A system of nonlinear set valued variational inclusions

Abstract

In this paper, we studied the existence theorems and techniques for finding the solutions of a system of nonlinear set valued variational inclusions in Hilbert spaces. To overcome the difficulties, due to the presence of a proper convex lower semicontinuous function ϕ and a mapping g which appeared in the considered problems, we have used the resolvent operator technique to suggest an iterative algorithm to compute approximate solutions of the system of nonlinear set valued variational inclusions. The convergence of the iterative sequences generated by algorithm is also proved.

AMS Mathematics subject classification

49J40; 47H06

Introduction

It is well known that variational inequality theory and complementarity problems are very powerful tools of current mathematical technology. In recent years, the classical variational inequality and complementarity problems have been extended and generalized to study a large variety of problems arising in economics, control problems, contact problems, mechanics, transportation, equilibrium problems, optimization theory, nonlinear programming, transportation equilibrium and engineering sciences, see (Aubin 1982; Baiocchi and Capelo 1984; Chang 1984; Giannessi and Maugeri 1995). Hassouni and Moudafi 2001 introduced and studied a class of mixed type variational inequalities with single valued mappings which was called variational inclusions. Since many authors have obtained important extension generalizations of the results in (Hassouni and Moudafi 2001) from various directions, see (Agarwal et al. 2011; Fang et al. 2005; Kassay and Kolumban 2000; Petrot 2010). Verma 1999; 2001a introduced and studied some system of variational inequalities with iterative algorithms to compute approximate solutions in Hilbert spaces.

Inspired and motivated by the research work going on this field, in this works, the methods for finding the common solutions of a system of nonlinear set valued variational inclusions involving different nonlinear operators and fixed point problem are considered and studied, via proximal method in the framework of Hilbert spaces.

Since the problems of a system of a nonlinear set valued variational inequalities and fixed point are both important, the results present in this paper are useful and can be viewed as an improvement and extension of the previously known results appearing in literature, which are improves the results of Chang et al. 2007 and also extends the results of Verma 2001b; 2002, Ahmad and Salahuddin 2012, Ding and Luo 2000, Inchan and Petrot 2011, Kim and Kim 2004, Kim and Hu 2008, Nie et al. 2003 and Suantai and Petrot 2011, etc.

Let H be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥ respectively and K be a nonempty closed convex subset of H. Let C B(H) be the family of all nonempty closed convex and bounded sets in H and ϕ:H→(−∞,+∞) be a proper convex lower semicontinuous function on H. Let N _i :H×H→H be a nonlinear function, g _i :K→H be a nonlinear operator, A _i ,B _i :K→C B(H) be the nonlinear set valued mappings and let r _i be a fixed positive real number for each i=1,2,3. Set $\Xi =\{N_1,N_2,N_3\},\mathfrak{A}=\{A_1,A_2,A_3\},\mathfrak{B}=\{B_1,B_2,B_3\},\wedge =\{g_1,g_2,g_3\}.$The system of nonlinear set valued variational inclusions involving three different nonlinear operators is defined as follows:

Find $(x^{\ast },y^{\ast },z^{\ast })\in H\times H\times H,u_3^{\ast }\in A_3\left(x^{\ast }\right),v_3^{\ast }\in B_3\left(x^{\ast }\right),u_2^{\ast }\in A_2\left(z^{\ast }\right),v_2^{\ast }\in B_2\left(z^{\ast }\right),u_1^{\ast }\in A_1\left(y^{\ast }\right),v_1^{\ast }\in B_1\left(y^{\ast }\right),$ such that

$$\begin{array}{l}\left\{\begin{array}{l}〈r_1{N}_1(u_1^{\ast },v_1^{\ast })+g_1\left(x^{\ast }\right)-g_1\left(y^{\ast }\right),g_1\left(x\right)-g_1\left(x^{\ast }\right)〉-r_1\varphi \left(g_1\right(x^{\ast }\left)\right)+r_1\varphi \left(g_1\right(x\left)\right)\ge 0,g_1\left(x\right)\in K,\\ 〈r_2{N}_2(u_2^{\ast },v_2^{\ast })+g_2\left(y^{\ast }\right)-g_2\left(z^{\ast }\right),g_2\left(x\right)-g_2\left(y^{\ast }\right)〉-r_2\varphi \left(g_2\right(y^{\ast }\left)\right)+r_2\varphi \left(g_2\right(x\left)\right)\ge 0,g_2\left(x\right)\in K,\\ 〈r_3{N}_3(u_3^{\ast },v_3^{\ast })+g_3\left(z^{\ast }\right)-g_3\left(x^{\ast }\right),g_3\left(x\right)-g_3\left(z^{\ast }\right)〉-r_3\varphi \left(g_3\right(z^{\ast }\left)\right)+r_3\varphi \left(g_3\right(x\left)\right)\ge 0,g_3\left(x\right)\in \mathrm{K.}\end{array}\right.\end{array}$$

(1)

We denote the set of all solutions $(x^{\ast },y^{\ast },z^{\ast },u_1^{\ast },v_1^{\ast },u_2^{\ast },v_2^{\ast },u_3^{\ast },v_3^{\ast })$ of problem (1) by SNSVVID$(\Xi ,\mathfrak{A},\mathfrak{B},\wedge ,K)$.

We first recall some basic concepts and well known results.

