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The wellposedness for a system of generalized quasivariational inclusion problems
Journal of Inequalities and Applications volume 2014, Article number: 321 (2014)
Abstract
We introduce the concept of LevitinPolyak wellposedness for a system of generalized quasivariational inclusion problems and show some characterizations of LevitinPolyak wellposedness for the system of generalized quasivariational inclusion problems under some suitable conditions. We also give some results concerned with the Hadamard wellposedness for the system of generalized quasivariational inclusion problems.
MSC:49J40, 49K40, 90C31.
1 Introduction
Wellposedness plays a crucial role in the theory and methodology of scalar optimization problems. In 1966, Tykhonov [1] first introduced the concept of wellposedness for a global minimizing problem, which has become known as Tykhonov wellposedness. Soon after, LevitinPolyak [2] strengthened the concept of Tykhonov wellposedness, already known as the LevitinPolyak (for short, LP) wellposedness. Subsequently, some authors studied the LP wellposedness for convex scalar optimization problems with functional constraints [3], vector optimization problems [4], variational inequality problems [5], generalized mixed variational inequality problems [6], generalized quasivariational inequality problems [7], generalized vector variational inequality problems [8], equilibrium problems [9], vector equilibrium problems [10], generalized vector quasiequilibrium problems [11] and generalized quasivariational inclusion and disclusion problems [12]. Another important notion of the wellposedness for a minimizing problem is the wellposedness by perturbations or the extended wellposedness due to Zolezzi [13]. The notion of the wellposedness by perturbations establishes a form of continuous dependence of the solutions upon a parameter. Recently, Lemaire et al. [14] introduced the wellposedness by perturbations for variational inequalities and Fang et al. [15] considered the wellposedness by perturbations for mixed variational inequalities in Banach spaces. For more details about the wellposedness by perturbations, we refer readers to [16, 17] and the references therein.
On the other hand, for optimization problems, there is another concept of wellposedness, which has become known as the Hadamard wellposedness. The concept of Hadamard wellposedness 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 wellposedness can be found in [18–20]. Recently, the concept of Hadamard wellposedness has been extended to vector optimization problems and vector equilibrium problems. Li and Zhang [21] investigated the Hadamard wellposedness for vector optimization problems. Zeng et al. [22] obtained a sufficient condition for the Hadamard wellposedness of a setvalued optimization problem. Salamon [23] investigated the generalized Hadamard wellposedness for parametric vector equilibrium problems with trifunctions.
Very recently, Lin and Chuang [24] studied the wellposedness in the generalized sense for variational inclusion problems and variational disclusion problems, the wellposedness for optimization problems with variational inclusion problems, variational disclusion problems as constraints. Motivated by Lin, Wang et al. [25] investigated the wellposedness for generalized quasivariational inclusion problems and for optimization problems with generalized quasivariational inclusion problems as constraints. A system of generalized quasivariational inclusion problems, which consists of a family of generalized quasivariational inclusion problems defined on a product set, was first introduced by Lin [26]. It is well known that the system of generalized quasivariational inclusion problems contains the system of variational inequalities, the system of equilibrium problems, the system of vector equilibrium problems, the system of vector quasiequilibrium problems, the system of generalized vector quasiequilibrium 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 LevitinPolyak and Hadamard wellposedness for the system of generalized quasivariational inclusion problems. Therefore, it is very interesting to generalize the concept of LevitinPolyak and Hadamard wellposedness to the system of generalized quasivariational inclusion problems.
Motivated and inspired by research work mentioned above, in this paper, we study the LP and Hadamard wellposedness for the system of generalized quasivariational inclusion problems. This paper is organized as follows. In Section 2, we introduce the concept of LP wellposedness for the system of generalized quasivariational inclusion problems. Some characterizations of the LP wellposedness for the system of generalized quasivariational inclusion problems are shown in Section 3. Some results concerned with Hadamard wellposedness for the system of generalized quasivariational 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 setvalued 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 quasivariational inclusion problems (for short, SQVIP).
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}={({\overline{y}}_{i})}_{i\in I}\in {T}_{i}(\overline{x}) satisfying
for all {z}_{i}\in {A}_{i}(\overline{x}). We denote by S the solution set of (SQVIP).
If the mapping {G}_{i}:K\times Y\times {K}_{i}\to {2}^{{Z}_{i}} is perturbed by a parameter p\in P, that is, {G}_{i}:P\times K\times Y\times {K}_{i}\to {2}^{{Z}_{i}} such that, for some {p}^{\ast}\in P, {G}_{i}({p}^{\ast},x,y,{z}_{i})={G}_{i}(x,y,{z}_{i}) for all (x,y,{z}_{i})\in K\times Y\times {K}_{i}, then, for any p\in P, we define a parametric system of generalized quasivariational inclusion problem (for short, PSQVIP): 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}={({\overline{y}}_{i})}_{i\in I}\in {T}_{i}(\overline{x}) satisfying
for all {z}_{i}\in {A}_{i}(\overline{x}).
Some special cases of (SQVIP) are as follows:

(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 singlevalued 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})
for all {z}_{i}\in {A}_{i}(\overline{x}), which has been studied by Peng and Wu [34] and the references therein.

(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 setvalued 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 setvalued vector quasiequilibrium 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})\u2288int{C}_{i}(\overline{x})
for all {z}_{i}\in {A}_{i}(\overline{x}).

(III)
If the index set I is a single set, then (SQVIP) reduces to the generalized quasivariational inclusion problem studied in Wang et al. [12, 25] and the references therein.
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
and
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 wellposed 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 wellposed 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]
Let {E}_{1}, {E}_{2} be two topological spaces. A setvalued mapping F:{E}_{1}\to {2}^{{E}_{2}} is said to be:

(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 setvalued 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]
Let A be a nonempty subset of X, the measure of noncompactness μ of the set A is defined by
Definition 2.6 [38]
Let 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 ab\parallel.
3 The LevitinPolyak wellposedness for (SQVIP)
In this section, we discuss some metric characterizations of the LP wellposedness for (SQVIP). First, we introduce the following LP approximating solution set for (SQVIP):
for all \delta ,\u03f5>0, where B({p}^{\ast},\delta ) denotes the closed ball centered at {p}^{\ast} with radius δ.
Clearly, we have the following:

(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 compactvalued, {A}_{i}:K\to {2}^{{X}_{i}} be closedvalued 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
and
for all {z}_{i}\in {A}_{i}(\overline{x}). Clearly, {p}^{n}\to {p}^{\ast}. Since \{{y}_{i}^{n}\}\subseteq {T}_{i}(\overline{x}) and {T}_{i}(\overline{x}) is a compact set, there exist a subsequence \{{y}_{i}^{{n}_{k}}\} of \{{y}_{i}^{n}\} and {\overline{y}}_{i}\in {T}_{i}(\overline{x}) such that {y}_{i}^{{n}_{k}}\to {\overline{y}}_{i} and, for each k\in N,
for all {z}_{i}\in {A}_{i}(\overline{x}). For all {z}_{i}\in {A}_{i}(\overline{x}), there exists {\lambda}_{k}\in {B}^{+}(0,\frac{1}{{n}_{k}}) such that
for all {z}_{i}\in {A}_{i}(\overline{x}). Clearly, {\lambda}_{k}\to 0. Since (p,y)\mapsto {G}_{i}(p,\overline{x},y,{z}_{i}) is closed for all (\overline{x},{z}_{i})\in K\times {K}_{i}, this together with (2) implies that
for all {z}_{i}\in {A}_{i}(\overline{x}). Since {A}_{i} is closedvalued, 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
Then it is easy to see that all the conditions of Proposition 3.1 are satisfied. By Proposition 3.1, S={\bigcap}_{\delta >0,\u03f5>0}\mathrm{\Omega}(\delta ,\u03f5). Indeed, for all \delta ,\u03f5>0,
and
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 finitedimensional space;

(ii)
{T}_{i}:K\to {2}^{{Y}_{i}} is u.s.c. and compactvalued;

