# The well-posedness for a system of generalized quasi-variational inclusion problems

- Xiao-bo Li
^{1}, - Ravi P Agarwal
^{2, 3}, - Yeol Je Cho
^{3, 4}Email author and - Nan-jing Huang
^{1}

**2014**:321

https://doi.org/10.1186/1029-242X-2014-321

© Li et al.; licensee Springer 2014

**Received: **19 March 2014

**Accepted: **22 July 2014

**Published: **22 August 2014

## Abstract

We introduce the concept of Levitin-Polyak well-posedness for a system of generalized quasi-variational inclusion problems and show some characterizations of Levitin-Polyak well-posedness for the system of generalized quasi-variational inclusion problems under some suitable conditions. We also give some results concerned with the Hadamard well-posedness for the system of generalized quasi-variational inclusion problems.

**MSC:**49J40, 49K40, 90C31.

## Keywords

## 1 Introduction

Well-posedness plays a crucial role in the theory and methodology of scalar optimization problems. In 1966, Tykhonov [1] first introduced the concept of well-posedness for a global minimizing problem, which has become known as Tykhonov well-posedness. Soon after, Levitin-Polyak [2] strengthened the concept of Tykhonov well-posedness, already known as the Levitin-Polyak (for short, LP) well-posedness. Subsequently, some authors studied the LP well-posedness for convex scalar optimization problems with functional constraints [3], vector optimization problems [4], variational inequality problems [5], generalized mixed variational inequality problems [6], generalized quasi-variational inequality problems [7], generalized vector variational inequality problems [8], equilibrium problems [9], vector equilibrium problems [10], generalized vector quasi-equilibrium problems [11] and generalized quasi-variational inclusion and disclusion problems [12]. Another important notion of the well-posedness for a minimizing problem is the well-posedness by perturbations or the extended well-posedness due to Zolezzi [13]. The notion of the well-posedness by perturbations establishes a form of continuous dependence of the solutions upon a parameter. Recently, Lemaire *et al.* [14] introduced the well-posedness by perturbations for variational inequalities and Fang *et al.* [15] considered the well-posedness by perturbations for mixed variational inequalities in Banach spaces. For more details about the well-posedness by perturbations, we refer readers to [16, 17] and the references therein.

On the other hand, for optimization problems, there is another concept of well-posedness, which has become known as the Hadamard well-posedness. The concept of Hadamard well-posedness was inspired by the classical idea of Hadamard, which goes back to the beginning of the last century. It requires the existence and uniqueness of the optimal solution together with continuous dependence on the problem data. Some results about the Hadamard well-posedness can be found in [18–20]. Recently, the concept of Hadamard well-posedness has been extended to vector optimization problems and vector equilibrium problems. Li and Zhang [21] investigated the Hadamard well-posedness for vector optimization problems. Zeng *et al.* [22] obtained a sufficient condition for the Hadamard well-posedness of a set-valued optimization problem. Salamon [23] investigated the generalized Hadamard well-posedness for parametric vector equilibrium problems with trifunctions.

Very recently, Lin and Chuang [24] studied the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems as constraints. Motivated by Lin, Wang *et al.* [25] investigated the well-posedness for generalized quasi-variational inclusion problems and for optimization problems with generalized quasi-variational inclusion problems as constraints. A system of generalized quasi-variational inclusion problems, which consists of a family of generalized quasi-variational inclusion problems defined on a product set, was first introduced by Lin [26]. It is well known that the system of generalized quasi-variational inclusion problems contains the system of variational inequalities, the system of equilibrium problems, the system of vector equilibrium problems, the system of vector quasi-equilibrium problems, the system of generalized vector quasi-equilibrium problems, the system of variational inclusions problems and variational disclusions problems as special cases. For more details, one can refer to [27–33] and the references therein. Nonetheless, to the best of our knowledge, there is no paper dealing with the Levitin-Polyak and Hadamard well-posedness for the system of generalized quasi-variational inclusion problems. Therefore, it is very interesting to generalize the concept of Levitin-Polyak and Hadamard well-posedness to the system of generalized quasi-variational inclusion problems.