Definition 1

A mapping g:H→H is said to be

• (i) monotone, if

  $$〈g\left(x\right)-g\left(y\right),x-y〉\ge 0\forall x,y\in H;$$

• (ii) strictly monotone, if g is monotone and

  $$〈g\left(x\right)-g\left(y\right),x-y〉=0\mathit{\text{ifandonlyif}}x=y;$$

• (iii) υ-strongly monotone, if there exists a constant υ>0 such that

  $$〈g\left(x\right)-g\left(y\right),x-y〉\ge \upsilon \parallel x-y{\parallel }^2,\forall x,y\in H;$$

• (iv) Lipschitz continuous, if there exists a constant υ>0 such that

  $$\parallel g\left(x\right)-g\left(y\right)\parallel \le \upsilon \parallel x-y\parallel ,\forall x,y\in \mathrm{H.}$$

Definition 2

A set valued mapping A:H→2^H is said to be υ -strongly monotone, if there exists a constant υ>0 such that

$$\begin{array}{l}〈w_1-w_2,x-y〉\ge \upsilon \parallel x-y{\parallel }^2,\forall x,y\in H,w_1\in A\left(x\right),w_2\in \mathrm{Ay.}\end{array}$$

Definition 3

A set valued mapping A:H→C B(H) is said to be τ-Lipschitz continuous if there exists a constant τ>0 such that

$$\begin{array}{l}\mathcal{\mathscr{H}}(\mathit{\text{Ax}},\mathit{\text{Ay}})\le \tau \parallel x-y\parallel ,\forall x,y\in H,\end{array}$$

where (·,·) is the Hausdorff metric on CB().

Definition 4

(Brezis 1973)

If M is maximal monotone operator on H then for any λ>0the resolvent operator associated with M is defined by

$$\begin{array}{l}{J}_M\left(x\right)={(I+\mathrm{\lambda M})}^{-1}\left(x\right),\forall x\in \mathrm{H.}\end{array}$$

It is well know that a monotone operator is maximal iff its resolvent operator is defined every where. Furthermore the resolvent operator is single valued and nonexpansive. In particular the subdifferential ∂ ϕ of a proper convex lower semicontinuous function ϕ:H→(−∞,+∞) is a maximal monotone operator.

Lemma 1

(Brezis 1973) The points u,z∈H satisfies the inequality

$$\begin{array}{l}〈u-z,x-u〉+\mathrm{\lambda \varphi }\left(x\right)-\mathrm{\lambda \varphi }\left(u\right)\ge 0,\forall x\in H,\end{array}$$

if and only if

$$\begin{array}{l}u=J_{\varphi }^{\lambda }\left(z\right),\end{array}$$

where $J_{\varphi }^{\lambda }={(I+\mathrm{\lambda \partial \varphi })}^{-1}$ is a resolvent operator and λ>0 is a constant.

For any $x,y\in H,J_{\varphi }^{\lambda }$ is nonexpansive, i.e.,

$$\begin{array}{l}\parallel J_{\varphi }^{\lambda }\left(x\right)-J_{\varphi }^{\lambda }\left(y\right)\parallel \le \parallel x-y\parallel ,\forall x,y\in \mathrm{H.}\end{array}$$

Assume that g:H→H is a surjective mapping and from Lemma 1 and (1) we have the following proximal point problem:

$$\begin{array}{l}\left\{\begin{array}{ll}{g}_1\left(x^{\ast }\right)& =J_{\varphi }^{r_1}\left[g_1\right(y^{\ast })-r_1{N}_1(u_1^{\ast },v_1^{\ast }\left)\right],\\ g_2\left(y^{\ast }\right)& =J_{\varphi }^{r_2}\left[g_2\right(z^{\ast })-r_2{N}_2(u_2^{\ast },v_2^{\ast }\left)\right],\\ g_3\left(z^{\ast }\right)& =J_{\varphi }^{r_3}\left[g_3\right(x^{\ast })-r_3{N}_3(u_3^{\ast },v_3^{\ast }\left)\right],\end{array}\right.\end{array}$$

(2)

provided K⊂g _i (H) for each i=1,2,3.

Lemma 2

(Weng 1991)

Let {a _n },{b _n } and {c _n } be three sequences of nonnegative real numbers such that

$$\begin{array}{l}{a}_{n+1}\le (1-t_n)a_n+b_n+c_n\forall n>n_0,\end{array}$$

where n₀ is a nonnegative integer, {t _n } is a sequence in (0,1) with ${\sum }_{n=0}^{\infty }{t}_n=+\infty ,{\text{lim}}_{n\to \infty }{b}_n=0\left(t_n\right)$ and ${\sum }_{n=0}^{\infty }{c}_n<+\infty$. Than a _n →0 as n→+∞.

Definition 5

Let A,B:H→2^H be set valued mappings and N:H×H→H be a nonlinear mapping.

• (i) N is said to be A-strongly monotone with respect to the first argument, if there exists a constant υ>0 such that for all x,y∈H

  $$\begin{array}{l}〈N(u_1,w)-N(u_2,w),x-y〉\ge \upsilon \parallel x-y{\parallel }^2\forall u_1\in A\left(x\right),u_2\in A\left(y\right),w\in H;\end{array}$$

• (ii) N is said to be B-relaxed monotone with respect to the second argument, if there exists a constant ξ>0 such that for all x,y∈H,v ₁∈B(x),v ₂∈B(y)

  $$\begin{array}{l}〈N(u,v_1)-N(u,v_2),x-y〉\ge -\xi \parallel x-y{\parallel }^2,\forall u\in \mathrm{H.}\end{array}$$

Main results

We begin with some observations which are related to the problem (1).

