n^2=A_2{x}_n^1,x_n^3=A_3{x}_n^2,$ for all $n\in \mathbb{N},0<{\lambda }_i<2\eta ,$ for all $i=1,2,3$ and ${\gamma }_j>0,$ for all $j=2,3$. Suppose that the conditions (i)–(v) are true;

(i)

${lim}_{n+\to \infty }{\alpha }_n=0,{\sum }_{n=1}^{+\infty }{\alpha }_n=+\infty$;

(ii)

${\gamma }_2\left(1+{\gamma }_3\right)\le {\displaystyle \frac{1}{L}}$, where $L=max\left\{L_{A_1},L_{A_2}\right\}\le 1where{L}_{A_1}{L}_{A_2}$ are spectral radius of $A_2{A}_2^{*},A_3{A}_3^{*}$, respectively;

(iii)

$0<a\le {\beta }_n,{\gamma }_n,{\delta }_n\le b<1,forsomea,b>0,foralln\in \mathbb{N}$;

(iv)

${\sum }_{n=1}^{+\infty }{\lambda }_n<+\infty and0<{\lambda }_n<1-\kappa ,foralln\in \mathbb{N}$;

(v)

${\sum }_{n=1}^{+\infty }\left|{\alpha }_{n+i}-{\alpha }_n\right|,{\sum }_{n=1}^{+\infty }\left|{\beta }_{n+i}-{\beta }_n\right|,{\sum }_{n=1}^{+\infty }\left|{\gamma }_{n+1}-{\gamma }_n\right|<+\infty .$

Then the sequence $\left\{x_n\right\}$ converges strongly to $z_0=P_{\Omega \cap F\left(\vartheta \right)\cap F\left(\varrho \right)}u$.

Proof. 

Put $M=a\left(I-\vartheta \right)+\left(1-a\right)\left(I-\varrho \right)$ and $u_n=P_{C_1}(I-{\lambda }_1{B}_1)(x_n^1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2+{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I-{\lambda }_n{B}_3))x_n^3))$. Thus, we can rewrite $x_n$ as follows:

$$\begin{array}{c}\hfill x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{P}_{C_1}\left(I-{\lambda }_{n}M\right)x_n+{\delta }_n{u}_n,\end{array}$$

(16)

for all $n\ge 1$.

From determining the definition of $u_n$ put $w_n=\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_n^2+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I-{\lambda }_3{B}_3\right)\right)x_n^3$ and $z_n=\left(I_3-P_{C_3}\left(I-{\lambda }_3{B}_3\right)\right)x_n^3$, we have

$$\begin{array}{c}\hfill u_n=P_{C_1}\left(I_1-{\lambda }_1{B}_1\right)(x_n-{\gamma }_2{A}_2^{*}{w}_n).\end{array}$$

