A new system of generalized quasi-variational-like inclusions with noncompact valued mappings

Abstract

In this article, we introduce and study a new system of generalized quasi-variational-like inclusions with noncompact valued mappings. By using the η- proximal mapping technique, we prove the existence of solutions and the convergence of some new N-step iterative algorithms for this system of generalized quasi-variational-like inclusions. Our results extend and improve some known results in the literature.

Mathematics Subject Classification 2000: 49H09; 49J40; 49H10.

1 Introduction

Let H be a real Hilbert space, and CB(H) be the family of all nonempty bounded closed subsets of H. We will consider the following problem:

For i, j = 1, 2, . . . , N, let A _ij : H → CB(H), η _i : H × H → H, g _i : H → H, $T_i:\underset{N}{\underbrace{H\times H\times \cdots \times H}}\to H$ be nonlinear mappings, and let φ _i : H → R ∪ {+∞} be real function.

Find$x_1^{*},x_2^{*},\dots ,x_N^{*}\in H$, $u_{11}^{*}\in A_{11}{x}_1^{*},u_{12}^{*}\in A_{12}{x}_2^{*},\dots ,u_{1N}^{*}\in A_{1N}{x}_N^{*},\dots ,u_{N1}^{*}\in A_{N1}{x}_1^{*},u_{N2}^{*}\in A_{N2}{x}_2^{*},\dots ,u_{NN}^{*}\in A_{NN}{x}_N^{*}$ such that

$$⟨T_i\left(u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*}\right),{\eta }_i\left(x,g_i\left(x_i^{*}\right)\right)⟩\ge {\phi }_i\left(g_i\left(x_i^{*}\right)\right)-{\phi }_i\left(x\right),\forall x\in H,i=1,2,\dots N.$$

(1.1)

Problem (1.1) is called the set-valued nonlinear generalized quasi-variational-like inclusions.

Various special cases of the problem (1.1) had been studied by many authors before. Here, we mention some of them as follows:

• (1) If N = 2, A ₁₁ = A ₁₂ = A, A ₂₁ = A ₂₂ = B, T ₁ = T, T ₁(A(·), B(·)) : H → CB(H), then the problem (1.1) reduces to find x* ∈ H, u* ∈ Ax*, v* ∈ Bx* such that

  $$⟨T\left(u^{*},v^{*}\right),\eta \left(x,g\left(x^{*}\right)\right)⟩\ge \phi \left(g\left(x^{*}\right)\right)-\phi \left(x\right),\forall x\in H.$$

  (1.2)

Problem (1.2) was introduced and studied by Ding [1] in 2001.

• (2) If N = 2, A ₁₁ = A ₁₂ = A ₂₁ = A ₂₂ = I, g _i = I (identical operator), η(x, y) = x-y, φ ₁ = φ ₂ = φ, T : H × H → H, T ₁(A ₁₁ x, A ₁₂ y) = ρ ₁ T (A ₁₂ y, A ₁₁ x) + A ₁₁ x - A ₁₂ y, T ₂(A ₂₁ x, A ₂₂ y) = ρ ₂ T (A ₂₁ x, A ₂₂ y) + A ₂₂ y - A ₂₁ x, then the problem (1.1) reduces to find x*, y* ∈ H such that

  $$\left\{\begin{array}{cc}\hfill ⟨{\rho }_{1}T\left(y^{*},x^{*}\right)+x^{*}-y^{*},x-x^{*}⟩+\phi \left(x\right)-\phi \left(x^{*}\right)\ge 0,\forall x\in H,\hfill & \hfill {\rho }_1>0;\hfill \\ \hfill ⟨{\rho }_{2}T\left(x^{*},y^{*}\right)+y^{*}-x^{*},x-y^{*}⟩+\phi \left(x\right)-\phi \left(y^{*}\right)\ge 0,\forall x\in H,\hfill & \hfill {\rho }_2>0.\hfill \end{array}\right.$$

  (1.3)

Problem (1.3) was studied by He and Gu [2] in 2009.

• (3) Let K ⊂ H be a closed convex subset, φ(x) = I _K (x), the problem (1.3) reduces to find x*, y* ∈ K such that

  $$\left\{\begin{array}{cc}\hfill ⟨{\rho }_{1}T\left(y^{*},x^{*}\right)+x^{*}-y^{*},x-x^{*}⟩\ge 0,\forall x\in K,\hfill & \hfill {\rho }_1>0;\hfill \\ \hfill ⟨{\rho }_{2}T\left(x^{*},y^{*}\right)+y^{*}-x^{*},x-y^{*}⟩\ge 0,\forall x\in K,\hfill & \hfill {\rho }_2>0.\hfill \end{array}\right.$$

  (1.4)

Problem (1.4) was inspected and studied by Chang [3], Verma [4, 5] and Huang [6].

