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 Open Access
Several types of wellposedness for generalized vector quasiequilibrium problems with their relations
 Dening Qu^{1, 2}Email author and
 Caozong Cheng^{1}
https://doi.org/10.1186/1687181220148
© Qu and Cheng; licensee Springer. 2014
 Received: 23 September 2013
 Accepted: 5 December 2013
 Published: 7 January 2014
Abstract
The conceptions of (generalized) Tykhonov wellposedness for generalized vector quasiequilibrium problems, (generalized) Hadamard wellposedness for parametrically generalized vector quasiequilibrium problems and (generalized) Tykhonov wellposedness for parametrical system of generalized vector quasiequilibrium problems are introduced. The metric characterizations and/or sufficient criteria of the proposed wellposedness are presented, and the relations between (generalized) Tykhonov wellposedness for generalized vector quasiequilibrium problems and that for constrained minimizing problems are discussed. Finally, the relations among several types of the wellposedness are exhibited in detail.
MSC:49K40, 90C31, 90C33.
Keywords
 wellposedness
 generalized vector quasiequilibrium problem
 setvalued objective mapping
 relation
1 Introduction and preliminaries
Wellposedness is very important for both theory and numerical method of many problems such as optimization problems, optimal control, variations, mathematical programming, fixedpoint problems, variational inequality, variational inclusion problems and equilibrium problems (in short, EPs), since it guarantees that for any approximating solution sequence of one of mentioned problems, there must exist a subsequence converging to some correlative solution. The classical concept of wellposedness for unconstrained optimization problem was introduced by Tykhonov [1] in Banach space in 1966. In the same year, this notion was extended to the case of constrained optimization problems by Levitin and Polyak [2]. Ever since then, various types of wellposedness for scalar or vector optimization problems with unconstraint or constraints have been widely focused on. More details on wellposedness for optimization problems, optimal control, variations and mathematical programming and for vector optimization problems can be found in the monographs [3–5] and [6], respectively. In the other directions, some kinds of wellposedness were introduced for other problems, such as fixedpoint problems [7–13], variational inequality problems [11–18], vector variational inequality problems [19], variational inclusion problems [10–12, 20–22], complementary problems [23, 24], Nash EPs in the game with two players [25–27] or nplayers [13, 28–30] and ParetoNash EPs in the game with finite or infinite players [31], and many significant results related to them were obtained.
As understood by Blum and Oetti [32], EPs contain many problems as special cases, for example, optimization problems, fixedpoint problems, variational inequality problems, complementary problems and Nash EPs. The discussion on various aspects, such as existence of solutions, iterative algorithms and stability of solutions, etc. for these problems can be classified to the corresponding discussion for general EPs. Some results on different types of wellposedness for EPs were obtained. For instance, Long et al. [33] and Zaslavski [34] introduced the notions of generalized LevitinPolyak wellposedness for explicit constrained EPs and generic wellposedness for EPs, respectively. Bianchi et al. [35] defined T_{opt} and T_{vi}wellposedness for EPs and proposed the conception of Hadamard wellposedness for parametrical EPs to unify two notions as above. Fang and Hu [36] and Wang and Cheng [37] defined wellposedness for parametrical systems of EPs which are the generalizations of Stampacchia/Minty type variational inequalities and quasivariationallike inequalities. In addition, Fang et al. [38] introduced generalized wellposedness for a parametrical system of EPs. The sufficient and necessary conditions and metric characterizations of corresponding wellposedness were investigated in [33–38].
Recently, multifarious conceptions of wellposedness for vector equilibrium problems (in short, VEPs) and the related results have been recorded in many literature works. For example, the conceptions of (generalized) LevitinPolyak wellposedness for VEPs [39, 40], convex symmetric vector quasiequilibrium problems (VQEPs for brevity) [41], VQEPs without constraints [42] and VQEPs with functional constraints [43–45] were introduced respectively, and their criteria and/or metric characterizations were discussed. Besides, the notions of M and Bwellposedness for VEPs were presented in [46] and their sufficient conditions were given. The generalized Tykhonov wellposedness for system of VEPs was studied by Peng and Wu [47]. Also, for the wellposedness of parametric strong VQEPs, refer to [48].