Remark 1

If (x^∗,y^∗,z^∗)∈ SNSVVID$(\Xi ,\mathfrak{A},\mathfrak{B},\wedge ,K)$, by (2) we have that

$$\begin{array}{l}{x}^{\ast }=x^{\ast }-g_1\left(x^{\ast }\right)+J_{\varphi }^{r_1}\left[g_1\left(y^{\ast }\right)-r_1{N}_1(u_1^{\ast },v_1^{\ast })\right].\end{array}$$

(3)

provided K⊂g₁(H).

Consequently if S is a Lipschitz mapping such that x^∗∈F(S), then it follows from (3) that

$$\begin{array}{l}{x}^{\ast }=S\left(x^{\ast }\right)=S(x^{\ast }-g_1(x^{\ast })+J_{\varphi }^{r_1}\left[g_1\left(y^{\ast }\right)-r_1{N}_1(u_1^{\ast },v_1^{\ast })\right]).\end{array}$$

(4)

By virtue of (4) and Nadler’s Theorem (Nadler 1969), we suggest the following iterative algorithm.

Algorithm 1 Let ε _n be a sequence of nonnegative real number with ε _n →0 as n→∞. Let r₁,r₂,r₃ be three given positive real numbers in (0,1). For arbitrary chosen initial x₀∈H, compute the sequences {x _n },{y _n } and {z _n } in H, such that

$$\begin{array}{l}\left\{\begin{array}{ll}{g}_3\left(z_n\right)& =J_{\varphi }^{r_3}\left[g_3\left(x_n\right)-r_3{N}_3(u_{n,3},v_{n,3})\right],\\ g_2\left(y_n\right)& =J_{\varphi }^{r_2}\left[g_2\left(z_n\right)-r_2{N}_2(u_{n,2},v_{n,2})\right],\forall n\ge 1\\ x_{n+1}& =(1-{\alpha }_n)x_n+{\alpha }_{n}S(x_n-g_1(x_n)+J_{\varphi }^{r_1}\left[g_1\left(y_n\right)-r_1{N}_1(u_{n,1},v_{n,1})\right]),\end{array}\right.\end{array}$$

(5)

where

$$\begin{array}{l}\left\{\begin{array}{l}{u}_{n,3}\in A_3\left(x_n\right),u_{n-1,3}\in A_3\left(x_{n-1}\right):\parallel u_{n,3}-u_{n-1,3}\parallel \le (1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(A_3\right(x_n),A_3(x_{n-1}\left)\right),\\ v_{n,3}\in B_3\left(x_n\right),v_{n-1,3}\in B_3\left(x_{n-1}\right):\parallel v_{n,3}-v_{n-1,3}\parallel \le (1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(B_3\right(x_n),B_3(x_{n-1}\left)\right),\\ u_{n,2}\in A_2\left(z_n\right),u_{n-1,2}\in A_2\left(z_{n-1}\right):\parallel u_{n,2}-u_{n-1,2}\parallel \le (1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(A_2\right(z_n),A_2(z_{n-1}\left)\right),\\ v_{n,2}\in B_2\left(z_n\right),v_{n-1,2}\in B_2\left(z_{n-1}\right):\parallel v_{n,2}-v_{n-1,2}\parallel \le (1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(B_2\right(z_n),B_2(z_{n-1}\left)\right),\\ u_{n,1}\in A_1\left(y_n\right),u_{n-1,1}\in A_1\left(y_{n-1}\right):\parallel u_{n,1}-u_{n-1,1}\parallel \le (1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(A_1\right(y_n),A_1(y_{n-1}\left)\right),\\ v_{n,1}\in B_1\left(y_n\right),v_{n-1,1}\in B_1\left(y_{n-1}\right):\parallel v_{n,1}-v_{n-1,1}\parallel \le (1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(B_1\right(y_n),B_1(y_{n-1}\left)\right),\end{array}\right.\end{array}$$

(6)

and {α _n } is a sequence in (0,1) and S:H→H is a mapping.

Theorem 1

Let K be a nonempty closed and convex subset of a real Hilbert space H and ϕ:H→(−∞,+∞) be a proper convex lower semicontinuous function. Let A _i :H→2^H be a μ _i -Lipschitz continuous mapping with μ _i <1 and B _i :H→2^H be a σ _i -Lipschitz continuous mapping with σ _i <1, i=1,2,3. Let N _i :H×H→H be a ρ _i -Lipschitz continuous with respect to the first variable and η _i -Lipschitz continuous with respect to the second variable and N _i be A _i -strongly monotone with constant υ _i >0 and B _i -relaxed monotone with constant ξ _i >0, i=1,2,3. Let g _i :H→H be a λ _i -strongly monotone and γ _i -Lipschitz continuous mapping, i=1,2,3. Let S:H→H be a τ-Lipschitz continuous mapping with 0<τ≤1. If $\mathit{\text{SNSVVID}}(\Xi ,\mathfrak{A},\mathfrak{B},\wedge ,K)\cap F\left(S\right)\ne \varnothing$, and the following conditions are satisfied:

• (i)
  $$\begin{array}{l}{h}_i\in \left[0,\right.\left.\frac{({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)-\sqrt{{({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)}^2-{({\upsilon }_i-{\xi }_i)}^2}}{2({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)}\right)\bigcup \left[\right.\left.\frac{({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)+\sqrt{{({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)}^2-{({\upsilon }_i-{\xi }_i)}^2}}{2({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)},1\right)\end{array}$$

  where $h_i=\sqrt{1-2{\lambda }_i+{\gamma }_i^2},i=1,2,3$;

• (ii)
  $$\begin{array}{l}|r_i-\frac{{\upsilon }_i-{\xi }_i}{{({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)}^2}|<\frac{\sqrt{{({\upsilon }_i-{\xi }_i)}^2-{({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)}^2\left(4h_i\right)(1-h_i)}}{{({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i)}^2},i=1,2,3;\end{array}$$

• (iii) for each i=1,2,3

  $$\begin{array}{l}\frac{{\Phi }_{n,N_i}\left(r_i\right)+h_i}{1-h_i}\le \frac{{\Phi }_{N_i}\left(r_i\right)+h_i}{1-h_i}<1,\end{array}$$

  where

  $$\begin{array}{l}\left\{\begin{array}{l}{\Phi }_{N_i}\left(r_i\right)=\sqrt{1-2r_i({\upsilon }_i-{\xi }_i)+r_i^2{\left(\right({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i\left)\right(1+M\left)\right)}^2};\\ {\Phi }_{n,N_i}\left(r_i\right)=\sqrt{1-2r_i({\upsilon }_i-{\xi }_i)+r_i^2{\left(\right({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i\left)\right(1+{\epsilon }_n\left)\right)}^2};\end{array}\right.\end{array}$$

  (7)

  where M= supn≥1ε _n .

• (iv) {α _n }⊂(0,1) such that ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$.

Then the sequences {x _n },{y _n },{z _n },{u_n,i},{v_n,i} suggested by Algorithm 1 converge strongly to $x^{\ast },y^{\ast },z^{\ast },u_i^{\ast },v_i^{\ast }i=1,2,3$ respectively, and $(x^{\ast },y^{\ast },z^{\ast },u_i^{\ast },v_i^{\ast })\in$ SNSVVID$(\Xi ,\mathfrak{A},\mathfrak{B},\wedge ,K)$, x^∗∈F(S).

Proof. Let $(x^{\ast },y^{\ast },z^{\ast },u_i^{\ast },v_i^{\ast })\in \mathit{\text{SNSVVID}}(\Xi ,\mathfrak{A},\mathfrak{B},\wedge ,K)$ and x^∗∈F(S). By (2) and (4) we have

$$\begin{array}{l}\left\{\begin{array}{l}{g}_3\left(z^{\ast }\right)=J_{\varphi }^{r_3}\left[g_3\right(x^{\ast })-r_3{N}_3(u_3^{\ast },v_3^{\ast }\left)\right],\\ g_2\left(y^{\ast }\right)=J_{\varphi }^{r_2}\left[g_2\right(z^{\ast })-r_2{N}_2(u_2^{\ast },v_2^{\ast }\left)\right],\\ x^{\ast }=(1-{\alpha }_n)x^{\ast }+{\alpha }_{n}S(x^{\ast }-g_1(x^{\ast })+J_{\varphi }^{r_1}[g_1\left(y^{\ast }\right)-r_1{N}_1(u_1^{\ast },v_1^{\ast })\left]\right)\end{array}\right.\end{array}$$

(8)

Consequently, by (5) and (6), we have

$$\begin{array}{l}\begin{array}{l}\parallel x_{n+1}-x^{\ast }\parallel \\ =\parallel (1-{\alpha }_n)x_n+{\alpha }_{n}S(x_n-g_1(x_n)+J_{\varphi }^{r_1}[g_1\left(y_n\right)-r_1{N}_1(n_{n,1},v_{n,1})\left]\right)-x^{\ast }\parallel \\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n\parallel S(x_n-g_1(x_n)+J_{\varphi }^{r_1}[g_1\left(y_n\right)-r_1{N}_1(u_{n,1},v_{n,1})\left]\right)\\ -S(x^{\ast }-g_1(x^{\ast })+J_{\varphi }^{r_1}[g_1\left(y^{\ast }\right)-r_1{N}_1(u_1^{\ast },v_1^{\ast })\left]\right)\parallel \\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n\tau [\parallel x_n-x^{\ast }-(g_1\left(x_n\right)-g_1\left(x^{\ast }\right))\parallel \\ +\parallel J_{\varphi }^{r_1}\left[g_1\right(y_n)-r_1{N}_1(u_{n,1},v_{n,1}\left)\right]-J_{\varphi }^{r_1}\left[g_1\right(y^{\ast })-r_1{N}_1(u_1^{\ast },v_1^{\ast }\left)\right]\parallel ]\\ \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel +{\alpha }_n\tau [\parallel x_n-x^{\ast }-(g_1\left(x_n\right)-g_1\left(x^{\ast }\right))\parallel \\ +\parallel y_n-y^{\ast }-\left(g_1\right(y_n)-g_1(y^{\ast }\left)\right)\parallel +\parallel y_n-y^{\ast }-r_1\left(N_1\right(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }\left)\right)\parallel ].\end{array}\end{array}$$

(9)