• (4) If N = 2, A ₁₁ = A ₁₂ = A ₂₁ = A ₂₂ = I, g : H → H, T ₁(x, y) = ρ ₁ Ty + g(x) - g(y), T ₂(x, y) = ρ ₂ Tx + g(y) - g(x), (ρ ₁, ρ ₂ > 0), η(x, y) = g(x) - g(y), then the problem (1.1) reduces to find x*, y* ∈ H such that

  $$\{\begin{array}{ccc}〈{\rho }_{1}T{y}^{*}+g(x^{*})-g(y^{*}),& g(x)-g(x^{*})〉\ge 0,& \forall x\in H;\\ 〈{\rho }_{2}T{x}^{*}+g(y^{*})-g(x^{*}),& g(x)-g(y^{*})〉\ge 0,& \forall x\in H.\end{array}$$

  (1.5)

Problem (1.5) was introduced and studied by Hajjafar and Verma [7].

• (5) If N = 3, η(x, y) = x - y, then the problem (1.1) reduces to find $x_1^{*},x_2^{*},x_3^{*}\in H,u_{i1}^{*}\in A_{i1}{x}_1^{*},u_{i2}^{*}\in A_{i2}{x}_2^{*},u_{i3}^{*}\in A_{i3}{x}_3^{*}\left(i=1,2,3\right)$ such that

  $$\left\{\begin{array}{cc}\hfill ⟨T_1\left(u_{11}^{*},u_{12}^{*},u_{13}^{*}\right),\hfill & \hfill x-g_1\left(x_1^{*}\right)⟩\ge {\phi }_1\left(g_1\left(x_1^{*}\right)\right)-{\phi }_1\left(x\right),\forall x\in H;\hfill \\ \hfill ⟨T_2\left(u_{21}^{*},u_{22}^{*},u_{23}^{*}\right),\hfill & \hfill x-g_2\left(x_2^{*}\right)⟩\ge {\phi }_2\left(g_2\left(x_2^{*}\right)\right)-{\phi }_2\left(x\right),\forall x\in H;\hfill \\ \hfill ⟨T_3\left(u_{31}^{*},u_{32}^{*},u_{33}^{*}\right),\hfill & \hfill x-g_3\left(x_3^{*}\right)⟩\ge {\phi }_3\left(g_3\left(x_3^{*}\right)\right)-{\phi }_3\left(x\right),\forall x\in H.\hfill \end{array}\right.$$

  (1.6)

Problem (1.6) was studied by Kazmi et al. [8].

For more special cases, please refer to [1–9] and the references therein.

Remark 1.1. Yang [10] pointed out a fact for the problem (1.4) discussed in reference [5], namely, if the problem (1.4) has a solution (x*, y*), then x* = y*. Therefore, actually, the problem(1.4) is a single variational inequality:

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

In this article, we study the problem (1.1). By using the η proximal mapping technique, we prove the existence of solutions and approximate the solutions by some new N - step iterative algorithms. Our results extend and improve some known results in the references [1–9].

2 Preliminaries

In this article, we need the following concepts and lemmas.

Definition 2.1[1] A mapping g : H → H is said to be

• (i) ξ-strongly monotone if there exists a constant ξ > 0 such that

  $$⟨g\left(x\right)-g\left(y\right),x-y⟩\ge \xi {∥x-y∥}^2,\forall x,y\in H.$$

• (ii) ζ-Lipschitz continuous if there exists a constant ζ > 0 such that

  $$∥g\left(x\right)-g\left(y\right)∥\le \zeta ∥x-y∥,\forall x,y\in H.$$

Definition 2.2[1] A mapping η : H × H → H is said to be

• (i) σ - strongly monotone if there exists a constant σ > 0 such that

  $$⟨x-y,\eta \left(x,y\right)⟩\ge \sigma {∥x-y∥}^2,\forall x,y\in H;$$

• (ii) τ - Lipschitz continuous if there exists a constant τ > 0 such that

  $$∥\eta \left(x,y\right)∥\le \tau ∥x-y∥,\forall x,y\in H.$$

Definition 2.3[1, 11] Let A : H → CB(H) be a set-valued mapping, $T:\underset{N}{\underbrace{H\times H\times \cdots \times H}}\to H$ is said to be