(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,
and there exists {y}_{i}^{n}\in {T}_{i}({x}^{n}) such that
for all {z}_{i}\in {A}_{i}({x}^{n}). Since P is a finitedimensional 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,
Assume that the left inequality does not hold. Then there exists \gamma >0 such that
Thus there exist an increasing sequence \{{n}_{k}\} and a sequence \{{u}_{i}^{k}\} with {u}_{i}^{k}\in {A}_{i}({x}^{{n}_{k}}) such that
Since, for each i\in I, {A}_{i} is closed and (s,s)subcontinuous, the sequence \{{u}_{i}^{k}\} has a subsequence, which is still denoted by \{{u}_{i}^{k}\}, converging strongly to a point {\overline{u}}_{i}\in {A}_{i}(\overline{x}). It follows that, for each i\in I,
which is a contradiction. Thus, for each i\in I, {d}_{i}(\overline{x},{A}_{i}(\overline{x}))\le \u03f5. Since, for each i\in I, {T}_{i}:K\to {2}^{{Y}_{i}} is u.s.c. and compactvalued, there exist a subsequence \{{y}_{i}^{{n}_{k}}\} of \{{y}_{i}^{n}\} and {\overline{y}}_{i}\in {T}_{i}(\overline{x}) such that {y}_{i}^{{n}_{k}}\to {\overline{y}}_{i}. For any {\overline{z}}_{i}\in {A}_{i}(\overline{x}), since {A}_{i} is l.s.c., there exists a sequence \{{z}_{i}^{k}\} with {z}_{i}^{k}\in {A}_{i}({x}^{{n}_{k}}) such that {z}_{i}^{k}\to {\overline{z}}_{i} and, for each k\in \mathbf{N},
Since {G}_{i} is closed and {e}_{i} is continuous, we obtain
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 compactvalued;

(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 wellposed by perturbations if and only if, for any \delta ,\u03f5>0,
as (\delta ,\u03f5)\to (0,0).
Proof Suppose that (SQVIP) is LP wellposed 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.
Now, we show that
as (\delta ,\u03f5)\to (0,0). If not, then there exist \gamma >0, sequences \{{\delta}_{n}\} and \{{\u03f5}_{n}\} of nonnegative real numbers with ({\delta}_{n},{\u03f5}_{n})\to (0,0), and the sequences \{{x}^{n}\} and \{{\overline{x}}^{n}\} with {x}^{n},{\overline{x}}^{n}\in \mathrm{\Omega}(\delta ,\u03f5) satisfying
for all n\in \mathbf{N}. Since {x}^{n},{\overline{x}}^{n}\in \mathrm{\Omega}(\delta ,\u03f5), there exist {p}^{n},{\overline{p}}^{n}\in B({p}^{\ast},{\delta}_{n}) and {y}_{i}^{n}\in {T}_{i}({x}^{n}) and {\overline{y}}_{i}^{n}\in {T}_{i}({\overline{x}}^{n}) such that
for all {z}_{i}\in {A}_{i}({x}^{n}) and
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 wellposed 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).
Conversely, suppose that (4) holds. Let \{{p}^{n}\}\subseteq P be any sequence with {p}^{n}\to {p}^{\ast} and \{{x}^{n}\} be the LP approximating solution sequence for (SQVIP) corresponding to \{{p}^{n}\}. Then there exist a sequence \{{\u03f5}_{n}\} of nonnegative real numbers with {\u03f5}_{n}\to 0 and {y}_{i}^{n}\in {T}_{i}({x}^{n}) such that
and
for all n\in \mathbf{N}. Set {\delta}_{n}={d}_{0}({p}^{n},{p}^{\ast}). Then {p}^{n}\in B({p}^{\ast},{\delta}_{n}) and {x}^{n}\in \mathrm{\Omega}({\delta}_{n},{\u03f5}_{n}) and {\delta}_{n}\to 0. It follows from (4) that \{{x}^{n}\} is a Cauchy sequence and so it converges strongly to a point \overline{x}\in K. By the similar arguments as in the proof of Proposition 3.2, we can show that {\overline{x}}_{i}\in {A}_{i}(\overline{x}) and there exists {\overline{y}}_{i}\in {T}_{i}(\overline{x}) such that
for all {z}_{i}\in {A}_{i}(\overline{x}). Thus \overline{x} is a solution of (SQVIP).
Finally, to complete the proof, it is sufficient to prove that (SQVIP) has a unique solution. If (SQVIP) has two distinct solutions 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
Then A is (s,s)subcontinuous, l.s.c. and closed, T is u.s.c. and compactvalued and G is closed. For any \delta ,\u03f5>0, we have
and
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 finitedimensional space. We assume that