Since N₁(·,·) is ρ₁-Lipschitz continuous with respect to the first variable and η₁-Lipschitz continuous with respect to the second variable, and A₁ is μ₁-Lipschitz continuous, and B₁ is σ₁-Lipschitz continuous, we have

$$\begin{array}{l}\parallel N_1(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast })\parallel \le {\rho }_1\parallel u_{n,1}-u_1^{\ast }\parallel +{\eta }_1\parallel v_{n,1}-v_1^{\ast }\parallel \\ \le {\rho }_1(1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(A_1\right(y_n),A_1(y^{\ast }\left)\right)+{\eta }_1(1+{\epsilon }_n)\mathcal{\mathscr{H}}\left(B_1\right(y_n),B_1(y^{\ast }\left)\right)\\ \le {\rho }_1{\mu }_1(1+{\epsilon }_n)\parallel y_n-y^{\ast }\parallel +{\eta }_1{\sigma }_1(1+{\epsilon }_n)\parallel y_n-y^{\ast }\parallel \\ \le ({\rho }_1{\mu }_1+{\eta }_1{\sigma }_1)(1+{\epsilon }_n)\parallel y_n-y^{\ast }\parallel .\end{array}$$

(10)

Since N₁ is A₁-strongly monotone with constant υ₁>0 and B₁-relaxed monotone with constant ξ _i >0, it follows from (10) that

$$\begin{array}{l}\parallel y_n-y^{\ast }-r_1\left(N_1\right(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }\left)\right){\parallel }^2\\ =& \parallel y_n-y^{\ast }{\parallel }^2-2r_1〈N_1(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }),y_n-y^{\ast }〉\\ +r_1^2\parallel N_1(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }){\parallel }^2\\ =& \parallel y_n-y^{\ast }{\parallel }^2-2r_1〈N_1(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_{n,1}),y_n-y^{\ast }〉\\ -2r_1〈N_1(u_1^{\ast },v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }),y_n-y^{\ast }〉+r_1^2\parallel N_1(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }){\parallel }^2\\ \le & \parallel y_n-y^{\ast }{\parallel }^2-2r_1{\upsilon }_1\parallel y_n-y^{\ast }{\parallel }^2+2r_1{\xi }_1\parallel y_n-y^{\ast }{\parallel }^2\\ +r_1^2{\left(\right({\rho }_1{\mu }_1+{\eta }_1{\sigma }_1\left)\right(1+{\epsilon }_n\left)\right)}^2\parallel y_n-y^{\ast }{\parallel }^2\\ \le & (1-2r_1{\upsilon }_1+2r_1{\xi }_1+r_1^2{\left(\right({\rho }_1{\mu }_1+{\eta }_1{\sigma }_1\left)\right(1+{\epsilon }_n\left)\right)}^2)\parallel y_n-y^{\ast }{\parallel }^2\end{array}$$

i.e.,

$$\begin{array}{l}\parallel y_n-y^{\ast }-r_1\left(N_1\right(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }\left)\right){\parallel }^2\le {\left({{\Phi }_n}_{N_1}\right(r_1\left)\right)}^2\parallel y_n-y^{\ast }{\parallel }^2,\end{array}$$

(11)

where

$$\begin{array}{l}{\Phi }_{n,N_1}\left(r_1\right):=\sqrt{1-2r_1({\upsilon }_1-{\xi }_1)+r_1^2{\left(\right({\rho }_1{\mu }_1+{\eta }_1{\sigma }_1\left)\right(1+{\epsilon }_n\left)\right)}^2}.\end{array}$$

Note that

$$\begin{array}{lll}\parallel y_n& -& y^{\ast }\parallel =\parallel y_n-y^{\ast }-\left[g_2\right(y_n)-g_2(y^{\ast }\left)\right]+\left[g_2\right(y_n)-g_2(y^{\ast }\left)\right]\parallel \\ \le & \parallel y_n-y^{\ast }-\left[g_2\right(y_n)-g_2(y^{\ast }\left)\right]\parallel +\parallel g_2\left(y_n\right)-g_2\left(y^{\ast }\right)\parallel .\end{array}$$

(12)

Since g₂ is λ₂-strongly monotone and γ₂-Lipschitz continuous mapping, we have

$$\begin{array}{ll}\parallel & y_n-y^{\ast }-\left[g_2\right(y_n)-g_2(y^{\ast }\left)\right]{\parallel }^2\\ =& \parallel y_n-y^{\ast }{\parallel }^2-2〈g_2\left(y_n\right)-g_2\left(y^{\ast }\right),y_n-y^{\ast }〉+\parallel g_2\left(y_n\right)-g_2\left(y^{\ast }\right){\parallel }^2\\ \le & \parallel y_n-y^{\ast }{\parallel }^2-2{\lambda }_2\parallel y_n-y^{\ast }{\parallel }^2+{\gamma }_2^2\parallel y_n-y^{\ast }{\parallel }^2\\ \le & (1-2{\lambda }_2+{\gamma }_2^2)\parallel y_n-y^{\ast }{\parallel }^2\\ =& {\left(h_2\right)}^2\parallel y_n-y^{\ast }{\parallel }^2,\end{array}$$

(13)

where $h_2=\sqrt{1-2{\lambda }_2+{\gamma }_2^2}$.