• (i) α - (A, g)-strongly monotone in the i th argument if α > 0 such that

  $$⟨T\left(\dots ,u_i,\dots \right)-T\left(\dots ,v_i,\dots \right),g\left(x\right)-g\left(y\right)⟩\ge \alpha {∥x-y∥}^2,\forall x,y\in H,u_i\in Ax,v_i\in Ay.$$

• (ii) (s ₁, s ₂, . . . , s _N )-Lipschitz continuous if there exist constants s ₁, s ₂, . . . , s _N > 0 such that for all x _i , y _i ∈ H, i = 1, 2, . . . , N,

  $$∥T\left(x_1,x_2,\dots ,x_N\right)-T\left(y_1,y_2,\dots ,y_N\right)∥\le s_1∥x_1-y_1∥+s_2∥x_2-y_2∥+\cdots +s_N∥x_N-y_N∥.$$

• (iii) A set-valued A is said to be δ - H - Lipschitz continuous if there exists a constant δ > 0 such that

  $$H\left(Ax,Ay\right)\le \delta ∥x-y∥,\forall x,y\in H,$$

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

Definition 2.4[1] A functional f : H ×H → R∪{+∞} is said to be 0-diagonally quasi-concave (in short,0-DQCV) in x, if for any finite set {x₁, . . . , x _N } ⊂ H and for any $y={\sum }_{i=1}^n{\lambda }_i{x}_i$ with λ _i ≥ 0 and ${\sum }_{i=1}^n{\lambda }_i=1$,

$$\underset{1\le i\le n}{\text{min}}f\left(x_i,y\right)\le 0.$$

Definition 2.5[1] Let η : H × H → H be a single-valued mapping. A proper functional φ : H → R∪{+∞} is said to be η- subdifferentiable at a point x ∈ H, if there exists a point f * ∈ H such that

$$⟨f^{*},\eta \left(y,x\right)⟩\le \phi \left(y\right)-\phi \left(x\right),\forall y\in H,$$

where f * is called a η- subgradient of φ at x. The set of all η- subgradients of φ at x is denoted by ∂ _η φ(x). We have

$${\partial }_{\eta }\phi \left(x\right)=\left\{f^{*}\in H,⟨f^{*},\eta \left(y,x\right)⟩\le \phi \left(y\right)-\phi \left(x\right),\forall y\in H.\right\}$$

(2.1)

Definition 2.6[1] Let η, φ be according to Definition 2.5, if for each x ∈ H and ρ > 0,there exists a unique point u ∈ H such that

$$⟨u-x,\eta \left(y,u\right)⟩\ge \rho \phi \left(u\right)-\rho \phi \left(y\right),\forall y\in H,$$

(2.2)

then the mapping $x\mapsto u$ denoted by $J_{\phi }^{\rho }$, is said to be η- proximal mapping of φ. By (2.1) and the definition of $J_{\phi }^{\rho }$, we have x - u ∈ ρ∂ _η φ(x), it follows that

$$J_{\phi }^{\rho }\left(x\right)={\left(I+\rho {\partial }_{\eta }\phi \right)}^{-1}\left(x\right).$$

Lemma 2.1[1] Let η : H × H → H be continuous and σ- strongly monotone such that η(x, y) = -η(y, x) for all x, y ∈ H. And for any given x ∈ H, the function h(y, u) = 〈x - u, η(y, u)〉 is 0-DQCV in y. Let φ : H → R ∪ {+∞} be a lower semicontinuous η- subdifferentiable proper functional on H, then for any given ρ > 0 and x ∈ H there exists a unique u ∈ H such that

$$⟨u-x,\eta \left(y,u\right)⟩\ge \rho \phi \left(u\right)-\rho \phi \left(y\right),\forall y\in H.$$

i.e., $u=J_{\phi }^{\rho }\left(x\right)$.

Lemma 2.2 Let η : H × H → H be σ- strongly monotone and τ - Lipschitz continuous such that η(x, y) = -η(y, x). Let h(y, u), φ, ρ be according to Lemma 2.1, then the η- proximal mapping $J_{\phi }^{\rho }\left(x\right)$ of φ is $\frac{\tau }{\sigma }$- Lipschitz continuous.