Motivated and inspired by research work mentioned above, in this paper, we study the LP and Hadamard well-posedness for the system of generalized quasi-variational inclusion problems. This paper is organized as follows. In Section 2, we introduce the concept of LP well-posedness for the system of generalized quasi-variational inclusion problems. Some characterizations of the LP well-posedness for the system of generalized quasi-variational inclusion problems are shown in Section 3. Some results concerned with Hadamard well-posedness for the system of generalized quasi-variational inclusion problems are given in Section 4.

## 2 Preliminaries

Let *I* be an index set and $(P,{d}_{0})$ be a metric space. For each $i\in I$, let ${X}_{i}$ be a metric space, ${Y}_{i}$ and ${Z}_{i}$ be Hausdorff topological vector spaces, ${K}_{i}\subset {X}_{i}$ be a nonempty closed and convex subset. Set $X={\mathrm{\Pi}}_{i\in I}{X}_{i}$, $K={\mathrm{\Pi}}_{i\in I}{K}_{i}$ and $Y={\mathrm{\Pi}}_{i\in I}{Y}_{i}$. For each $i\in I$, let ${A}_{i}:K\to {2}^{{X}_{i}}$, ${T}_{i}:K\to {2}^{{Y}_{i}}$ and ${G}_{i}:K\times Y\times {K}_{i}\to {2}^{{Z}_{i}}$ be set-valued mappings. Let ${e}_{i}:K\to {Z}_{i}$ be a continuous mapping. Throughout this paper, unless otherwise specified, we use these notations and assumptions.

Now, we consider the following system of *generalized quasi*-*variational inclusion problems* (for short, SQVIP).

for all ${z}_{i}\in {A}_{i}(\overline{x})$. We denote by *S* the solution set of (SQVIP).

for all ${z}_{i}\in {A}_{i}(\overline{x})$.

- (I)If, for each $i\in I$, ${F}_{i}:K\times Y\times {K}_{i}\to {Z}_{i}$ is a mapping, ${C}_{i}:K\to {2}^{{Z}_{i}}$ is a pointed, closed and convex cone with $int{C}_{i}(x)\ne \mathrm{\varnothing}$ for every $x\in K$, ${G}_{i}(x,y,{z}_{i})$ reduces to a single-valued mapping and ${G}_{i}(x,y,{z}_{i})={F}_{i}(\overline{x},\overline{y},{z}_{i})+{Z}_{i}\mathrm{\setminus}int{C}_{i}(x)$ for all $(x,y,{z}_{i})\in K\times Y\times {K}_{i}$, then (SQVIP) reduces to the system of vector equilibrium problems: Find $\overline{x}\in {({\overline{x}}_{i})}_{i\in I}\in K$ such that, for each $i\in I$, ${\overline{x}}_{i}\in {A}_{i}(\overline{x})$ and there exists ${\overline{y}}_{i}\in {T}_{i}(\overline{x})$ satisfying${F}_{i}(\overline{x},\overline{y},{z}_{i})\notin -int{C}_{i}(\overline{x})$

- (II)If, for each $i\in I$, ${F}_{i}:K\times Y\times {K}_{i}\to {2}^{{Z}_{i}}$ and ${\mathrm{\Psi}}_{i}:K\times {K}_{i}\to {2}^{{Z}_{i}}$ are set-valued mappings, ${C}_{i}:K\to {2}^{{Z}_{i}}$ is a pointed, closed and convex cone with $int{C}_{i}(x)\ne \mathrm{\varnothing}$ for all $x\in K$, ${G}_{i}(x,y,{z}_{i})={F}_{i}(x,y,{z}_{i})+{\mathrm{\Psi}}_{i}(x,{z}_{i})+{Z}_{i}\mathrm{\setminus}int{C}_{i}(x)$ for all $(x,y,{z}_{i})\in K\times Y\times {K}_{i}$, then (SQVIP) reduces to the system of set-valued vector quasi-equilibrium problems of Chen
*et al.*[35]: Find $\overline{x}={({\overline{x}}_{i})}_{i\in I}\in K$ such that, for each $i\in I$, ${\overline{x}}_{i}\in {A}_{i}(\overline{x})$ and there exists ${\overline{y}}_{i}\in {T}_{i}(\overline{x})$ satisfying${F}_{i}(\overline{x},\overline{y},{z}_{i})+{\mathrm{\Psi}}_{i}(x,{z}_{i})\u2288-int{C}_{i}(\overline{x})$