Up to the present, there are few literature works to record the wellposedness for EPs involving setvalued objective mappings. The aim of this article is to explore wellposed VEPs with setvalued objective mappings. This paper is organized as follows. A generalized nonlinear scalarization function, which will be used to construct gap functions of generalized vector quasiequilibrium problems (in short, GVQEPs), is introduced in this section. The metric characterizations and sufficient criteria of (generalized) Tykhonov wellposedness, (G)TWPness for brevity, for GVQEPs are presented by applying Kuratowski noncompactness measure, and the relations between (G)TWPness for GVQEPs and that for constrained minimizing problems are exhibited in Section 2. The sufficient conditions of (generalized) Hadamard wellposedness ((G)HWPness, for brevity) for parametrically GVQEPs are proposed in Section 3. The metric characterizations and sufficient criteria of (G)TWPness for parametrical system of GVQEPs are presented in Section 4. Finally, the relations among the types of proposed wellposedness are illuminated in detail in Section 5.
We first recall some notions and concepts. ℝ, ${\mathbb{R}}_{+}$ and ℕ denote the sets of real numbers, nonnegative real numbers and positive integers, respectively, and $\mathcal{N}(\ast )$ denotes the collection of all open neighborhoods of ∗, where ∗ is a point or a set in a topological space.
Definition 1.1 Let X be a topological space and $E\subset X$ be a nonempty subset. A realvalued function $g:E\to \mathbb{R}$ is said to be upper semicontinuous on E if $\{x\in E:g(x)<\lambda \}$ is open for each $\lambda \in \mathbb{R}$; lower semicontinuous on E if $\{x\in E:g(x)>\lambda \}$ is open for each $\lambda \in \mathbb{R}$.
Definition 1.2 ([49])
Let X and Y be topological spaces and $E\subset X$ be a nonempty subset. A setvalued mapping $G:E\to {2}^{Y}$ is said to be upper semicontinuous at ${x}_{0}\in E$ if for any $N\in \mathcal{N}(G({x}_{0}))$, there exists $B\in \mathcal{N}({x}_{0})$ such that $G(x)\subset N$ for all $x\in B$; lower semicontinuous at ${x}_{0}\in E$ if for any ${y}_{0}\in G({x}_{0})$ and any $N\in \mathcal{N}({y}_{0})$, there exists $B\in \mathcal{N}({x}_{0})$ such that $G(x)\cap N\ne \mathrm{\varnothing}$ for all $x\in B$; upper semicontinuous (resp., lower semicontinuous) on E if G is upper semicontinuous (resp., lower semicontinuous) at each $x\in E$; closed if its graph $Graph(G)=\{(x,y)\in E\times Y:y\in G(x)\}$ is closed in $E\times Y$.
 (i)
Let X be a topological space and $E\subset X$ be a nonempty subset. An extended realvalued function $h:E\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is said to be levelcompact on E if $\{x\in E:h(x)\le \lambda \}$ is compact for any $\lambda \in \mathbb{R}$.
 (ii)Further suppose that $(X,\parallel \cdot \parallel )$ is a finitedimensional normed linear space. h is said to be levelbounded on E if E is bounded or$\underset{x\in E,\parallel x\parallel \to +\mathrm{\infty}}{lim}h(x)=+\mathrm{\infty}.$
respectively, where $d(x,B)=inf\{d(x,y):y\in B\}$ is the distance from x to B.
 (i)
$\alpha (A)=0$ if A is compact;
 (ii)
$\alpha (B)\le \alpha (A)+2\epsilon $, where $B=\{a\in X:d(a,A)<\epsilon \}$;
 (iii)
$\alpha (A)=\alpha (clA)$, where clA is the closure of A.
A subset D of a linear space Y is called a cone if $\lambda x\in D$ for all $x\in D$ and $\lambda >0$. Let D be a cone in Y and $A\subset Y$. D is called proper if $D\ne Y$. A is called Dclosed [52] if $A+clD$ is closed and Dbounded [52] if for each neighborhood U of zero in Y, there exists $\lambda >0$ such that $A\subset \lambda U+D$. Obviously, any compact subset in Y is both Dclosed and Dbounded. Let X and Y be nonempty sets. A setvalued mapping $G:X\to {2}^{Y}$ is said to be strict if $G(x)\ne \mathrm{\varnothing}$ for any $x\in X$.