3 Main results

Theorem 3.1$\left(x_1^{*},x_2^{*},\dots ,x_N^{*};u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*},i=1,2,\dots ,N.\right)$ is a solution of problem (1.1) if and only if $\left(x_1^{*},x_2^{*},\dots ,x_N^{*};u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*},i=1,2,\dots ,N.\right)$ satisfies the following relation: For every i = 1, 2, . . . N,

$$g_i\left(x_i^{*}\right)=J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^{*}\right)-{\rho }_i{T}_i\left(u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*}\right)\right),$$

(3.1)

where $J_{{\phi }^i}^{{\rho }_i}={\left(I+{\rho }_i{\partial }_{\eta }{\phi }_i\right)}^{-1},{\rho }_i>0.$

Proof. Assume the $\left(x_1^{*},x_2^{*},\dots ,x_N^{*};u_{11}^{*},u_{12}^{*},\dots ,u_{1N}^{*},\dots ,u_{N1}^{*},u_{N2}^{*},\dots ,u_{NN}^{*}\right)$ satisfies relation (3.1). Since $J_{{\phi }_{{}^i}}^{{\rho }_i}={\left(I+{\rho }_i{\partial }_{{\eta }_i}{\phi }_i\right)}^{-1}$, we have

$$g_i\left(x_i^{*}\right)+{\rho }_i{\partial }_{{\eta }_i}{\phi }_i\left(g_i\left(x_i^{*}\right)\right)\in g_i\left(x_i^{*}\right)-{\rho }_i{T}_i\left(u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*}\right).$$

i.e.,

$$-T_1\left(u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*}\right)\in {\partial }_{{\eta }_i}{\phi }_i\left(g_i\left(x_i^{*}\right)\right).$$

By the Definition 2.5 of η _i - subdifferential, the above relation holds if and only if

$$-⟨T_i\left(u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*}\right),{\eta }_i\left(x,g_i\left(x_i^{*}\right)\right)⟩\le {\phi }_i\left(x\right)-{\phi }_i\left(g_i\left(x_i^{*}\right)\right),\forall x\in H,$$

and hence

$⟨T_i\left(u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*}\right),{\eta }_i\left(x,g_i\left(x_i^{*}\right)\right)⟩\ge {\phi }_i\left(g_i\left(x_i^{*}\right)\right)-{\phi }_i\left(x\right),\forall x\in H,$i = 1, 2, . . . , N. i.e., $\left(x_1^{*},x_2^{*},\dots ,x_N^{*};u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*},i=1,2,\dots ,N.\right)$ is a solution of the problem (1.1). □

Now, we give iterative algorithms of problem (1.1).

Algorithm(I) For given $x_1^0,x_2^0,\dots ,x_N^0\in H,u_{i1}^0\in A_{i1}{x}_1^0,u_{i2}^0\in A_{i2}{x}_2^0,\dots ,u_{iN}^0\in A_{iN}{x}_N^0$, let

$$\begin{array}{ll}\hfill x_1^1=x_1^0-g_1\left(x_1^0\right)+J_{{\phi }_1}^{{\rho }_1}(g_1\left(x_1^0\right)& -{\rho }_1{T}_1\left(u_{11}^0,u_{12}^0,\dots ,u_{1N}^0\right));\\ \hfill x_2^1=x_2^0-g_2\left(x_2^0\right)+J_{{\phi }_2}^{{\rho }_2}(g_2\left(x_2^0\right)& -{\rho }_2{T}_2\left(u_{21}^0,u_{22}^0,\dots ,u_{2N}^0\right));\\ ⋮\\ \hfill x_N^1=x_N^0-g_N\left(x_N^0\right)+J_{{\phi }_{{}^N}}^{{\rho }_N}(g_N\left(x_N^0\right)& -{\rho }_N{T}_N\left(u_{N1}^0,u_{N2}^0,\dots ,u_{NN}^0\right)).\end{array}$$

By Nadler [12], for i = 1, 2, . . . , N, j = 1, 2, . . . , N, there exists $u_{ij}^1\in A_{ij}{x}_j^1$ such that

$$∥u_{ij}^0-u_{ij}^1∥\le \left(1+1\right)H\left(A_{ij}{x}_j^0,A_{ij}{x}_j^1\right),j=1,2,\dots ,N.$$

Let

$$x_i^2=x_i^1-g_i\left(x_i^1\right)+J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^1\right)-{\rho }_i{T}_i\left(u_{i1}^1,u_{i2}^1,\dots ,u_{iN}^1\right)\right),i=1,2,\dots ,N.$$