**Definition 2.1**Let ${p}^{\ast}\in P$ and $\{{p}^{n}\}\subset P$ be a sequence such that ${p}^{n}\to {p}^{\ast}$. A sequence $\{{x}^{n}\}\subset K$ is called a

*LP approximating solution sequence*for (SQVIP) corresponding to $\{{p}^{n}\}$ if, for each $i\in I$ and $n\in \mathbf{N}$, there exists a sequence of nonnegative real numbers $\{{\u03f5}^{n}\}$ with ${\u03f5}^{n}\to 0$ and ${y}_{i}^{n}\in {T}_{i}({x}^{n})$ such that

for all ${z}_{i}\in {A}_{i}({x}^{n})$, where ${B}^{+}(0,{\u03f5}^{n})$ denote the closed interval $[0,{\u03f5}^{n}]$.

**Definition 2.2**(1) (SQVIP) is said to be

*LP well*-

*posed by perturbations*if it has a unique solution and, for all $\{{p}^{n}\}\subset P$ with ${p}^{n}\to {p}^{\ast}$, every LP approximating solution sequence for (SQVIP) corresponding to $\{{p}^{n}\}$ converges strongly to the unique solution.

- (2)
(SQVIP) is said to be

*generalized LP well*-*posed by perturbations*if the solution set*S*for (SQVIP) is nonempty and, for all sequences $\{{p}^{n}\}\subset P$ with ${p}^{n}\to {p}^{\ast}$, every LP approximating solution sequence for (SQVIP) corresponding to $\{{p}^{n}\}$ has some subsequence which converges strongly to some point of*S*.

**Definition 2.3** [36]

- (1)
*upper semicontinuous*(for short, u.s.c.) at $x\in {E}_{1}$ if, for any neighborhood*V*of $F(x)$, there exists a neighborhood*U*of*x*such that $F(\overline{x})\subset V$ for all $\overline{x}\in U$; - (2)
*lower semicontinuous*(for short, l.s.c.) at $x\in {E}_{1}$ if, for each open set*V*in ${E}_{2}$ with $F(x)\cap V\ne \mathrm{\varnothing}$, there exists an open neighborhood $U(x)$ of*x*such that $F({x}^{\prime})\cap V\ne \mathrm{\varnothing}$ for all ${x}^{\prime}\in U(x)$; - (3)
*u.s.c.*(resp.,*l.s.c.*) on ${E}_{1}$ if it is u.s.c. (resp., l.s.c.) on every point $x\in {E}_{1}$; - (4)
*continuous*on ${E}_{1}$ if it is both u.s.c. and l.s.c. on ${E}_{1}$; - (5)
*closed*if the graph of*F*is closed,*i.e.*, the set $gph(F)=\{(x,y)\in {E}_{1}\times {E}_{2}:y\in F(x)\}$ is closed in ${E}_{1}\times {E}_{2}$.

**Definition 2.4** [37]

Let ${Z}_{1}$ and ${Z}_{2}$ be two metric spaces. A set-valued mapping $F:{Z}_{1}\to {2}^{{Z}_{2}}$ is said to be $(s,s)$-*subcontinuous* if, for any sequence $\{{x}_{n}\}$ converging strongly in ${Z}_{1}$, the sequence $\{{y}_{n}\}$ with ${y}_{n}\in F({x}_{n})$ has a strongly convergent subsequence.