(i)
{T}_{i}:K\to {2}^{{Y}_{i}} is u.s.c. and compactvalued;

(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 wellposed by perturbations if and only if, for any \delta ,\u03f5>0,
as (\delta ,\u03f5)\to (0,0).
Proof Suppose that (SQVIP) is generalized LP wellposed by perturbations. Then S is nonempty.
Now, we prove that 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 wellposed 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
Since S is compact,
In order to prove \mu (\delta ,\mathrm{\Omega}(\u03f5))\to 0, we need to prove that
as (\delta ,\u03f5)\to (0,0). Assume that this is not true. Then there exist \alpha >0, and the sequences \{{\delta}_{n}\} and \{{\u03f5}_{n}\} of nonnegative real numbers with ({\delta}_{n},{\u03f5}_{n})\to (0,0) and \{{x}^{n}\} with {x}^{n}\in \mathrm{\Omega}({\delta}_{n},{\u03f5}_{n}) such that, for n sufficiently large,
Since {x}^{n}\in \mathrm{\Omega}({\delta}_{n},{\u03f5}_{n}), there exists {p}^{n}\in B({p}^{\ast},{\delta}_{n}) such that, for each i\in I, {d}_{i}({x}_{i}^{n},{A}_{i}({x}^{n}))\le {\u03f5}_{n} and there exists {y}_{i}^{n}\in {T}_{i}({x}^{n}) satisfying
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 wellposedness 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).
Conversely, suppose that (9) holds. From Propositions 3.1 and 3.2, \mathrm{\Omega}(\delta ,\u03f5) is closed for any \delta ,\u03f5>0 and S={\bigcap}_{\delta >0,\u03f5>0}\mathrm{\Omega}(\delta ,\u03f5). Since \mu (\mathrm{\Omega}(\delta ,\u03f5))\to 0 as (\delta ,\u03f5)\to (0,0), theorem on p.412 in [38] can be applied and one concludes that the set S is nonempty compact and
as (\delta ,\u03f5)\to (0,0). Let \{{p}^{n}\} be any sequence in 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
and
for all {z}_{i}\in {A}_{i}({x}^{n}). For each n\in \mathbf{N}, let {\delta}_{n}={d}_{0}({p}^{n},{p}^{\ast}). Then {p}^{n}\in B({p}^{\ast},{\delta}_{n}) and {x}^{n}\in \mathrm{\Omega}({\delta}_{n},{\u03f5}_{n}). Thus it follows from (11) that
as ({\delta}_{n},{\u03f5}_{n})\to (0,0). The compactness of S implies that (SQVIP) is generalized LP wellposed 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
Then all the conditions of Theorem 3.2 are satisfied. It follows that, for any \delta ,\u03f5>0,
and
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 wellposedness by perturbations.
4 The Hadamard wellposedness for (SQVIP)
In this section, for each i\in I, we assume that {X}_{i}, {Y}_{i} and {Z}_{i} are finitedimensional spaces, {K}_{i}\subset {X}_{i} is a nonempty closed and convex subset.
For each i\in I, let {M}_{i} be the collection of all ({A}_{i},{T}_{i},{G}_{i}) such that

(i)
{A}_{i}:K\to {2}^{{X}_{i}} is continuous and bounded compactconvexvalued;

(ii)
{T}_{i}:K\to {2}^{{Y}_{i}} is u.s.c. and bounded compactconvexvalued;