On the other hand, by (2) and (5), we have

$$\begin{array}{ll}\parallel & g_2\left(y_n\right)-g_2\left(y^{\ast }\right)\parallel \\ =& \parallel J_{\varphi }^{r_2}\left[g_2\right(z_n)-r_2{N}_2(u_{n,2},v_{n,2}\left)\right]-J_{\varphi }^{r_2}\left[g_2\right(z^{\ast })-r_2{N}_2(u_2^{\ast },v_2^{\ast }\left)\right]\parallel \\ \le & \parallel g_2\left(z_n\right)-g_2\left(z^{\ast }\right)-r_2\left(N_2\right(u_{n,2},v_{n,2})-N_2(u_2^{\ast },v_2^{\ast }\left)\right)\parallel \\ \le & \parallel z_n-z^{\ast }-\left(g_2\right(z_n)-g_2(z^{\ast }\left)\right)\parallel +\parallel z_n-z^{\ast }-r_2\left(N_2\right(u_{n,2},v_{n,2})-N_2(u_2^{\ast },v_2^{\ast }\left)\right)\parallel .\end{array}$$

(14)

In view of the assumptions of N₂,A₂,B₂, g₂ and by using the same method as given in the proofs in (11) and (13), we can obtain that

$$\begin{array}{l}\parallel z_n-z^{\ast }-r_2\left(N_2\right(u_{n,2},v_{n,2})-N_2(u_2^{\ast },v_2^{\ast }\left)\right){\parallel }^2\le {\left({\Phi }_{n,N_2}\right(r_2\left)\right)}^2\parallel z_n-z^{\ast }{\parallel }^2,\end{array}$$

(15)

where

$$\begin{array}{l}\left({\Phi }_{n,N_2}\right(r_2)=\sqrt{1-2r_2({\upsilon }_2-{\xi }_2)+r_2^2{\left(\right({\rho }_2{\mu }_2+{\eta }_2{\sigma }_2\left)\right(1+{\epsilon }_n\left)\right)}^2}\end{array}$$

and

$$\begin{array}{l}\parallel z_n-z^{\ast }-\left(g_2\right(z_n)-g_2(z^{\ast }\left)\right){\parallel }^2\le {\left(h_2\right)}^2\parallel z_n-z^{\ast }{\parallel }^2.\end{array}$$

(16)

From (15), (16) and (14), we have

$$\begin{array}{l}\parallel g_2\left(y_n\right)-g_2\left(y^{\ast }\right)\parallel \le \left({\Phi }_{n,N_2}\right(r_2)+h_2)\parallel z_n-z^{\ast }\parallel .\end{array}$$

(17)

Combining (12), (13) and (17) we obtained

$$\begin{array}{l}\parallel y_n-y^{\ast }\parallel \le h_2\parallel y_n-y^{\ast }\parallel +\left({{\Phi }_n}_{N_2}\right(r_2)+h_2)\parallel z_n-z^{\ast }\parallel .\end{array}$$

(18)

Observe that

$$\begin{array}{lll}\parallel z_n-z^{\ast }\parallel & =& \parallel z_n-z^{\ast }-\left[g_3\right(z_n)-g_3(z^{\ast }\left)\right]+\left[g_3\right(z_n)-g_3(z^{\ast }\left)\right]\parallel \\ \le & \parallel z_n-z^{\ast }-\left[g_3\right(z_n)-g_3(z^{\ast }\left)\right]\parallel +\parallel g_3\left(z_n\right)-g_3\left(z^{\ast }\right)\parallel .\end{array}$$

(19)

and in view of (2) and (5), we have

$$\begin{array}{lll}\parallel g_3\left(z_n\right)-g_3\left(z^{\ast }\right)\parallel & \le & \parallel x_n-x^{\ast }-\left[g_3\right(x_n)-g_3(x^{\ast }\left)\right]\parallel \\ +\parallel x_n-x^{\ast }-r_3\left(N_3\right(u_{n,3},v_{n,3})-N_3(u_3^{\ast },v_3^{\ast }\left)\right)\parallel .\end{array}$$

(20)

By using the assumptions on N₃,A₃,B₃ and g₃, we have

$$\begin{array}{l}\parallel x_n-x^{\ast }-r_3\left(N_3\right(u_{n,3},v_{n,3})-N_3(u_3^{\ast },v_3^{\ast }\left)\right){\parallel }^2\le {\left({\Phi }_{n,N_3}\right(r_3\left)\right)}^2\parallel x_n-x^{\ast }{\parallel }^2.\end{array}$$

(21)

where

$$\begin{array}{lll}{\Phi }_{n,N_3}\left(r_3\right)& =& \sqrt{1-2r_3({\upsilon }_3-{\xi }_3)+r_3^2{\left(\right({\rho }_3{\mu }_3+{\eta }_3{\sigma }_3\left)\right(1+{\epsilon }_n\left)\right)}^2}\\ \parallel x_n-x^{\ast }& -& \left[g_3\right(x_n)-g_3(x^{\ast }\left)\right]{\parallel }^2\le {\left(h_3\right)}^2\parallel x_n-x^{\ast }{\parallel }^2.\end{array}$$

(22)

$$\begin{array}{l}\parallel z_n-z^{\ast }-\left[g_3\right(z_n)-g_3(z^{\ast }\left)\right]{\parallel }^2\le {\left(h_3\right)}^2\parallel z_n-z^{\ast }{\parallel }^2.\end{array}$$