**Definition 2.5** [38]

*A*be a nonempty subset of

*X*, the measure of noncompactness

*μ*of the set

*A*is defined by

**Definition 2.6** [38]

*A*and

*B*be two nonempty subsets of a Banach space

*X*. The

*Hausdorff metric*$\mathcal{H}(\cdot ,\cdot )$ between

*A*and

*B*is defined by

where $e(A,B)={sup}_{a\in A}d(a,B)$ with $d(a,B)={inf}_{b\in B}\parallel a-b\parallel $.

## 3 The Levitin-Polyak well-posedness for (SQVIP)

for all $\delta ,\u03f5>0$, where $B({p}^{\ast},\delta )$ denotes the closed ball centered at ${p}^{\ast}$ with radius *δ*.

- (1)
$S\subseteq \mathrm{\Omega}(\delta ,\u03f5)$ for all $\delta ,\u03f5>0$;

- (2)
if $0<{\delta}_{1}<{\delta}_{2}$ and $0<{\u03f5}_{1}<{\u03f5}_{2}$, then $\mathrm{\Omega}({\delta}_{1},{\u03f5}_{1})\subseteq \mathrm{\Omega}({\delta}_{2},{\u03f5}_{2})$.

Next, we present some properties of $\mathrm{\Omega}(\delta ,\u03f5)$.

**Proposition 3.1** *For each* $i\in I$, *let* ${T}_{i}:K\to {2}^{{Y}_{i}}$ *be compact*-*valued*, ${A}_{i}:K\to {2}^{{X}_{i}}$ *be closed*-*valued and* $(p,y)\to {G}_{i}(p,x,y,{z}_{i})$ *be closed for all* $(x,{z}_{i})\in K\times {K}_{i}$. *Then* $S={\bigcap}_{\delta >0,\u03f5>0}\mathrm{\Omega}(\delta ,\u03f5)$.

*Proof*Clearly, $S\subseteq {\bigcap}_{\delta >0,\u03f5>0}\mathrm{\Omega}(\delta ,\u03f5)$. Hence we only need to show that ${\bigcap}_{\delta >0,\u03f5>0}\mathrm{\Omega}(\delta ,\u03f5)\subseteq S$. If not, then there exists $\overline{x}\in {\bigcap}_{\delta >0,\u03f5>0}\mathrm{\Omega}(\delta ,\u03f5)$ such that $\overline{x}\notin S$. Thus, for any $\delta >0$ and $\u03f5>0$, we have $\overline{x}\in \mathrm{\Omega}(\delta ,\u03f5)\mathrm{\setminus}S$. For each $i\in I$ and $n\in \mathbf{N}$, it follows that $\overline{x}\in \mathrm{\Omega}(\frac{1}{n},\frac{1}{n})\mathrm{\setminus}S$ and there exist ${p}^{n}\in B({p}^{\ast},\frac{1}{n})$ and ${y}_{i}^{n}\in {T}_{i}(\overline{x})$ such that

for all ${z}_{i}\in {A}_{i}(\overline{x})$. Since ${A}_{i}$ is closed-valued, it follows from (1) that ${\overline{x}}_{i}\in {A}_{i}(\overline{x})$ and so $\overline{x}\in S$, which is a contradiction. This completes the proof. □

**Example 3.1**Let

*I*be a single set, $P=[-1,1]$, $X=Y=Z=\mathbb{R}=(-\mathrm{\infty},+\mathrm{\infty})$ and $K=[0,+\mathrm{\infty})$. For any $(p,x,y,z)\in P\times K\times Y\times K$, let

Therefore, ${\bigcap}_{\delta >0,\u03f5>0}\mathrm{\Omega}(\delta ,\u03f5)=[0,1]=S$.

**Proposition 3.2**

*For each*$i\in I$,

*assume that*

- (i)
*P**is a finite*-*dimensional space*; - (ii)
${T}_{i}:K\to {2}^{{Y}_{i}}$

*is u*.*s*.*c*.*and compact*-*valued*; - (iii)
${A}_{i}:K\to {2}^{{X}_{i}}$

*is*$(s,s)$-*subcontinuous*,*l*.*s*.*c*.*and closed*; - (iv)
${G}_{i}:P\times K\times Y\times {K}_{i}\to {2}^{{Z}_{i}}$

*is closed*.