In order to construct gap functions of GVQEPs, a generalized nonlinear scalarization function of a setvalued mapping and its properties are listed.
In view of Lemma 3.1 in [53], we can define a general nonlinear scalarization function as follows.
It is easy to find differences between the generalized nonlinear scalarization function ${\zeta}_{G}$ and the general nonlinear scalarization function ${\xi}_{G}$ given by Qu and Cheng [54]. But if $X=Y=Z=E=F$ and $G(u)=\{u\}$ for all $u\in F$, then both ${\zeta}_{G}$ and ${\xi}_{G}$ reduce simultaneously to the nonlinear scalarization function of a singlevalued mapping introduced by Chen and Yang [55]. According to Proposition 3.1 in [53], we have the following.
 (i)
${\zeta}_{G}(x,u)<\lambda \u27faG(u)\cap (\lambda e(x)intC(x))\ne \mathrm{\varnothing}$.
 (ii)
${\zeta}_{G}(x,u)\le \lambda \u27faG(u)\cap (\lambda e(x)C(x))\ne \mathrm{\varnothing}$.
2 (G)TWPness for GVQEPs
where $f:E\times F\to {2}^{Y}$, $P:E\to {2}^{E}$ and $Q:E\to {2}^{F}$ are strict setvalued mappings. Ω denotes the solution set of (GVQEP).
(resp., ${\mathrm{\Omega}}_{2}(\epsilon )=\{x\in E:x\text{satisfies (2.1)(2.3)}\}$).
A sequence $\{{x}_{n}\}$ is called a type I approximating solution sequence, ASS1 for brevity (resp., type II approximating solution sequence, ASS2 for brevity) of (GVQEP) if there exists $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}({\epsilon}_{n})$ (resp., ${x}_{n}\in {\mathrm{\Omega}}_{2}({\epsilon}_{n})$).
Definition 2.2 (GVQEP) is said to be generalized type I Tykhonov wellposed, GTWP1 for brevity (resp., generalized type II Tykhonov wellposed, GTWP2 for brevity) if $\mathrm{\Omega}\ne \mathrm{\varnothing}$ and for any ASS1 (resp., ASS2) $\{{x}_{n}\}$ of (GVQEP), there exists a subsequence $\{{x}_{{n}_{i}}\}$ such that ${x}_{{n}_{i}}\to \overline{x}\in \mathrm{\Omega}$; to be type I Tykhonov wellposed, TWP1 for brevity (resp., type II Tykhonov wellposed, TWP2 for brevity) if it is GTWP1 (resp., GTWP2) and Ω is a singleton.
When (GVQEP) reduces to (VEP), generalized type I Tykhonov wellposedness (GTWPness1 for brevity) and generalized type II Tykhonov wellposedness (GTWPness2 for brevity) for (GVQEP) become type I LevitinPolyak wellposedness and type II LevitinPolyak wellposedness for (VEP), respectively, which were discussed by Li and Li [39] in the case that X and Y are locally convex topological vector spaces, where X is equipped with a metric d compatible with its topology, F is a nonempty closed convex subset, and f is a continuous mapping.
Remark 2.1 (i) An ASS2 of (GVQEP) must be its ASS1. So GTWPness1 (resp., TWPness1) for (GVQEP) implies its GTWPness2 (resp., TWPness2), where TWPness1 and TWPness2 are the abbreviations of type I Tykhonov wellposedness and type II Tykhonov wellposedness, respectively.
Then $d(\overline{x},P(\overline{x}))\le \epsilon $ for any $\epsilon \ge 0$. It follows from (2.4) and (1.1) that (2.2) holds.