(23)

Substituting (21) and (22) into (20), we have

$$\begin{array}{l}\parallel g_3\left(z_n\right)-g_3\left(z^{\ast }\right)\parallel \le \left({\Phi }_{n,N_3}\right(r_3)+h_3)\parallel x_n-x^{\ast }\parallel .\end{array}$$

(24)

Combining (19), (23) and (24), it yields that

$$\begin{array}{l}\parallel z_n-z^{\ast }\parallel \le h_3\parallel z_n-z^{\ast }\parallel +\left({\Phi }_{n,N_3}\right(r_3)+h_3)\parallel x_n-x^{\ast }\parallel .\end{array}$$

(25)

This imply that

$$\begin{array}{l}\parallel z_n-z^{\ast }\parallel \le \frac{\left({\Phi }_{n,N_3}\right(r_3)+h_3)}{1-h_3}\parallel x_n-x^{\ast }\parallel .\end{array}$$

(26)

Substituting (26) into (18) we have

$$\begin{array}{ll}\parallel & y_n-y^{\ast }\parallel \le h_2\parallel y_n-y^{\ast }\parallel +\frac{\left({\Phi }_{n,N_2}\right(r_2)+h_2)\left({\Phi }_{n,N_3}\right(r_3)+h_3)}{1-h_3}\parallel x_n-x^{\ast }\parallel ,\end{array}$$

(27)

that is

$$\begin{array}{l}\parallel y_n-y^{\ast }\parallel \le \frac{\left({\Phi }_{n,N_2}\right(r_2)+h_2)\left({\Phi }_{n,N_3}\right(r_3)+h_3)}{(1-h_2)(1-h_3)}\parallel x_n-x^{\ast }\parallel .\end{array}$$

(28)

From (11) and (28), we get

$$\begin{array}{ll}\parallel & y_n-y^{\ast }-r_1\left[N_1\right(u_{n,1},v_{n,1})-N_1(u_1^{\ast },v_1^{\ast }\left)\right]\parallel \\ \le & \frac{\left({\Phi }_{n,N_1}\right(r_1\left)\right)\left({\Phi }_{n,N_2}\right(r_2)+h_2)\left({\Phi }_{n,N_3}\right(r_3)+h_3)}{(1-h_2)(1-h_3)}\parallel x_n-x^{\ast }\parallel .\end{array}$$

(29)

On the other hand, since g₁ is λ₁-strongly monotone and γ₁-Lipschitz continuous mapping, we have

$$\begin{array}{ll}\parallel x_n-x^{\ast }-& \left(g_1\right(x_n)-g_1(x^{\ast }\left)\right){\parallel }^2=\left|\right|x_n-x^{\ast }|{|}^2+|\left|g_1\right(x_n)-g_1(x^{\ast }\left)\right|{|}^2\\ -2〈x_n-x^{\ast },g_1\left(x_n\right)-g_1\left(x^{\ast }\right)〉\\ \le (1-2{\lambda }_1+{\gamma }_1^2)\left|\right|x_n-x^{\ast }|{|}^2=h_1^2||x_n-x^{\ast }|{|}^2,\end{array}$$

i.e.,

$$\begin{array}{l}\parallel x_n-x^{\ast }-\left(g_1\right(x_n)-g_1(x^{\ast }\left)\right)\parallel \le h_1\parallel x_n-x^{\ast }\parallel .\end{array}$$

(30)

Similarly, we have

$$\begin{array}{l}\parallel y_n-y^{\ast }-\left(g_1\right(y_n)-g_1(y^{\ast }\left)\right)\parallel \le h_1\parallel y_n-y^{\ast }\parallel .\end{array}$$

(31)

Substituting (28) into (31), we have

$$\begin{array}{ll}\parallel & y_n-y^{\ast }-\left(g_1\right(y_n)-g_1(y^{\ast }\left)\right)\parallel \\ \le & h_1\frac{\left({\Phi }_{n,N_2}\right(r_2)+h_2)\left({\Phi }_{n,N_3}\right(r_3)+h_3)}{(1-h_2)(1-h_3)}\parallel x_n-x^{\ast }\parallel .\end{array}$$

(32)

Set

$$\begin{array}{l}{\ell }_n=\frac{\left({\Phi }_{n,N_2}\right(r_2)+h_2)\left({\Phi }_{n,N_3}\right(r_3)+h_3)}{(1-h_2)(1-h_3)}.\end{array}$$

(33)

Substituting (30), (31), (32) and (33) into (9), we get

$$\begin{array}{ll}\parallel & x_{n+1}-x^{\ast }\parallel \le (1-{\alpha }_n(1-\tau (h_1+h_1{\ell }_n+{\Phi }_{n,N_1}(r_1\left){\ell }_n\right)\left)\right)\parallel x_n-x^{\ast }\parallel .\end{array}$$

(34)