For every $n\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill {\parallel u_n-u_{n-1}\parallel }^2& \le & {\parallel x_n-{\gamma }_2{A}_2^{*}{w}_n-x_{n-1}+{\gamma }_2{A}_2^{*}{w}_{n-1}\parallel }^2\hfill \\ & =& {\parallel x_n-x_{n-1}-{\gamma }_2{A}_2^{*}\parallel w_n-w_{n-1}\parallel }^2\hfill \\ & =& {\parallel x_n-x_{n-1}\parallel }^2-2{\gamma }_2\langle A_2{x}_n-A_2{x}_{n-1},w_n-w_{n-1}\rangle +{\gamma }_2^2{\parallel A_2^{*}(w_n-w_{n-1})\parallel }^2\hfill \\ & =& {\parallel x_n-x_{n-1}\parallel }^2-2{\gamma }_2\langle x_n^2-x_{n-1}^2,(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_n^2\hfill \\ & & +{\gamma }_3{A}_3^{*}{z}_n-(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_{n-1}^2-{\gamma }_3{A}_3^{*}{z}_{n-1}\rangle \hfill \\ & & +{\gamma }_2^2{\parallel A_2^{*}(w_n-w_{n-1})\parallel }^2\hfill \\ & =& {\parallel x_n-x_{n-1}\parallel }^2+2{\gamma }_2\langle x_{n-1}^2-x_n^2,(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_n^2\hfill \\ & & -(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_{n-1}^2\rangle +2{\gamma }_2{\gamma }_3\langle x_{n-1}^3-x_n^3,z_n\hfill \\ & & -z_{n-1}\rangle +{\gamma }_2^2{\parallel A_2^{*}(w_n-w_{n-1})\parallel }^2\hfill \\ & \le & {\parallel x_n-x_{n-1}\parallel }^2+2{\gamma }_2\langle x_{n-1}^2-x_n^2,(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_n^2\hfill \\ & & -(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_{n-1}^2\rangle +2{\gamma }_2{\gamma }_3\langle x_{n-1}^3-x_n^3,z_n-z_{n-1}\rangle \hfill \\ & & +{\gamma }_2^{2}L{\parallel (I_2-P_{C_2}(I_2-{\lambda }_2{B}_2\left)\right)x_n^2-(I_2-P_{C_2}(I_2-{\lambda }_2{B}_2\left)\right)x_{n-1}^2+{\gamma }_3{A}_3^{*}(z_n-z_{n-1}\parallel }^2\hfill \\ & \le & {\parallel x_n-x_{n-1}\parallel }^2+2{\gamma }_2\langle x_{n-1}^2-x_n^2,(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_n^2\hfill \\ & & -(I_2-P_{C_2}(I-{\lambda }_2{B}_2\left)\right)x_{n-1}^2\rangle +2{\gamma }_2{\gamma }_3\langle x_{n-1}^3-x_n^3,z_n-z_{n-1}\rangle \hfill \\ & & +{\gamma }_2^{2}L({\parallel (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2-(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2\parallel }^2\hfill \\ & & +{\gamma }_3^{2}L{\parallel z_n-z_{n-1}\parallel }^2+2{\gamma }_3\langle (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2\hfill \\ & & -(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2,A^{*}(z_n-z_{n-1})\rangle )\hfill \\ & \hfill \le & {∥x_n-x_{n-1}∥}^2+2{\gamma }_2\langle x_{n-1}^2-x_n^2,(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right))x_n^2-\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_{n-1}^2\rangle \hfill \\ & & +2{\gamma }_2{\gamma }_3\langle x_{n-1}^3-x_n^3,(I_3-P_{C_3}\left(I-{\lambda }_3{B}_3\right))x_n^3-\left(I_3-P_{C_3}\left(I-{\lambda }_3{B}_3\right)\right)x_{n-1}^3\rangle \hfill \\ & & +{\gamma }_2^{2}L(\parallel (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2-(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2{\parallel }^2+{\gamma }_3^{2}L{\parallel z_n-z_{n-1}\parallel }^2\hfill \\ & & +{\gamma }_3\parallel (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2-(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2{\parallel }^2+{\gamma }_{3}L\parallel z_n-z_{n-1}{\parallel }^2)\hfill \\ & \hfill \le & {∥x_n-x_{n-1}∥}^2+2{\gamma }_2(-{∥\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_n^2-\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_{n-1}^2∥}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{∥\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_n^2-\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_{n-1}^2∥}^2)\hfill \\ & & +2{\gamma }_2{\gamma }_3(-{∥z_n-z_{n-1}∥}^2+{\displaystyle \frac{1}{2}}{∥z_n-z_{n-1}∥}^2)\hfill \\ & & +{\gamma }_2^{2}L(\parallel (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2-(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2{\parallel }^2+{\gamma }_3^{2}L{\parallel z_n-z_{n-1}\parallel }^2\hfill \\ & & +{\gamma }_3\parallel (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2-(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2{\parallel }^2+{\gamma }_{3}L\parallel z_n-z_{n-1}\parallel )\hfill \\ & \hfill =& {∥x_n-x_{n-1}∥}^2-{\gamma }_2{∥\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_n^2-\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_{n-1}^2∥}^2\hfill \\ & & -{\gamma }_2{\gamma }_3{∥z_n-z_{n-1}∥}^2+{\gamma }_2^{2}L{\parallel (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2-(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2\parallel }^2\hfill \\ & & +{\gamma }_2^2{\gamma }_3^2{L}^2\parallel z_n-z_{n-1}{\parallel }^2+{\gamma }_2^2{\gamma }_{3}L{\parallel (I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2-(I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_{n-1}^2\parallel }^2\hfill \\ & & +{\gamma }_2^2{\gamma }_3{L}^2\parallel z_n-z_{n-1}\parallel \hfill \\ & \hfill =& {∥x_n-x_{n-1}∥}^2-{\gamma }_2(1-{\gamma }_{2}L\left(1+{\gamma }_3\right))\parallel \left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_n^2\hfill \\ & & -\left(I_2-P_{C_2}\left(I-{\lambda }_2{B}_2\right)\right)x_{n-1}^2{\parallel }^2-{\gamma }_2{\gamma }_3(1-{\gamma }_2{L}^2\left(1+{\gamma }_3\right)){∥z_n-z_{n-1}∥}^2\hfill \\ & \hfill \le & {∥x_n-x_{n-1}∥}^2.\hfill \end{array}$$