(iii) (GVQEP) is GTWP1 if and only if Ω is nonempty compact and $d({x}_{n},\mathrm{\Omega})\to 0$ for its any ASS1 $\{{x}_{n}\}$. Assume that Ω is compact, (GVQEP) is GTWP2 if and only if $\mathrm{\Omega}\ne \mathrm{\varnothing}$ and $d({x}_{n},\mathrm{\Omega})\to 0$ for its any ASS2 $\{{x}_{n}\}$. In addition, (GVQEP) is TWP1 (resp., TWP2) if and only if $\mathrm{\Omega}=\{\overline{x}\}$ and $d({x}_{n},\overline{x})\to 0$ for any ASS1 (resp., ASS2) $\{{x}_{n}\}$ of (GVQEP).
The following example shows that neither the GTWPness1 for (GVQEP) nor the compactness of Ω can be deduced from the GTWPness2 for (GVQEP).
Obviously, $\mathrm{\Omega}=\mathbb{R}$ is noncompact and so (GVQEP) is not GTWP1 by Remark 2.1(iii). However, (GVQEP) is GTWP2. In fact, for any ASS2 $\{{x}_{n}\}$ of (GVQEP), let $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{2}({\epsilon}_{n})$. For ${x}_{n}$ and ${\epsilon}_{n}$, (2.1) and (2.2) hold trivially.
The fact ${x}_{n}\to 0\in \mathrm{\Omega}$ proceeds from (2.5) and ${\epsilon}_{n}\to 0$.
2.1 Metric characterization of (G)TWPness for (GVQEP)
The metric characterizations of (G)TWPness for GVQEPs are depicted by using Kuratowski noncompactness measure and the corresponding results are obtained as follows.

(a1) f is lower semicontinuous on $E\times F$;

(a2) P is compactvalued and upper semicontinuous on E;

(a3) Q is lower semicontinuous on E;

(a4) W is upper semicontinuous on E, where $W:E\to {2}^{Y}$ is defined as $W(x)=Y\setminus intC(x)$ for all $x\in E$;

(a5) e is continuous on E.
 (i)
${\mathrm{\Omega}}_{1}(\epsilon )$ is closed for each $\epsilon >0$;
 (ii)
$\mathrm{\Omega}={\bigcap}_{\epsilon >0}{\mathrm{\Omega}}_{1}(\epsilon )$.
Thus $\overline{x}\in {\mathrm{\Omega}}_{1}(\epsilon )$ and ${\mathrm{\Omega}}_{1}(\epsilon )$ is closed.
(ii) $\mathrm{\Omega}\subset {\bigcap}_{\epsilon >0}{\mathrm{\Omega}}_{1}(\epsilon )$ stems easily from Remark 2.1(ii). For any $\overline{x}\in {\bigcap}_{\epsilon >0}{\mathrm{\Omega}}_{1}(\epsilon )$, (2.8) and (2.10) hold for any $\epsilon >0$. Then $\overline{x}\in P(\overline{x})$ by (a2), and $y+\epsilon e(\overline{x})\in W(\overline{x})$ for any $y\in f(\overline{x},z)$ and $z\in Q(\overline{x})$ by (2.10). Letting $\epsilon \to 0$, we have $y\in W(\overline{x})$, and so $f(\overline{x},z)\cap (intC(\overline{x}))=\mathrm{\varnothing}$. Consequently, $\overline{x}\in \mathrm{\Omega}$ and ${\bigcap}_{\epsilon >0}{\mathrm{\Omega}}_{1}(\epsilon )\subset \mathrm{\Omega}$. □
Lemma 2.2 Suppose that (a1)(a5) and
(a6) $0\in f(x,Q(x))$ for all $x\in E$
 (i)
${\mathrm{\Omega}}_{2}(\epsilon )$ is closed for each $\epsilon >0$;
 (ii)
$\mathrm{\Omega}={\bigcap}_{\epsilon >0}{\mathrm{\Omega}}_{2}(\epsilon )$.
Proof (i) For each $\epsilon >0$, let $\{{x}_{n}\}\subset {\mathrm{\Omega}}_{2}(\epsilon )$ with ${x}_{n}\to \overline{x}$. It is enough to testify that $\overline{x}$ satisfies (2.3) by Lemma 2.1(i). As a matter of fact, $0\in f(\overline{x},\tilde{z})$ for some $\tilde{z}\in Q(\overline{x})$ according to (a6). As a result, $f(\overline{x},\tilde{z})\cap (\epsilon e(\overline{x})C(\overline{x}))\ne \mathrm{\varnothing}$ owing to (1.1).