Since

$$\begin{array}{lll}{\Phi }_{n,N_i}\left(r_i\right)& :=& \sqrt{1-2r_i({\upsilon }_i-{\xi }_n)+r_i^2{\left(\right({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i\left)\right(1+{\epsilon }_n\left)\right)}^2}\\ \le & \sqrt{1-2r_i({\upsilon }_i-{\xi }_n)+r_i^2{\left(\right({\rho }_i{\mu }_i+{\eta }_i{\sigma }_i\left)\right(1+M\left)\right)}^2}:={\Phi }_{N_i}\left(r_i\right),\end{array}$$

letting $\ell :=\frac{\left({\Phi }_{N_2}\right(r_2)+h_2)\left({\Phi }_{N_3}\right(r_3)+h_3)}{(1-h_2)(1-h_3)}$, then we have ℓ _n ≤ℓ. Therefore from (34) we have that

$$\begin{array}{ll}\parallel & x_{n+1}-x^{\ast }\parallel \le (1-{\alpha }_n(1-\tau (h_1+h_1\ell +{\Phi }_{N_1}(r_1\left)\ell \right)\left)\right)\parallel x_n-x^{\ast }\parallel .\end{array}$$

(35)

By condition (iii)

$$\begin{array}{l}{\prod }_{i=1}^3\frac{{\Phi }_{N_i}\left(r_i\right)+h_i}{1-h_i}<1,\end{array}$$

(36)

this imply that

$$\begin{array}{l}\ell <\frac{1-h_1}{{\Phi }_{N_1}\left(r_1\right)+h_1}\end{array}$$

(37)

that is

$$\begin{array}{l}I:=h_1+h_1\ell +{\Phi }_{N_1}\left(r_1\right)\ell <1.\end{array}$$

(38)

Put

$$\begin{array}{l}\left\{\begin{array}{ll}{a}_n& =\parallel x_n-x^{\ast }\parallel \\ t_n& ={\alpha }_n(1-\mathrm{\tau I}).\end{array}\right.\end{array}$$

(39)

By the assumption that 0<τ≤1, it follows that

$$\begin{array}{l}\mathrm{\tau I}\in (0,1).\end{array}$$

This imply that t _n ∈(0,1). From assumption (iv) we have

$$\begin{array}{l}{\sum }_{n=0}^{\infty }{t}_n=\mathrm{\infty .}\end{array}$$

These show that all conditions in Lemma 2 are satisfied. Hence x _n →x^∗ as n→∞. Consequently from (26) and (28), we have z _n →z^∗ and y _n →y^∗ as n→∞, respectively. Moreover since A _i is μ _i -Lipschitz continuous and B _i is σ _i -Lipschitz continuous with μ _i <1, σ _i <1, we can also prove that {u_n,i} and {v_n,i}, i=1,2,3 are Cauchy sequences. Thus there exists $u_i^{\ast },v_i^{\ast }\in H$ such that $u_{n,i}\to u_i^{\ast },v_{n,i}\to v_i^{\ast },(i=1,2,3)$ as n→∞. Moreover by using the continuity of mappings $A_i,B_i,g_i,N_i,J_{\varphi }^{r_i}$, i=1,2,3, it follows from (5) that

$$\begin{array}{l}{g}_3\left(z^{\ast }\right)=J_{\varphi }^{r_3}\left[g_3\left(x^{\ast }\right)-r_3{N}_3(u_3^{\ast },v_3^{\ast })\right],\end{array}$$

$$\begin{array}{l}{g}_2\left(y^{\ast }\right)=J_{\varphi }^{r_2}\left[g_2\right(z^{\ast })-r_2{N}_2(u_2^{\ast },v_2^{\ast }\left)\right],\end{array}$$

$$\begin{array}{l}{x}^{\ast }=S(x^{\ast }-g_1(x^{\ast })+J_{\varphi }^{r_1}[g_1\left(y^{\ast }\right)-r_1{N}_1(u_1^{\ast },v_1^{\ast })\left]\right).\end{array}$$

Hence from Lemma 2 it follows that $(x^{\ast },y^{\ast },z^{\ast },u_i^{\ast },v_i^{\ast })\in$ SNSVVID$(\Xi ,\mathfrak{A},\mathfrak{B},\wedge ,K).$ Finally we prove that $u_i^{\ast }\in A_i\left(y^{\ast }\right)$ and $v_i^{\ast }\in B_1\left(y^{\ast }\right)$ Indeed we have

$$\begin{array}{lll}d(u_1^{\ast },A_1(y^{\ast }\left)\right)& =& inf\{\parallel \underset{1}{\overset{\ast }{u}}-w\parallel :w\in A_1(y^{\ast }\left)\right\}\\ \le & \parallel u_1^{\ast }-u_{n,1}\parallel +d(u_{n,1},A_1(y^{\ast }\left)\right)\\ \le & \parallel u_1^{\ast }-u_{n,1}\parallel +\mathcal{\mathscr{H}}\left(A_1\right(y_n),A_1(y^{\ast }\left)\right)\\ \le & \parallel u_1^{\ast }-u_{n,1}\parallel +{\mu }_1\parallel y_n-y^{\ast }\parallel \to 0\text{as}n\to \mathrm{\infty .}\end{array}$$

That is $d(u_1^{\ast },A_1(y^{\ast }\left)\right)=0$. Since A₁(y^∗)∈C B(H), we must have $u_1^{\ast }\in A_1\left(y^{\ast }\right)$. Similarly we can show that $u_2^{\ast }\in A_2\left(z^{\ast }\right),u_3^{\ast }\in A_3\left(x^{\ast }\right),v_1^{\ast }\in B_1\left(y^{\ast }\right),v_2^{\ast }\in B_2\left(z^{\ast }\right)$ and $v_3^{\ast }\in B_3\left(x^{\ast }\right).$ This complete the proof. â–