(17)

Let ${\varpi }^{*}\in \Omega \cap F\left(\varrho \right)\cap F\left(\vartheta \right)$. From Lemma 6 and utilization of (8), we have

$$\begin{array}{ccc}∥x_{n+1}-{\varpi }^{*}∥& \le & {\alpha }_n∥u-{\varpi }^{*}∥+{\beta }_n∥x_n-{\varpi }^{*}∥+{\gamma }_n∥P_{C_1}\left(I-{\lambda }_{n}M\right)x_n-{\varpi }^{*}∥\hfill \\ & & +{\delta }_n∥u_n-{\varpi }^{*}∥\hfill \\ & \le & {\alpha }_n∥u-{\varpi }^{*}∥+(1-{\alpha }_n)∥x_n-{\varpi }^{*}∥\hfill \\ & \le & \tilde{M},\hfill \end{array}$$

(18)

where $\tilde{M}=max\{∥u-{\varpi }^{*}∥,∥x_1-{\varpi }^{*}∥\}$. By induction we can conclude that the sequence $\left\{x_n\right\}$ is bounded and so are $\left\{u_n\right\}$ and $\left\{P_{C_1}\left(I-{\lambda }_{n}M\right)x_n\right\}$.

From determining the definition of $x_n$ and (17), we have

$$\begin{array}{ccc}\hfill \parallel x_{n+1}-x_n\parallel & =& \parallel {\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{P}_{C1}\left(I-{\lambda }_{n}M\right)x_n+{\delta }_n{u}_n\hfill \\ & & -{\alpha }_{n-1}u-{\beta }_{n-1}{x}_{n-1}-{\gamma }_{n-1}{P}_{C1}\left(I-{\lambda }_{n-1}M\right)x_{n-1}-{\delta }_{n-1}{u}_{n-1}\parallel \hfill \\ & \le & \left|{\alpha }_n-{\alpha }_{n-1}\right|∥u∥+{\beta }_n∥x_n-x_{n-1}∥+\left|{\beta }_n-{\beta }_{n-1}\right|∥x_{n-1}∥+{\delta }_n∥u_n-u_{n-1}∥\hfill \\ & & +\left|{\delta }_n-{\delta }_{n-1}\right|∥u_{n-1}∥+{\gamma }_n∥P_{C_1}\left(I-{\lambda }_{n}M\right)x_n-P_{C_1}\left(I-{\lambda }_{n-1}M\right)x_{n-1}∥\hfill \\ & & +\left|{\gamma }_n-{\gamma }_{n-1}\right|∥P_{C_1}\left(I_1-{\lambda }_{n-1}M\right)x_{n-1}∥\hfill \\ & \le & \left|{\alpha }_n-{\alpha }_{n-1}\right|∥u∥+{\beta }_n∥x_n-x_{n-1}∥+\left|{\beta }_n-{\beta }_{n-1}\right|∥x_{n-1}∥+{\delta }_n∥u_n-u_{n-1}∥\hfill \\ & & +\left|{\delta }_n-{\delta }_{n-1}\right|∥u_{n-1}∥+{\gamma }_n∥x_n-x_{n-1}∥+\parallel {\lambda }_{n-1}M{x}_{n-1}-{\lambda }_{n}M{x}_n\parallel \hfill \\ & & +\left|{\gamma }_n-{\gamma }_{n-1}\right|∥P_{C_1}\left(I_1-{\lambda }_{n-1}M\right)x_{n-1}∥\hfill \\ & \le & \left|{\alpha }_n-{\alpha }_{n-1}\right|∥u∥+{\beta }_n∥x_n-x_{n-1}∥+\left|{\beta }_n-{\beta }_{n-1}\right|∥x_{n-1}∥+{\delta }_n∥x_n-x_{n-1}∥\hfill \\ & & +\left|{\delta }_n-{\delta }_{n-1}\right|∥u_{n-1}∥+{\gamma }_n∥x_n-x_{n-1}∥+{\lambda }_{n-1}\parallel M{x}_{n-1}\parallel +{\lambda }_n\parallel M{x}_n\parallel \hfill \\ & & +\left|{\gamma }_n-{\gamma }_{n-1}\right|∥P_{C_1}\left(I_1-{\lambda }_{n-1}M\right)x_{n-1}∥\hfill \\ & =& (1-{\alpha }_n)∥x_n-x_{n-1}∥+\left|{\alpha }_n-{\alpha }_{n-1}\right|∥u∥+\left|{\beta }_n-{\beta }_{n-1}\right|∥x_{n-1}∥\hfill \\ & & +\left|{\delta }_n-{\delta }_{n-1}\right|∥u_{n-1}∥+{\lambda }_{n-1}\parallel M{x}_{n-1}\parallel +{\lambda }_n\parallel M{x}_n\parallel \hfill \\ & & +\left|{\gamma }_n-{\gamma }_{n-1}\right|∥P_{C_1}\left(I_1-{\lambda }_{n-1}M\right)x_{n-1}∥.\hfill \end{array}$$

From the conditions $\left(i\right),\left(iv\right),\left(v\right)$ and Lemma 3, we have

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}∥x_{n+1}-x_n∥=0\end{array}$$

(19)

Applying (8) and the definition of $x_n$, we have

$$\begin{array}{ccc}\hfill {∥x_{n+1}-{\varpi }^{*}∥}^2& \le & {\alpha }_n{∥u-{\varpi }^{*}∥}^2+{\beta }_n{∥x_n-{\varpi }^{*}∥}^2+{\gamma }_n{∥P_{C_1}\left(I-{\lambda }_{n}M\right)x_n-{\varpi }^{*}∥}^2\hfill \\ & & +{\delta }_n{∥u_n-{\varpi }^{*}∥}^2-{\gamma }_n{\beta }_n{∥P_{C_1}\left(I-{\lambda }_{n}M\right)x_n-x_n∥}^2-{\delta }_n{\beta }_n{∥u_n-x_n∥}^2\hfill \\ & \le & {\alpha }_n{∥u-{\varpi }^{*}∥}^2+{∥x_n-{\varpi }^{*}∥}^2-{\gamma }_n{\beta }_n{∥P_{C1}\left(I-{\lambda }_{n}M\right)x_n-x_n∥}^2\hfill \\ & & -{\delta }_n{\beta }_n{∥u_n-x_n∥}^2,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill {\gamma }_n{\beta }_n{∥P_{C_1}\left(I-{\lambda }_{n}M\right)x_n-x_n∥}^2+{\delta }_n{\beta }_n{∥u_n-x_n∥}^2& \le & (\parallel x_{n+1}-{\varpi }^{*}\parallel +\parallel x_n-{\varpi }^{*}\parallel )\parallel x_{n+1}-x_n\parallel \hfill \\ & & +{\alpha }_n{∥u-{\varpi }^{*}∥}^2.\hfill \end{array}$$

From the conditions (i), (iii) and (19) we can conclude the following results

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}∥u_n-x_n∥=\underset{n\to +\infty }{lim}∥P_{C1}\left(I-{\lambda }_{n}M\right)x_n-x_n∥=0.\end{array}$$

(20)