(ii) $\mathrm{\Omega}={\bigcap}_{\epsilon >0}{\mathrm{\Omega}}_{1}(\epsilon )\supset {\bigcap}_{\epsilon >0}{\mathrm{\Omega}}_{2}(\epsilon )$ by Lemma 2.1(ii). We only need to show that $\overline{x}$ satisfies (2.3) for any $\overline{x}\in \mathrm{\Omega}$ and $\epsilon >0$, while this can be deduced easily from the proof of (i). □
 (i)If (GVQEP) is GTWP 1, then${\mathrm{\Omega}}_{1}(\epsilon )\ne \mathrm{\varnothing}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\epsilon 0\mathit{\text{and}}\underset{\epsilon \to 0}{lim}\alpha ({\mathrm{\Omega}}_{1}(\epsilon ))=0.$(2.11)
 (ii)
If (a1)(a5) hold, then (2.11) implies that (GVQEP) is GTWP 1.
It is enough to testify that $\tilde{e}({\mathrm{\Omega}}_{1}(\epsilon ),\mathrm{\Omega})\to 0$ as $\epsilon \to 0$. Otherwise, there exist $r>0$, ${\epsilon}_{n}\downarrow 0$ and ${x}_{n}\in {\mathrm{\Omega}}_{1}({\epsilon}_{n})$ such that $d({x}_{n},\mathrm{\Omega})\ge r$ for all $n\in \mathbb{N}$. Clearly, $\{{x}_{n}\}$ is an ASS1 of (GVQEP). Thus $d({x}_{n},\mathrm{\Omega})\to 0$ by Remark 2.1(iii), which contradicts $d({x}_{n},\mathrm{\Omega})\ge r$ for all $n\in \mathbb{N}$.
by Kuratowski theorem [51]. Resultingly, $d({x}_{n},\mathrm{\Omega})\to 0$ and (GVQEP) is GTWP1 by Remark 2.1(iii). □
Similarly, the following result can be proved by using Lemma 2.2.
 (i)If (GVQEP) is GTWP 2, then${\mathrm{\Omega}}_{2}(\epsilon )\ne \mathrm{\varnothing}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\epsilon 0\mathit{\text{and}}\underset{\epsilon \to 0}{lim}\alpha ({\mathrm{\Omega}}_{2}(\epsilon ))=0.$(2.12)
 (ii)
If (a1)(a6) are satisfied, then (2.12) implies that (GVQEP) is GTWP 2.
When Ω is a singleton, the following corollary that shows the metric information of TWPness1 and TWPness2 for (GVQEP) follows from Theorems 2.1 and 2.2.
 (i)
If (GVQEP) is TWP 1 (resp., TWP 2), then (2.11) (resp., (2.12)) holds.
 (ii)
If (a1)(a5) (resp., (a1)(a6)) hold, then (2.11) (resp., (2.12)) implies that (GVQEP) is TWP 1 (resp., TWP 2).
2.2 Relations between (G)TWPness for (GVQEP) and that for constrained minimizing problem
where $\varphi :E\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is a proper function and $P:E\to {2}^{E}$ is a strict setvalued mapping. The optimal set and optimal value of (CMP) are denoted by argminϕ and $\tilde{\nu}$, respectively. In this subsection, the equivalent relations between (G)TWPness for (GVQEP) and that for (CMP) are discussed, where a gap function of (GVQEP) is taken as the objective function ϕ of (CMP).
Definition 2.4 (CMP) is said to be GTWP1 (resp., GTWP2) if $argmin\varphi \ne \mathrm{\varnothing}$ and for any MS1 (resp., MS2) $\{{x}_{n}\}$ of (CMP), there exists a subsequence $\{{x}_{{n}_{i}}\}$ such that ${x}_{{n}_{i}}\to \overline{x}\in argmin\varphi $; to be TWP1 (resp., TWP2) if it is GTWP1 (resp., GTWP2) and argminϕ is a singleton.