*Then* $\mathrm{\Omega}(\delta ,\u03f5)$ *is closed for any* $\delta ,\u03f5>0$.

*Proof*For any $\delta ,\u03f5\ge 0$, let $\{{x}^{n}\}\subset \mathrm{\Omega}(\delta ,\u03f5)$ and ${x}^{n}\to \overline{x}$. Then there exists ${p}^{n}\in B({p}^{\ast},\delta )$ such that, for each $i\in I$,

*P*is a finite-dimensional space, we can suppose that ${p}^{n}\to \overline{p}\in B({p}^{\ast},\delta )$. In order to prove that $\overline{x}\in \mathrm{\Omega}(\delta ,\u03f5)$, we first prove that, for each $i\in I$,

for all ${\overline{z}}_{i}\in {A}_{i}(\overline{x})$. Thus $\overline{x}\in \mathrm{\Omega}(\delta ,\u03f5)$ and so $\mathrm{\Omega}(\delta ,\u03f5)$ is closed. This completes the proof. □

**Remark 3.1** If *I* is a single set and $x\in A(x)$ for all $x\in K$, then Propositions 3.1 and 3.2 can be considered as a generalization of Properties 3.1 and 3.2 of [25], respectively.

In this paper, let $d(x,y)={sup}_{i\in I}{d}_{i}({x}_{i},{y}_{i})$ for all $x,y\in X$. It is clear that $(X,d)$ is a metric space.

**Theorem 3.1**

*For each*$i\in I$,

*let*${X}_{i}$

*be complete*.

*We assume that*

- (i)
${T}_{i}:K\to {2}^{{Y}_{i}}$

*is u*.*s*.*c*.*and compact*-*valued*; - (ii)
${A}_{i}:K\to {2}^{{X}_{i}}$

*is*$(s,s)$-*subcontinuous*,*l*.*s*.*c*.*and closed*; - (iii)
${G}_{i}:P\times K\times Y\times {K}_{i}\to {2}^{{Z}_{i}}$

*is closed*.

*Then*(SQVIP)

*is LP well*-

*posed by perturbations if and only if*,

*for any*$\delta ,\u03f5>0$,

*as* $(\delta ,\u03f5)\to (0,0)$.

*Proof* Suppose that (SQVIP) is LP well-posed by perturbations. Then (SQVIP) has a unique solution ${x}^{\ast}\in \mathrm{\Omega}(\delta ,\u03f5)$ for any $\delta ,\u03f5>0$. This implies that $\mathrm{\Omega}(\delta ,\u03f5)\ne \mathrm{\varnothing}$ for any $\delta ,\u03f5>0$.

for all ${\overline{z}}_{i}\in {A}_{i}({\overline{x}}^{n})$. Clearly, ${p}^{n}\to {p}^{\ast}$ and ${\overline{p}}^{n}\to {p}^{\ast}$. Thus $\{{x}^{n}\}$ and $\{{\overline{x}}^{n}\}$ are both the LP approximating solution sequences for (SQVIP) corresponding to $\{{p}^{n}\}$ and $\{{\overline{p}}^{n}\}$, respectively. Since (SQVIP) is LP well-posed by perturbations, $\{{x}^{n}\}$ and $\{{\overline{x}}^{n}\}$ have to converge strongly to the unique solution ${x}^{\ast}$ of (SQVIP), which is a contradiction to (5).

for all ${z}_{i}\in {A}_{i}(\overline{x})$. Thus $\overline{x}$ is a solution of (SQVIP).

*x*and $\overline{x}$, then it is easy to see that $x,\overline{x}\in \mathrm{\Omega}(\delta ,\u03f5)$ for any $\delta ,\u03f5>0$. It follows that

for all $\delta ,\u03f5>0$, which contradicts (4). Thus (SQVIP) has a unique solution. This completes the proof. □

**Remark 3.2** If *I* is a single set, $x\in A(x)$ for all $x\in K$, then Theorem 3.1 can be seen as a generalization of Theorem 3.1 of [25].