(iii)
{G}_{i}:K\times Y\times {K}_{i}\to {2}^{{Z}_{i}} is u.s.c. and bounded compactconvexvalued.
Definition 4.1 [39]
A sequence \{{D}_{n}\} of nonempty subsets of {\mathbb{R}}^{n} is said to be convergent to D in the sense of PainlevéKuratowski (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 setvalued mappings {F}_{n}:{\mathbb{R}}^{k}\to {2}^{{\mathbb{R}}^{h}} is said to be convergent to a setvalued mapping F:{\mathbb{R}}^{k}\to {2}^{{\mathbb{R}}^{h}} in the sense of PainlevéKuratowski (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 PainlevéKuratowski (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 wellposedness 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:
Find \overline{x}\in K such that, for each i\in I, there exists {\overline{y}}_{i}\in {T}_{i}^{n}(\overline{x}) satisfying
for all {z}_{i}\in {A}_{i}^{n}(\overline{x}).
Definition 4.3 (SQVIP) is said to be Hadamard wellposed 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 PainlevéKuratowski, 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
for all {z}_{i}\in {A}_{i}^{1}({x}^{n}). For each i\in I, since {y}_{i}^{n}\in {T}_{i}^{1}({x}^{n}), {T}_{i}^{1}(\cdot ) is u.s.c. and compactvalued, this implies that there exist {y}_{i}^{\ast}\in {T}_{i}^{1}({x}^{\ast}) and a subsequence \{{y}_{i}^{{n}_{k}}\} of \{{y}_{i}^{n}\} such that {y}_{i}^{{n}_{k}}\to {y}_{i}^{\ast}. Since {A}_{i}^{1}(\cdot ) is continuous and compactvalued, it follows that {A}_{i}^{1}(\cdot ) is closed, this implies that {x}_{i}^{\ast}\in {A}_{i}^{1}({x}^{\ast}). For each {\overline{z}}_{i}\in {A}_{i}^{1}({x}^{\ast}), since {A}_{i}^{1}(\cdot ) is continuous, there exists a sequence \{{z}_{i}^{n}\}\subseteq {K}_{i} with {z}_{i}^{n}\in {A}_{i}^{1}({x}^{n}) such that
Since {G}_{i}^{1} is u.s.c. and compactvalued, 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
From (13), it follows that there exists {x}^{n}\in {S}_{n} such that the sequence \{{x}^{n}\} has a subsequence, which is still denoted by \{{x}^{n}\}, converging strongly to {x}^{\ast}. For each i\in I, since {K}_{i} is compact, we know that 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
where B(S,\u03f5)={\bigcup}_{y\in S}B(y,\u03f5) and B(y,\u03f5) denotes the ball with the center 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
for all {z}_{i}^{n}\in {A}_{i}^{n}({x}^{n}). For each i\in I, since {T}_{i}^{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}{T}_{i}, we have
Again, since \{{x}^{n}\} is bounded and \{{T}_{i}^{n}\} is bounded, there exists a subsequence of \{{y}_{i}^{n}\} with {y}_{i}^{n}\in {T}_{i}^{n}({x}^{n}) converging strongly to a point {y}_{i}^{\ast}\in {Y}_{i}. This together with (14) implies that {y}_{i}^{\ast}\in {T}_{i}({x}^{\ast}). By similar arguments, we also know that {x}_{i}^{\ast}\in {A}_{i}({x}^{\ast}). Since {A}_{i}^{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}{A}_{i}, we have
By Theorem 5.37 of [39], for all {\overline{z}}_{i}\in {A}_{i}({x}^{\ast}), there exist a sequence \{{z}_{i}^{n}\} converging strongly to {\overline{z}}_{i} and \{{x}^{n}\} such that {z}_{i}^{n}\in {A}_{i}^{n}({x}^{n}) for all n\in \mathbf{N} and {x}^{n}\to {x}^{\ast}. It follows from (14) that there exists {g}_{i}^{n}\in {G}_{i}^{n}({x}^{n},{y}^{n},{z}_{i}^{n}) such that {g}_{i}^{n}=0. Since {G}_{i}^{n}\stackrel{\mathrm{P}.\mathrm{K}.}{\to}{G}_{i}, we have
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 wellposed.
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 wellposed. This completes the proof. □
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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 (NRF2013053358).
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Li, Xb., Agarwal, R.P., Cho, Y.J. et al. The wellposedness for a system of generalized quasivariational inclusion problems. J Inequal Appl 2014, 321 (2014). https://doi.org/10.1186/1029242X2014321
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DOI: https://doi.org/10.1186/1029242X2014321
Keywords
 system of generalized quasivariational inclusion problems
 LevitinPolyak wellposedness
 Hadamard wellposedness