By induction, we can define sequences $\left\{x_i^n\right\},\left\{u_{ij}^n\right\}$ satisfying

$$x_i^{n+1}=x_i^n-g_i\left(x_i^n\right)+J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^n\right)-{\rho }_i{T}_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{iN}^n\right)\right),$$

where for any i, j = 1, 2, . . . , N; n = 0, 1, 2, . . . ,

$$u_{ij}^n\in A_{ij}^n{x}_j^n,\Vert u_{ij}^n-u_{ij}^{n+1}\Vert \le \left(1+\frac{1}{n+1}\right)H\left(A_{ij}\left(x_j^n\right),A_{ij}\left(x_j^{n+1}\right)\right).$$

Theorem 3.2 Let H be a real Hilbert space. For i, j = 1, 2, . . . , N, let set-valued mapping A _ij : H → CB(H) be δ _ij - H - Lipschitz continuous. Let mapping η _i : H × H → H be σ _i - strongly monotone and τ _i -Lipschitz continuous such that η _i (x, y) = -η _i (y, x) for all x, y ∈ H and for any given x ∈ H, the function h _i (y, u) = 〈x - g _i (u), η _i (y, u)〉 is 0-DQCU in y. Let mapping g _i : H → H be ξ _i - strongly monotone and ζ _i - Lipschitz continuous, and $T_i:\underset{N}{\underbrace{H\times H\times \cdots \times H}}\to H$ be (s₁, . . . , s _N )- Lipschitz continuous and α _i - (A _ij , g _i )- strongly monotone in the i th argument. Let φ _i : H → R ∪ {+∞} be a lower semicontinuous η _i - subdifferentiable proper functional. If there exist ρ₁, . . . , ρ _N > 0 such that for all i = 1, 2, . . . , N

$${\left(1-2{\xi }_i+{\zeta }_i^2\right)}^{\frac{1}{2}}+\frac{{\tau }_i}{{\sigma }_i}{\left({\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2\right)}^{\frac{1}{2}}+\sum _{k=1,k\ne i}^N\frac{{\tau }_k}{{\sigma }_k}{\rho }_k{s}_i{\delta }_{ki}<1;$$

(3.2)

then the iterative sequences $\left\{x_1^n\right\},\dots ,\left\{x_N^n\right\},\left\{u_{11}^n\right\},\dots ,\left\{u_{1N}^n\right\},\dots ,\left\{u_{N1}^n\right\},\dots ,\left\{u_{NN}^n\right\}$, generated by algorithm (I) converge strongly to $x_1^{*},\dots ,x_N^{*},u_{11}^{*},\dots ,u_{1N}^{*},\dots ,u_{N1}^{*},\dots ,u_{NN}^{*}$, respectively, and $\left(x_1^{*},x_2^{*},\dots ,x_N^{*},u_{11}^{*},u_{12}^{*},\dots ,u_{1N}^{*},\dots ,u_{N1}^{*},u_{N2}^{*},\dots ,u_{NN}^{*}\right)$ is a solution of the problem (1.1).

Proof. For i = 1, 2, . . . , N, by algorithm (I) and Lemma 2.2, we have

$$\begin{array}{ll}\hfill ∥x_i^{n+1}-x_i^n∥& =∥x_i^n-g_i\left(x_i^n\right)+J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^n\right)-{\rho }_i{T}_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{iN}^n\right)\right)\\ -x_i^{n-1}+g_i\left(x_i^{n-1}\right)-J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^{n-1}\right)-{\rho }_i{T}_i\left(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{iN}^{n-1}\right)\right)∥\\ \le ∥x_i^n-x_i^{n-1}-g_i\left(x_i^n\right)+g_i\left(x_i^{n-1}\right)∥\\ +∥J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^n\right)-{\rho }_i{T}_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{iN}^n\right)\right)\\ -J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^{n-1}\right)-{\rho }_i{T}_i\left(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{iN}^{n-1}\right)\right)∥\\ \le ∥x_i^n-x_i^{n-1}-g_i\left(x_i^n\right)+g_i\left(x_i^{n-1}\right)∥\\ +\frac{{\tau }_i}{{\sigma }_i}∥g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)-{\rho }_i{T}_i\left(u_{i1}^n,\dots ,u_{iN}^n\right)+{\rho }_i{T}_i\left(u_{i1}^{n-1},\dots ,u_{iN}^{n-1}\right)∥.\end{array}$$