**Example 3.2**Let

*I*be a single set, $P=[-1,1]$, $X=Y=Z=\mathbb{R}=(-\mathrm{\infty},+\mathrm{\infty})$ and $K=[-1,0]$. For all $(p,x,y,z)\in P\times K\times Y\times K$, let

*A*is $(s,s)$-subcontinuous, l.s.c. and closed,

*T*is u.s.c. and compact-valued and

*G*is closed. For any $\delta ,\u03f5>0$, we have

for sufficiently small $\delta >0$. Therefore, $diam\mathrm{\Omega}(\delta ,\u03f5)\to 0$ as $(\delta ,\u03f5)\to (0,0)$.

**Theorem 3.2**

*For each*$i\in I$,

*let*${X}_{i}$

*be complete and*

*P*

*be a finite*-

*dimensional space*.

*We assume that*

- (i)
${T}_{i}:K\to {2}^{{Y}_{i}}$

*is u*.*s*.*c*.*and compact*-*valued*; - (ii)
${A}_{i}:K\to {2}^{{X}_{i}}$

*is*$(s,s)$-*subcontinuous*,*l*.*s*.*c*.*and closed*; - (iii)
${G}_{i}:P\times K\times Y\times {K}_{i}\to {2}^{{Z}_{i}}$

*is closed*.

*Then*(SQVIP)

*is generalized LP well*-

*posed by perturbations if and only if*,

*for any*$\delta ,\u03f5>0$,

*as* $(\delta ,\u03f5)\to (0,0)$.

*Proof* Suppose that (SQVIP) is generalized LP well-posed by perturbations. Then *S* is nonempty.

*S*is compact. Indeed, let $\{{x}^{n}\}$ be a sequence in

*S*. Then $\{{x}^{n}\}$ is the LP approximating solution sequence for (SQVIP) corresponding to $\{{p}^{\ast}\}$. Since (SQVIP) is generalized LP well-posed by perturbations, $\{{x}^{n}\}$ has a subsequence which converges strongly to a point of

*S*. This implies that

*S*is compact. For any $\delta ,\u03f5\ge 0$, since $S\subset \mathrm{\Omega}(\delta ,\u03f5)$, we have $\mathrm{\Omega}(\delta ,\u03f5)\ne \mathrm{\varnothing}$ and

*S*is compact,

*n*sufficiently large,

for all ${z}_{i}\in {A}_{i}({x}^{n})$, it follows that ${p}^{n}\to {p}^{\ast}$ and $\{{x}^{n}\}$ is the LP approximating solution sequence for (SQVIP) corresponding to $\{{p}^{n}\}$. By the generalized LP well-posedness by perturbations of (SQVIP), there exists a subsequence $\{{x}^{{n}_{k}}\}$ of $\{{x}^{n}\}$ which converges strongly to a point of *S*, which contradicts (10).

*S*is nonempty compact and

*P*with ${p}^{n}\to {p}^{\ast}$. If $\{{x}^{n}\}$ is the LP approximating solution sequence for (SQVIP) corresponding to $\{{p}^{n}\}$, then there exists a sequence $\{{\u03f5}_{n}\}$ of nonnegative numbers with ${\u03f5}_{n}\to 0$ and $\{{y}_{i}^{n}\}$ with ${y}_{i}^{n}\in {T}_{i}({x}^{n})$ such that

as $({\delta}_{n},{\u03f5}_{n})\to (0,0)$. The compactness of *S* implies that (SQVIP) is generalized LP well-posed by perturbations. This completes the proof. □

**Remark 3.3** If *I* is a single set, $x\in A(x)$ for every $x\in K$, then Theorem 3.2 can be considered as a generalization of Theorem 3.2 of [25].

**Example 3.3**Let

*I*be a single set, $P=[-1,1]$, $X=Y=Z=\mathbb{R}=(-\mathrm{\infty},+\mathrm{\infty})$ and $K=[0,+\mathrm{\infty})$. For all $(p,x,y,z)\in P\times K\times Y\times K$, let

for sufficiently small $\delta >0$. Therefore, $\mu (\mathrm{\Omega}(\delta ,\u03f5))\to 0$ as $(\delta ,\u03f5)\to (0,0)$. From Theorem 3.2, (SQVIP) is generalized LP well-posedness by perturbations.