Next, we show that

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; sup}\langle u-z_0,z_0-x_n\rangle \le 0,\end{array}$$

(21)

where $z_0=P_{\Omega \cap F\left(\varrho \right)\cap F\left(\vartheta \right)}u$. In order to prove this we may assume that

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; sup}\langle u-z_0,x_n-z_0\rangle =\underset{k\to +\infty }{lim}\langle u-z_0,x_{n_k}-z_0\rangle ,\end{array}$$

(22)

where $\left\{x_{n_k}\right\}$ is a subsequence of $\left\{x_n\right\}$. Since $\left\{x_n\right\}$ is bounded, we may assume that $x_{n_k}\rightharpoonup q$ as $k\to +\infty$. Assume that $q\notin F\left(\varrho \right)\cap F\left(\vartheta \right)$. From Lemma 6, we have $q\notin F\left(P_{C_1}\left(I-{\lambda }_{n_k}M\right)\right)$. By using properties of Opial’s condition and (20), we have

$$\begin{array}{ccc}\hfill \underset{k\to +\infty }{lim\; inf}∥x_{n_k}-q∥& <& \underset{k\to +\infty }{lim\; inf}∥x_{n_k}-P_{C_1}\left(I-{\lambda }_{n_k}M\right)q∥\hfill \\ & \le & \underset{k\to +\infty }{lim\; inf}(\parallel x_{n_k}-P_{C_1}\left(I-{\lambda }_{n_k}M\right)x_{n_k}\parallel \hfill \\ & & +\parallel P_{C_1}\left(I-{\lambda }_{n_k}M\right)x_{n_k}-P_{C_1}\left(I-{\lambda }_{n_k}M\right)q\parallel )\hfill \\ & \le & \underset{k\to +\infty }{lim\; inf}(\parallel x_{n_k}-q\parallel +{\lambda }_{n_k}\parallel M{x}_{n_k}-Mq\parallel )\hfill \\ & \le & \underset{k\to +\infty }{lim\; inf}∥x_{n_k}-q∥.\hfill \end{array}$$

This is a contradiction. Therefore $q\in F\left(\varrho \right)\cap F\left(\vartheta \right)$.

Assume $q\notin \Omega$. From Lemma 5, we have

$$\begin{array}{c}\hfill q\ne P_{C_1}\left(I-{\lambda }_1{B}_1\right)(q-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I-{\lambda }_2{B}_2))A_{2}q+{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I-{\lambda }_3{B}_3))A_3{A}_{2}q)).\end{array}$$

By using properties of Opial’s condition, and the definitions of $u_n$ and (20), we have

$$\begin{array}{ccc}\hfill \underset{k\to +\infty }{lim\; inf}\parallel x_{n_k}-q\parallel & <& \underset{k\to +\infty }{lim\; inf}\parallel x_{n_k}-P_{C_1}\left(I-{\lambda }_1{B}_1\right)(q-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I-{\lambda }_2{B}_2))A_{2}q\hfill \\ & & +{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I-{\lambda }_3{B}_3))A_3{A}_{2}q\left)\right)\parallel \hfill \\ & \le & \underset{k\to +\infty }{lim\; inf}(\parallel x_{n_k}-u_{n_k}\parallel \hfill \\ & & +\parallel u_{n_k}-P_{C_1}\left(I-{\lambda }_1{B}_1\right)(q-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I-{\lambda }_2{B}_2))A_{2}q\hfill \\ & & +{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I-{\lambda }_3{B}_3))A_3{A}_{2}q\left)\right)\parallel )\hfill \\ & \le & \underset{k\to +\infty }{lim\; inf}\parallel x_{n_k}-q\parallel .\hfill \end{array}$$

This is a contradiction. Then $q\in \Omega$. Therefore $q\in \Omega \cap F\left(\varrho \right)\cap F\left(\vartheta \right)$.