(3.3)

Since g _i is ξ _i - strongly monotone and ζ _i - Lipschitz continuous, we obtain

$$∥x_i^n-x_i^{n-1}-\left(g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)\right)∥\le \sqrt{1-2{\xi }_i+{\zeta }_i^2}∥x_i^n-x_i^{n-1}∥.$$

(3.4)

Notice that,

$$\begin{array}{l}∥g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)-{\rho }_i\left(T_i\left(u_{i1}^n,\dots ,u_{iN}^n\right)-T_i\left(u_{i1}^{n-1},\dots ,u_{iN}^{n-1}\right)\right)∥\\ \le ∥g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)-{\rho }_i\left(T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{ii}^n,u_{i,i+1}^n,\dots ,u_{iN}^n\right)\right.\\ \left.-T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{i,i}^{n-1},u_{i,i+1}^n,\dots ,u_{iN}^n\right)\right)∥\\ +{\rho }_i∥T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{i,i}^{n-1},u_{i,i+1}^n,\dots ,u_{iN}^n\right)-T_i\left(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{iN}^{n-1}\right)∥.\end{array}$$

(3.5)

Since T _i is (s₁, . . . , s _N )- Lipschitz continuous and α _i - (A _ij , g _i )- strongly monotone in the i th argument, we get

$$\begin{array}{l}∥g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)-{\rho }_i\left(T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{ii}^n,u_{i,i+1}^n,\dots ,u_{iN}^n\right)\right.\\ -T_i{\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{ii}^{n-1},u_{i,i+1}^n,\dots ,u_{iN}^n\right)∥}^2\\ =∥g_i\left(x_i^n\right)-g_i{\left(x_i^{n-1}\right)∥}^2-2{\rho }_i⟨g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right),T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{i,i}^n,u_{i,i+1}^n,\dots ,u_{i,N}^n\right)\\ -T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{i,i}^{n-1},u_{i,i+1}^n,\dots ,u_{iN}^n\right)⟩\\ +{\rho }_i^2∥T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{i,i}^n,u_{i,i+1}^n,\dots ,u_{iN}^n\right)\\ -T_i{\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{i,i}^{n-1},u_{i,i+1}^n,\dots ,u_{iN}^n\right)∥}^2\\ \le {\xi }_i^2{∥x_i^n-x_i^{n-1}∥}^2-2{\rho }_i{\alpha }_i∥x_i^n-{x_i^{n-1}∥}^2+{\rho }_i^2{s}_i^2∥u_{ii}^n-{u_{ii}^{n-1}∥}^2\\ \le \left({\xi }_i^2-2{\rho }_i{\alpha }_i\right)∥x_i^n-{x_i^{n-1}∥}^2+{\rho }_i^2{s}_i^2{\left(1+\frac{1}{n}\right)}^2{\left(H\left(A_{ii}{x}_i^n,A_{ii}{x}_i^{n-1}\right)\right)}^2\\ \le \left[{\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2{\left(1+\frac{1}{n}\right)}^2\right]{∥x_i^n-x_i^{n-1}∥}^2.\end{array}$$

(3.6)

Therefore,

$$\begin{array}{c}∥g_i\left(x_i^n\right)-g_i\left(x_i^{n-1}\right)-{\rho }_i\left(T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{ii}^n,u_{i,i+1}^n,\dots ,u_{iN}^n\right)\right.\\ -T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{ii}^{n-1},u_{i,i+1}^n,\dots ,u_N^n\right)∥\\ \le {\left[{\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2{\left(1+\frac{1}{n}\right)}^2\right]}^{\frac{1}{2}}∥x_i^n-x_i^{n-1}∥.\end{array}$$

(3.7)