## 4 The Hadamard well-posedness for (SQVIP)

In this section, for each $i\in I$, we assume that ${X}_{i}$, ${Y}_{i}$ and ${Z}_{i}$ are finite-dimensional spaces, ${K}_{i}\subset {X}_{i}$ is a nonempty closed and convex subset.

- (i)
${A}_{i}:K\to {2}^{{X}_{i}}$ is continuous and bounded compact-convex-valued;

- (ii)
${T}_{i}:K\to {2}^{{Y}_{i}}$ is u.s.c. and bounded compact-convex-valued;

- (iii)
${G}_{i}:K\times Y\times {K}_{i}\to {2}^{{Z}_{i}}$ is u.s.c. and bounded compact-convex-valued.

**Definition 4.1** [39]

*convergent to*

*D*

*in the sense of Painlevé*-

*Kuratowski*(for short, ${D}_{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}D$) if

where ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{D}_{n}$, the inner limit, consists of all possible limit points of the sequences $\{{x}_{n}\}$ with ${x}_{n}\in {D}_{n}$ for all $n\in \mathbf{N}$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{D}_{n}$, the outer limit, consists of all possible cluster points of such sequences.

**Definition 4.2** [22]

A sequence $\{{F}_{n}\}$ of nonempty set-valued mappings ${F}_{n}:{\mathbb{R}}^{k}\to {2}^{{\mathbb{R}}^{h}}$ is said to be *convergent to a set*-*valued mapping* $F:{\mathbb{R}}^{k}\to {2}^{{\mathbb{R}}^{h}}$ *in the sense of Painlevé*-*Kuratowski* (for short, ${F}_{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}F$) if $gph({F}_{n})\stackrel{\mathrm{P}.\mathrm{K}.}{\to}gph(F)$, where $gph({F}_{n})=\{(x,z)\in {\mathbb{R}}^{k}\times {\mathbb{R}}^{h}:x\in dom{F}_{n},z\in {F}_{n}(x)\}$ and $gph(F)=\{(x,z)\in {\mathbb{R}}^{k}\times {\mathbb{R}}^{h}:x\in domF,z\in F(x)\}$.

We say that, for each $i\in I$, a sequence $\{({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\}\subset {M}_{i}$ converges to $({A}_{i},{T}_{i},{G}_{i})\in {M}_{i}$ in the sense of Painlevé-Kuratowski (for short, $({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\stackrel{\mathrm{P}.\mathrm{K}.}{\to}({A}_{i},{T}_{i},{G}_{i})$) if ${A}_{i}^{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}{A}_{i}$, ${T}_{i}^{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}{T}_{i}$ and ${G}_{i}^{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}{G}_{i}$.

Next, we give the definition of the Hadamard well-posedness for (SQVIP). As mentioned above, we denote by *S* the solution set of (SQVIP) determined by $({A}_{i},{T}_{i},{G}_{i})$ for each $i\in I$. Similarly, we denote by ${S}_{n}$ the solution set of (SQVIP)_{
n
} determined by $({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})$ for each $i\in I$ and $n\in \mathbf{N}$, where (SQVIP)_{
n
} is formulated as follows:

for all ${z}_{i}\in {A}_{i}^{n}(\overline{x})$.

**Definition 4.3** (SQVIP) is said to be *Hadamard well*-*posed* if its solution set $S\ne \mathrm{\varnothing}$ and, when, for each $i\in I$, every sequence of pairs $\{({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\}\subset {M}_{i}$ converges to $({A}_{i},{T}_{i},{G}_{i})\in {M}_{i}$ in the sense of Painlevé-Kuratowski, any sequence $\{{x}^{n}\}$ satisfying ${x}^{n}\in {S}_{n}$ has a subsequence which converges strongly to a point in *S*.