From (22) and the well-known properties of metric projection, we have

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim\; sup}\langle u-z_0,x_n-z_0\rangle \le 0.\end{array}$$

From determining the definition of $x_n$, we can conclude that

$$\begin{array}{c}\hfill {∥x_{n+1}-z_0∥}^2\le \left(1-{\alpha }_n\right){∥x_n-z_0∥}^2+2{\alpha }_n\langle u-z_0,x_{n+1}-z_0\rangle ,\end{array}$$

where $z_0=P_{\Omega \cap F\left(\varrho \right)\cap F\left(\vartheta \right)}u$. From Lemma 4, we can conclude that the sequence $\left\{x_n\right\}$ converses strongly to $z_0=P_{\Omega \cap F\left(\varrho \right)\cap F\left(\vartheta \right)}u$. □

The following results were obtained directly from the main theorem.

Corollary 8.

For every $i=1,2,3$, let $C_i$ be a closed convex subset of a real Hilbert space $H_i$. Let $B_i:C_i\to H_i$ be ${\beta }_i$-inverse strongly monotonic mappings with $\eta ={min}_{i=1,2,3}\left\{{\beta }_i\right\}$ and let $A_2:H_1\to H_2,A_3:H_2\to l{H}_3$ be a bounded linear operator with the adjoint operator $A_2^{*}$ and $A_3^{*}$, respectively. Assume that ${\overline{x}}_1\in H_1,{\overline{x}}_2=A_2{\overline{x}}_1,{\overline{x}}_3=A_3{\overline{x}}_2$ and $\Omega \ne Ø$. Let $\vartheta ,\varrho :C\to C$ be nonspreading mappings, respectively. Assume that $\Omega \cap F\left(\vartheta \right)\cap F\left(\varrho \right)\ne Ø$ and let the sequence $\left\{x_n\right\}$ generated by $u,x_1\in C$, and

$$\begin{array}{ccc}\hfill x_{n+1}& =& {\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{P}_{C_1}\left(I_1-{\lambda }_n\left(a\left(I_1-\vartheta \right)+\left(1-a\right)\left(I_1-\varrho \right)\right)\right)x_n\hfill \\ & & +{\delta }_n{P}_{C_1}(I_1-{\lambda }_1{B}_1)(x_n^1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I-{\lambda }_2{B}_2))x_n^2+{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I-{\lambda }_3{B}_3))x_n^3)),\hfill \end{array}$$

for all $n\ge 1$ and $a\in (0,1)$, $I_i:H_i\to H_i$ are identity mappings, for all $i=1,2,3$, where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\}\subseteq \left[0,1\right]$ and ${\alpha }_n+{\beta }_n+{\gamma }_n+{\delta }_n=1$ and $x_n=x_n^1,x_n^2=A_2{x}_n^1,x_n^3=A_3{x}_n^2,$ for all $n\in \mathbb{N},0<{\lambda }_i<2\eta ,$ for all $i=1,2,3$ and ${\gamma }_j>0,$ for all $j=2,3$. Suppose that the conditions (i) to (v) are true;

(i)

${lim}_{n\to +\infty }{\alpha }_n=0,{\sum }_{n=1}^{+\infty }{\alpha }_n=+\infty$;

(ii)

${\gamma }_2\left(1+{\gamma }_3\right)\le {\displaystyle \frac{1}{L}}$, where $L=max\left\{L_{A_1},L_{A_2}\right\}\le 1$, where $L_{A_1},L_{A_2}$ are spectral radius of $A_2{A}_2^{*},A_3{A}_3^{*}$,

respectively;

(iii)

$0<a\le {\beta }_n,{\gamma }_n,{\delta }_n\le b<1$, for some $a,b>0$, for all $n\in \mathbb{N}$;

(iv)

${\sum }_{n=1}^{+\infty }{\lambda }_n<+\infty and0<{\lambda }_n<1-\kappa ,foralln\in \mathbb{N}$;

(v)

${\sum }_{n=1}^{+\infty }\left|{\alpha }_{n+i}-{\alpha }_n\right|,{\sum }_{n=1}^{+\infty }\left|{\beta }_{n+i}-{\beta }_n\right|,{\sum }_{n=1}^{+\infty }\left|{\gamma }_{n+1}-{\gamma }_n\right|<+\infty .$

Then the sequence $\left\{x_n\right\}$ converses strongly to $z_0=P_{\Omega \cap F\left(\vartheta \right)\cap F\left(\varrho \right)}u$.