And

$$\begin{array}{c}{\rho }_i∥T_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{i,i-1}^n,u_{ii}^{n-1},u_{i,i+1}^n,\dots ,u_{iN}^n\right)\\ -T_i\left(u_{i1}^{n-1},u_{i2}^{n-1},\dots ,u_{i,i-1}^{n-1},u_{ii}^{n-1},u_{i,i+1}^{n-1},\dots ,u_{iN}^{n-1}\right)∥\\ \le {\rho }_i\left(s_1∥u_{i1}^n-u_{i1}^{n-1}∥+s_2∥u_{i2}^n-u_{i2}^{n-1}∥+\cdots \right.\\ \left.+s_{i-1}∥u_{i,i-1}^n-u_{i,i-1}^{n-1}∥+s_{i+1}∥u_{i,i+1}^n-u_{i,i+1}^{n-1}∥+\cdots +s_N∥u_{iN}^n-u_{iN}^{n-1}∥\right)\\ \le {\rho }_i\left[s_1\left(1+\frac{1}{n}\right)H\left(A_{i1}{x}_1^n,A_{i1}{x}_1^{n-1}\right)\right]+s_2\left(1+\frac{1}{n}\right)H\left(A_{i2}{x}_2^n,A_{i2}{x}_2^{n-1}\right)+\cdots \\ +s_{i-1}\left(1+\frac{1}{n}\right)H\left(A_{i,i-1}{x}_{i-1}^n,A_{i,i-1}{x}_{i-1}^{n-1}\right)\\ \left.+s_{i+1}\left(1+\frac{1}{n}\right)H\left(A_{i,i+1}{x}_{i+1}^n,A_{i,i+1}{x}_{i+1}^{n-1}\right)+\cdots +s_N\left(1+\frac{1}{n}\right)H\left(A_{iN}{x}_N^n,A_{iN}{x}_N^{n-1}\right)\right]\\ \le {\rho }_i\left(1+\frac{1}{n}\right)\left[s_1{\delta }_{i1}∥x_1^n-x_1^{n-1}∥+s_2{\delta }_{i2}∥x_2^n-x_2^{n-1}∥+\cdots \right.\\ \left.+s_{i-1}{\delta }_{i,i-1}∥x_{i-1}^n-x_{i-1}^{n-1}∥+s_{i+1}{\delta }_{i,i+1}∥x_{i+1}^n-x_{i+1}^{n-1}∥+\cdots +s_N{\delta }_{iN}∥x_N^n-x_N^{n-1}∥\right].\end{array}$$

(3.8)

It follows from (3.3)-(3.8) that for every i = 1, 2, . . . ,N,

$$\begin{array}{ll}\hfill ∥x_i^{n+1}-x_i^n∥& \le \left[\sqrt{1-2{\xi }_i+{\zeta }_i^2}+\frac{{\tau }_i}{{\sigma }_i}{\left({\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2{\left(1+\frac{1}{n}\right)}^2\right)}^{\frac{1}{2}}\right]∥x_i^n-x_i^{n-1}∥\\ +{\rho }_i\frac{{\tau }_i}{{\sigma }_i}\left(1+\frac{1}{n}\right)\sum _{k=1,k\ne i}^N{s}_k{\delta }_{ik}∥x_k^n-x_k^{n-1}∥.\end{array}$$

(3.9)

So,

$$\begin{array}{l}∥x_1^{n+1}-x_1^n∥+∥x_2^{n+1}-x_2^n∥+\cdots +∥x_N^{n+1}-x_N^n∥\\ \le \sum _{i=1}^N\left\{\left[\sqrt{1-2{\xi }_i+{\zeta }_i^2}+\frac{{\tau }_i}{{\sigma }_i}{\left({\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2{\left(1+\frac{1}{n}\right)}^2\right)}^{\frac{1}{2}}\right.\right.∥x_i^n-x_i^{n-1}∥\\ \left.+{\rho }_i\frac{{\tau }_i}{{\sigma }_i}\sum _{k=1,k\ne i}^N{s}_k{\delta }_{ik}\left(1+\frac{1}{n}\right)∥x_k^n-x_k^{n-1}∥\right\}\\ =\sum _{i=1}^N{\left\{\sqrt{1-2{\xi }_i+{\zeta }_i^2}+\frac{{\tau }_i}{{\sigma }_i}\left({\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2{\left(1+\frac{1}{n}\right)}^2\right)\right.}^{\frac{1}{2}}\\ \left.+\sum _{k=1,k\ne i}^N{\rho }_k\frac{{\tau }_k}{{\sigma }_k}{s}_i{\delta }_{ki}\left(1+\frac{1}{n}\right)\right\}∥x_i^n-x_i^{n-1}∥\\ \le {\theta }_n\left(∥x_1^n-x_1^{n-1}∥+∥x_2^n-x_2^{n-1}∥+\cdots +∥x_N^n-x_N^{n-1}∥\right),\end{array}$$