 (i)
$g(x)\ge 0$ for all $x\in E$;
 (ii)
$x\in \{u\in E:g(u)=0\text{and}u\in P(u)\}$ if and only if $x\in \mathrm{\Omega}$.
Further suppose that f is compactvalued in this subsection.
Proof Clearly, $\varphi (x)>\mathrm{\infty}$ for all $x\in E$. Otherwise, $\varphi (\overline{x})=\mathrm{\infty}$ for some $\overline{x}\in E$. Then ${\zeta}_{f}(\overline{x},(\overline{x},\overline{z}))\ge +\mathrm{\infty}$ for all $\overline{z}\in Q(\overline{x})$, which contradicts the fact that ${\zeta}_{f}$ is realvalued.
⟺ $x\in P(x)$, ${\zeta}_{f}(x,(x,z))\ge 0$ for all $z\in Q(x)$ (By (2.17));
⟺ $x\in P(x)$, $f(x,z)\cap (intC(x))=\mathrm{\varnothing}$ for all $z\in Q(x)$ (By Lemma 1.2(i));
⟺ $x\in \mathrm{\Omega}$,
ϕ is a gap function of (GVQEP). □
In general, ϕ is required to be lower semicontinuous. It is natural to expect the lower semicontinuity of the constructed gap function. Now assume that ϕ appearing in the rest of this section is defined as (2.16).
Proposition 2.1 If (a1)(a5) hold, then ϕ is lower semicontinuous on E. If, further, (a6) holds and $\mathrm{\Omega}\ne \mathrm{\varnothing}$, then $dom\varphi \ne \mathrm{\varnothing}$.
that is, $\overline{x}\in L(\epsilon )$. If, further, (a6) holds and $\mathrm{\Omega}\ne \mathrm{\varnothing}$, then $dom\varphi \ne \mathrm{\varnothing}$ by Lemma 2.3 and Definition 2.5. □
Theorem 2.3 Assume that (a6) holds. Then (GVQEP) is GTWP 1 (resp., GTWP 2) if and only if so is (CMP) with the objective function ϕ.
 1∘
$\{{x}_{n}\}$ is an ASS1 of (GVQEP).
⟺ There exists $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}({\epsilon}_{n})$;
⟺ There exists $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that $d({x}_{n},P({x}_{n}))\le {\epsilon}_{n}$ and ${\zeta}_{f}({x}_{n},({x}_{n},{z}_{n}))\ge {\epsilon}_{n}$ for all ${z}_{n}\in Q({x}_{n})$. (By Lemma 1.2(i));
⟺ There exists $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that $d({x}_{n},P({x}_{n}))\le {\epsilon}_{n}$ and$\varphi ({x}_{n})=\underset{{z}_{n}\in Q({x}_{n})}{sup}{\zeta}_{f}({x}_{n},({x}_{n},{z}_{n}))\le {\epsilon}_{n};$⟺ ${lim}_{n\to +\mathrm{\infty}}d({x}_{n},P({x}_{n}))=0$ and ${lim\hspace{0.17em}sup}_{n\to +\mathrm{\infty}}\varphi ({x}_{n})\le \tilde{\nu}=0$;
⟺ $\{{x}_{n}\}$ is an MS1 of (CMP).
 2∘
$\{{x}_{n}\}$ is an ASS2 of (GVQEP).
⟺ There exists $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{2}({\epsilon}_{n})$;
⟺ There exists $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}({\epsilon}_{n})$ and $f({x}_{n},{\tilde{z}}_{n})\cap ({\epsilon}_{n}e({x}_{n})C({x}_{n}))\ne \mathrm{\varnothing}$ for some ${\tilde{z}}_{n}\in Q({x}_{n})$;
⟺ (A): There exists $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}({\epsilon}_{n})$ and ${\zeta}_{f}({x}_{n},({x}_{n},{\tilde{z}}_{n}))\le {\epsilon}_{n}$ for some ${\tilde{z}}_{n}\in Q({x}_{n})$. (By Lemma 1.2(ii));
⟺ (B): There exists $\{{\beta}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\beta}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}({\beta}_{n})$ and$\varphi ({x}_{n})=\underset{{z}_{n}\in Q({x}_{n})}{sup}{\zeta}_{f}({x}_{n},({x}_{n},{z}_{n}))\ge {\beta}_{n};$⟺ ${lim}_{n\to +\mathrm{\infty}}d({x}_{n},P({x}_{n}))=0$ and ${lim\hspace{0.17em}sup}_{n\to +\mathrm{\infty}}\varphi ({x}_{n})=0$. (By 1^{∘});
⟺ $\{{x}_{n}\}$ is an MS2 of (CMP).