3. Application

We have applied the problem (7) for the various fixed point problems in three Hilbert spaces as follows:

For every $i=1,2,3$, let $H_i$ be a real Hilbert space and $C_i$ be a nonempty closed convex subset of $H_i$. Let ${\vartheta }_i:C_i\to C_i$ be a mapping, for all $i=1,2,3$ and let $A_2:H_1\to H_2$ and $A_3:H_2\to H_3$. The fixed points problem in three Hilbert spaces is meant to find the point

$$\left\{\begin{array}{c}\begin{array}{c}{\varpi }_1^{*}\in C_1,\mathrm{such} \mathrm{that}{\varpi }_1^{*}\in F\left({\vartheta }_1\right)\mathrm{and}\hfill \\ {\varpi }_2^{*}=A_2{\varpi }_1^{*}\in C_2,\mathrm{such} \mathrm{that}{\varpi }_2^{*}\in F\left({\vartheta }_2\right)\mathrm{and}\hfill \\ {\varpi }_3^{*}=A_3{\varpi }_2^{*}\in C_3,\mathrm{such} \mathrm{that}{\varpi }_3^{*}\in F\left({\vartheta }_3\right).\hfill \end{array}\hfill \end{array}\right.$$

(23)

The set of the solutions of (23) is denoted by $\underline{\Omega }=\left\{{\varpi }^{*}=\left({\varpi }_1^{*},{\varpi }_2^{*},{\varpi }_3^{*}\right)\in C_1\times C_2\times C_3:{\varpi }_i^{*}\in F\left({\vartheta }_i\right)\right.,$ for all $i=1,2,3\}$. It is clear that $Var(C,I-T)=F\left(\vartheta \right)$, where $\vartheta :C\to C$ is a nonexpansive mapping with $F\left(\vartheta \right)\ne Ø$. By leveraging Lemma 5 and such knowledge, we have the following results:

Lemma 9.

For every $i=1,2,3$, let $H_i$ be a real Hilbert spaces and $C_i$ be a nonempty closed convex subset of $H_i$. Let ${\vartheta }_i:C_i\to C_i$ be nonexpansive mappings and let $A_2:H_1\to H_2,A_3:H_2\to H_3$ be a bounded linear operator with the adjoint operator $A_2^{*}$ and $A_3^{*}$, respectively. Assume that ${\overline{x}}_1\in C_1,A_2{\overline{x}}_1={\overline{x}}_2,A_3{\overline{x}}_2={\overline{x}}_3$ and $\underline{\Omega }\ne Ø$. The following are equivalent:

(i)

$\overline{x}\in \underline{\Omega },where\overline{x}=\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3\right)\in C_1\times C_2\times C_3$.

(ii)

${\overline{x}}_1=P_{C_1}\left(I_1-{\lambda }_1(I_1-{\vartheta }_1)\right)({\overline{x}}_1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}\left(I_2-{\lambda }_2(I_2-{\vartheta }_2)\right)){\overline{x}}_2$

$+{\gamma }_3{A}_3^{*}\left(I_3-P_{C_3}\left(I-{\lambda }_3(I_3-{\vartheta }_3)\right)\right){\overline{x}}_3\left)\right)$,

where $I_i:H_i\to H_i$ is an identity mapping, for all $i=1,2,3$, ${\gamma }_2(1+{\gamma }_3)\le {\displaystyle \frac{1}{L}}$, $L=max\{L_1,L_2\}\le 1$ which $L_1,L_2$ are spectral radii of $A_2{A}_2^{*}$ and $A_3{A}_3^{*}$, respectively, ${\lambda }_i\in \left(0,1\right),$ for all $i=1,2,3$ and ${\gamma }_2,{\gamma }_3\ge 0$

Proof. 

Given $F\left({\vartheta }_i\right)=Var\left(C,I_i-{\vartheta }_i\right)$ for all $i=1,2,3$; $(I_i-{\vartheta }_i)$—a ${\displaystyle \frac{1}{2}}$-inverse strongly monotonic; and Lemma 5, we can summarize the result of Lemma 9. □

As the direct benefits of Lemma 9, we get Corollary 10.