(3.10)

where

$${\theta }_n=\underset{i=1,2,\dots N}{\text{max}}\left\{\sqrt{1-2{\xi }_i+{\zeta }_i^2}+\frac{{\tau }_i}{{\sigma }_i}{\left({\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2{\left(1+\frac{1}{n}\right)}^2\right)}^{\frac{1}{2}}+\sum _{k=1,k\ne i}^N{\rho }_k\frac{{\tau }_k}{{\sigma }_k}{s}_i{\delta }_{ki}\left(1+\frac{1}{n}\right)\right\}$$

Letting

$$\theta =\underset{i=1,2,\dots N}{\text{max}}\left\{\sqrt{1-2{\xi }_i+{\zeta }_i^2}+\frac{{\tau }_i}{{\sigma }_i}{\left({\xi }_i^2-2{\rho }_i{\alpha }_i+{\rho }_i^2{s}_i^2{\delta }_{ii}^2\right)}^{\frac{1}{2}}+\sum _{k=1,k\ne i}^N{\rho }_k\frac{{\tau }_k}{{\sigma }_k}{s}_i{\delta }_{ki}\right\},$$

from (3.2) we have 0 < θ < 1, and hence $\left\{x_1^n\right\}\dots \left\{x_N^n\right\}$are also Cauchy sequences. Thus there exist $x_1^{*},\dots ,x_N^{*}\in H$ such that $x_i^n\to x_i^{*}\left(n\to \infty \right),$i = 1, 2, . . . ,N.

Now we prove $u_{ij}^n\to u_{ij}^{*}\left(n\to \infty \right),$for i = 1, 2, . . . ,N, j = 1, 2, . . . ,N. By $∥u_{ij}^n-u_{ij}^{n-1}∥\le \left(1+\frac{1}{n}\right)H\left(A_{ij}{x}_j^n,A_{ij}{x}_j^{n-1}\right)\le \left(1+\frac{1}{n}\right){\delta }_{ij}∥x_j^n-x_j^{n-1}∥.$

It follows that $\left\{u_{ij}^n\right\}$ are also Cauchy sequence. Therefore, there exist $u_{ij}^{*}\in H$ such that $u_{ij}^n\to u_{ij}^{*}\left(n\to \infty \right).$

Note that

$$\begin{array}{ll}\hfill d\left(u_{ij}^{*},A_{ij}{x}_j^{*}\right)& \le ∥u_{ij}^{*}-u_{ij}^n∥d\left(u_{ij}^n,A_{ij}{x}_j^{*}\right)\\ \le ∥u_{ij}^{*}-u_{ij}^n∥+H\left(A_{ij}{x}_j^n,A_{ij}{x}_j^{*}\right)\le ∥u_{ij}^{*}-u_{ij}^n∥+{\delta }_{ij}∥x_j^n-x_j^{*}∥\to 0\left(n\to \infty \right).\end{array}$$

We have $d\left(u_{ij}^{*},A_{ij}{x}_j^{*}\right)=0.$Since $A_{ij}{x}_j^{*}$ is closed, $u_{ij}^{*}\in A_{ij}{x}_j^{*}$, for each i = 1,2, . . . ,N, j = 1, 2, . . . ,N. By

$$x_i^{n+1}=x_i^n-g_i\left(x_i^n\right)+J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^n\right)-{\rho }_i{T}_i\left(u_{i1}^n,u_{i2}^n,\dots ,u_{iN}^n\right)\right),i=1,2,\dots ,N,$$

and the continuity of $g_i,J_{{\phi }^i}^{{\rho }_i},T_i,$ let n → ∞, we have that

$$0=-g_i\left(x_i^{*}\right)+J_{{\phi }_i}^{{\rho }_i}\left(g_i\left(x_i^{*}\right)-{\rho }_i{T}_i\left(u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*}\right)\right),$$

and

$$u_{i1}^{*}\in A_{i1}{x}_1^{*},u_{i2}^{*}\in A_{i2}{x}_2^{*},\dots ,u_{iN}^{*}\in A_{iN}{x}_N^{*}.$$

0By Theorem 3.1, $\left(x_1^{*},x_2^{*},\dots ,x_N^{*},u_{i1}^{*},u_{i2}^{*},\dots ,u_{iN}^{*},i=1,2,\dots ,N\right)$ is a solution of the problem (1.1). This completes the proof.□

Remark 3.1 For a suitable choice of T _i , A _ij , η _i , g _i and φ _i , Theorem 3.2 includes many known results of generalized quasi-variational-like inclusions as special cases (see [1–8]),where φ _i is nonconvex and A _ij is noncompact.