We can choose ${\gamma}_{n}\to 0$ and ${\tilde{z}}_{n}\in Q({x}_{n})$ is the corresponding point such that the above inequality holds. Therefore, (A) holds by taking ${\epsilon}_{n}={\beta}_{n}+{\gamma}_{n}$.
It follows from 1^{∘} (resp., 2^{∘}) that (GVQEP) is GTWP1 (resp., GTWP2) if and only if so is (CMP). □
Corollary 2.2 Assume that (a6) holds. Then (GVQEP) is TWP 1 (resp., TWP 2) if and only if so is (CMP) with the objective function ϕ.
When the assumptions in Theorem 2.3 are satisfied, we see that if (GVQEP) is GTWP1 (resp., GTWP2), then for any MS1 (resp., MS2) $\{{x}_{n}\}$ of (CMP) and for some $\overline{x}\in argmin\varphi =\mathrm{\Omega}$, ${lim\hspace{0.17em}sup}_{n\to +\mathrm{\infty}}\varphi ({x}_{n})\le \varphi (\overline{x})=\tilde{v}$ (resp., ${lim\hspace{0.17em}sup}_{n\to +\mathrm{\infty}}\varphi ({x}_{n})=\varphi (\overline{x})=\tilde{v}$) implies that $d({x}_{n},\overline{x})\to 0$, that is, $d({x}_{n},\mathrm{\Omega})\to 0$. It is reasonable to try estimating a bound below of $\varphi (x)\tilde{v}$ by using $d(x,\mathrm{\Omega})$. For the sake of this intention, a forcing function with parameter is introduced.
 (i)
(GVQEP) is GTWP 1;
 (ii)Ω is nonempty compact and there exists a forcing function with parameter $c:S\times T\to {\mathbb{R}}_{+}$ (where S is the parameter set) such that$\varphi (x)\ge c(d(x,P(x)),d(x,\mathrm{\Omega}))\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x\in E,$(2.20)
Proof By virtue of Lemma 2.3, ϕ is a gap function of (GVQEP).
there exists $\{{x}_{n}\}\subset E$ such that ${s}_{n}=d({x}_{n},P({x}_{n}))\to 0$, ${t}_{n}=d({x}_{n},\mathrm{\Omega})$ and $\varphi ({x}_{n})\to 0$ by the definition of infimum. Since $\tilde{v}=0$, $\{{x}_{n}\}$ is an MS1 of (CMP) and also an ASS1 of (GVQEP) in view of the proof of Theorem 2.3. Then (2.19) follows from the GTWPness1 for (GVQEP) and Remark 2.1(iii). Therefore, the assertion (ii) is true.
Setting ${s}_{n}=d({x}_{n},P({x}_{n}))$ and ${t}_{n}=d({x}_{n},\mathrm{\Omega})$, we have ${s}_{n}\to 0$. By the same argument given in the proof of Theorem 2.3, $\{{x}_{n}\}$ is an MS1 of (CMP). Therefore, ${lim\hspace{0.17em}sup}_{n\to +\mathrm{\infty}}\varphi ({x}_{n})\le 0$. On the other hand, ${lim\hspace{0.17em}inf}_{n\to +\mathrm{\infty}}\varphi ({x}_{n})\ge 0$ since $\varphi ({x}_{n})\ge 0$ for all $n\in \mathbb{N}$. Thus $\varphi ({x}_{n})\to 0$ and $c({s}_{n},{t}_{n})\to 0$, and so ${t}_{n}=d({x}_{n},\mathrm{\Omega})\to 0$ by (2.19). This, together with the compactness of Ω and Remark 2.1(iii), implies that (GVQEP) is GTWP1. □
Similarly, we can prove the following result by using Remark 2.1(iii).