Corollary 10.

For every $i=1,2,3$, let $C_i$ be a closed convex subset of a real Hilbert space $H_i$. Let ${\vartheta }_i:C_i\to C_i$ be a nonexpansive mapping and let $A_2:H_1\to H_2,A_3:H_2\to H_3$ be a bounded linear operator with the adjoint operator $A_2^{*}$ and $A_3^{*}$, respectively. Assume that ${\overline{x}}_1\in H_1,{\overline{x}}_2=A_2{\overline{x}}_1,{\overline{x}}_3=A_3{\overline{x}}_2$. Let $\vartheta ,\varrho :C\to C$ be nonspreading and κ-strictly pseudo-nonspreading mappings, respectively. Assume that $\underline{\Omega }\cap F\left(\vartheta \right)\cap F\left(\varrho \right)\ne Ø$ and let the sequence $\left\{x_n\right\}$ generated by $u,x_1\in C$, and

$$\begin{array}{ccc}\hfill x_{n+1}& =& {\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{P}_{C_1}\left(I_1-{\lambda }_n\left(a\left(I_1-\vartheta \right)+\left(1-a\right)\left(I_1-\varrho \right)\right)\right)x_n\hfill \\ & & +{\delta }_n{P}_{C_1}(I_1-{\lambda }_1(I_1-{\vartheta }_1))(x_n^1-{\gamma }_2{A}_2^{*}((I_2-P_{C_2}(I-{\lambda }_2(I_2-{\vartheta }_2)))x_n^2\hfill \\ & & +{\gamma }_3{A}_3^{*}(I_3-P_{C_3}(I-{\lambda }_3(I_3-{\vartheta }_3)))x_n^3\left)\right),\hfill \end{array}$$

for all $n\ge 1$ and $a\in (0,1)$, $I_i:H_i\to H_i$ is an identity mapping, for all $i=1,2,3$, where $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\},\left\{{\delta }_n\right\}\subseteq \left[0,1\right]$ and ${\alpha }_n+{\beta }_n+{\gamma }_n+{\delta }_n=1$ and $x_n=x_n^1,x_n^2=A_2{x}_n^1,x_n^3=A_3{x}_n^2,$ for all $n\in \mathbb{N},0<{\lambda }_i<1,$ for all $i=1,2,3$ and ${\gamma }_j>0,$ for all $j=2,3$. Suppose that the conditions (i) to (v) are true;

(i)

${lim}_{n\to +\infty }{\alpha }_n=0,{\sum }_{n=1}^{+\infty }{\alpha }_n=+\infty$;

(ii)

${\gamma }_2\left(1+{\gamma }_3\right)\le {\displaystyle \frac{1}{L}}$, where $L=max\left\{L_{A_1},L_{A_2}\right\}\le 1$, where $L_{A_1},L_{A_2}$ are spectral radii of $A_2{A}_2^{*},A_3{A}_3^{*}$, respectively;

(iii)

$0<a\le {\beta }_n,{\gamma }_n,{\delta }_n\le b<1,forsomea,b>0,foralln\in \mathbb{N}$;

(iv)

${\sum }_{n=1}^{+\infty }{\lambda }_n<+\infty and0<{\lambda }_n<1-\kappa ,foralln\in \mathbb{N}$;

(v)

${\sum }_{n=1}^{+\infty }\left|{\alpha }_{n+i}-{\alpha }_n\right|,{\sum }_{n=1}^{+\infty }\left|{\beta }_{n+i}-{\beta }_n\right|,{\sum }_{n=1}^{+\infty }\left|{\gamma }_{n+1}-{\gamma }_n\right|<+\infty .$

Then the sequence $\left\{x_n\right\}$ converses strongly to $z_0=P_{\underline{\Omega }\cap F\left(T\right)\cap F\left(S\right)}u$.

4. Conclusions

We have proposed a new split variational inequality in three Hilbert spaces. The convergence theorem for finding a common element of the set of solutions of such problems and the sets of fixed-points of discontinuous mappings are proved.