**Theorem 4.1** *For each* $i\in I$, *let* $({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\in {M}_{i}$ *for all* $n\in \mathbf{N}$. *Then the solution set* ${S}_{n}$ *for* (SQVIP)_{
n
} *is closed*.

*Proof*Without loss of generality, we suppose that $n=1$. Take any sequence $\{{x}^{n}\}\subset {S}_{1}$ satisfying ${x}^{n}\to {x}^{\ast}$. For each $i\in I$, since ${K}_{i}$ is closed, it follows that

*K*is closed and ${x}^{\ast}\in K$. Now, $\{{x}^{n}\}\subset {S}_{1}$ implies that, for each $i\in I$, there exists ${y}_{i}^{n}\in {T}_{i}^{1}({x}^{n})$ such that

Since ${G}_{i}^{1}$ is u.s.c. and compact-valued, we know that ${G}_{i}^{1}$ is closed, which implies that $0\in {G}_{i}^{1}({x}^{\ast},{y}^{\ast},{\overline{z}}_{i})$. Therefore, ${S}_{1}$ is closed. This completes the proof. □

**Theorem 4.2**

*For each*$i\in I$,

*let*${K}_{i}$

*be a nonempty compact subset of*${X}_{i}$, $({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\in {M}_{i}$

*for all*$n\in \mathbf{N}$, $({A}_{i},{T}_{i},{G}_{i})\in {M}_{i}$

*and*$({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\stackrel{\mathrm{P}.\mathrm{K}.}{\to}({A}_{i},{T}_{i},{G}_{i})$.

*Then*

*Proof*Suppose that (12) does not hold. Then there exists ${x}^{\ast}$ satisfying

*K*is compact. Again, from ${S}_{n}\subset K$, $S\subset K$ and Theorem 4.1, it follows that ${S}_{n}$ and

*S*are both compact. Thus, for

*n*sufficiently large, there exists $\u03f5>0$ satisfying

*y*and the radius

*ϵ*. It follows from ${x}^{n}\in {S}_{n}$ that, for each $i\in I$, there exists ${y}_{i}^{n}\in {T}_{i}^{n}({x}^{n})$ such that

Again, since $\{{x}^{n}\}$, $\{{y}^{n}\}$, $\{{z}^{n}\}$, and $\{{G}_{i}^{n}\}$ are bounded, it follows that there exists a subsequence of $\{{g}_{i}^{n}\}$ converging strongly to a point ${g}_{i}\in {Z}_{i}$ and so, from (16), ${g}_{i}\in {G}_{i}({x}^{\ast},{y}^{\ast},{\overline{z}}_{i})$. Since ${g}_{i}^{n}=0$ for all $n\in \mathbf{N}$, we get ${g}_{i}=0$. This implies that ${x}^{\ast}\in S$, which is a contradiction. This completes the proof. □

**Theorem 4.3** *For each* $i\in I$, *let* ${K}_{i}$ *be a nonempty compact subset of* ${X}_{i}$, $({A}_{i},{T}_{i},{G}_{i})\in {M}_{i}$ *and* $S\ne \mathrm{\varnothing}$. *Then* (SQVIP) *is Hadamard well*-*posed*.

*Proof*For each $i\in I$, let $\{({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\}\subset {M}_{i}$, $({A}_{i}^{n},{T}_{i}^{n},{G}_{i}^{n})\stackrel{\mathrm{P}.\mathrm{K}.}{\to}({A}_{i},{T}_{i},{G}_{i})$ and $\{{x}^{n}\}$ be a sequence satisfying ${x}^{n}\in {S}_{n}$. For each $i\in I$, by the compactness of ${K}_{i}$, we know that

*K*is compact. Again, from $\{{x}^{n}\}\subset K$ and the compactness of

*K*, it follows that ${x}^{n}\to {x}^{\ast}\in K$ and so, from Theorem 4.2,

Thus $\{{x}^{n}\}$ has a subsequence which converges strongly to an element in *S* and so (SQVIP) is Hadamard well-posed. This completes the proof. □

## Declarations

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (11171237) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013053358).

## Authors’ Affiliations

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