 (i)
(GVQEP) is GTWP 2;
 (ii)
$\mathrm{\Omega}\ne \mathrm{\varnothing}$ and there exists a forcing function with parameter $c:S\times T\to {\mathbb{R}}_{+}$ (where S is the parameter set) such that (2.20) holds, where S and T are defined by (2.21).
 (i)
(GVQEP) is TWP 1 (resp., TWP 2);
 (ii)
Ω is a singleton and there exists a forcing function with parameter $c:S\times T\to {\mathbb{R}}_{+}$ (where S is the parameter set) such that (2.20) holds, where S and T are defined by (2.21).
2.3 Sufficient criteria of (G)TWP for (GVQEP)
In this subsection, we shall list some sufficient criteria of (G)TWPness for (GVQEP).
Theorem 2.6 Let (a1)(a5) hold and $\mathrm{\Omega}\ne \mathrm{\varnothing}$. If
(b1) ${\mathrm{\Omega}}_{1}({\epsilon}_{0})$ is compact for some ${\epsilon}_{0}>0$
holds, then (GVQEP) is GTWP 1 and also GTWP 2.
Proof For any ASS1 $\{{x}_{n}\}$ of (GVQEP), let $\{{\epsilon}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}({\epsilon}_{n})$.
Since ${\epsilon}_{n}\to 0$, ${x}_{n}\in {\mathrm{\Omega}}_{1}({\epsilon}_{0})$ for sufficiently large $n\in \mathbb{N}$. So $\{{x}_{n}\}$ has a subsequence, still denoted by $\{{x}_{n}\}$, such that ${x}_{n}\to \overline{x}\in E$. It follows from (a2) that $\overline{x}\in P(\overline{x})$. For any $z\in Q(\overline{x})$, there exists ${\tilde{z}}_{n}\in Q({x}_{n})$ such that ${\tilde{z}}_{n}\to z$ by (a3). For each ${y}_{n}\in f({x}_{n},{\tilde{z}}_{n})$, we have ${y}_{n}\in {\epsilon}_{n}e({x}_{n})+W({x}_{n})$. In view of the lower semicontinuity of f, for any $y\in f(\overline{x},z)$, ${\tilde{y}}_{n}\in f({x}_{n},{\tilde{z}}_{n})$ can be chosen to satisfy ${\tilde{y}}_{n}\to y$ and ${\tilde{y}}_{n}+{\epsilon}_{n}e({x}_{n})\in W({x}_{n})$. By letting $n\to +\mathrm{\infty}$, $y\in W(\overline{x})$ by the closeness of W, and so $f(\overline{x},z)\cap (intC(\overline{x}))=\mathrm{\varnothing}$. Therefore, $\overline{x}\in \mathrm{\Omega}$. (GVQEP) is GTWP1 and also GTWP2. □
It is easy to see that the conclusion of Theorem 2.6 still holds if (b1) is replaced by ‘E is compact’. In addition, if f is compactvalued, then (b1) can also be substituted by:
(b2) ϕ is compactlevel on ${\mathrm{\Omega}}_{1}({\epsilon}_{0})$ for some ${\epsilon}_{0}>0$, or
(b3) X is a finitedimensional normed linear space and ϕ is levelbounded on E.
Indeed ${\mathrm{\Omega}}_{1}({\epsilon}_{0})=\{x\in {\mathrm{\Omega}}_{1}({\epsilon}_{0}):\varphi (x)\le \epsilon \}$ is compact for each $\epsilon \ge {\epsilon}_{0}$ by (b2) and so (b1) holds. Define $A(\epsilon )=\{x\in E:\varphi (x)\le \epsilon \}$ for each $\epsilon \in \mathbb{R}$. Then $A(\epsilon )$ is bounded by (b3), otherwise, there exists $\{{u}_{n}\}\subset A(\epsilon )\subset E$ such that $\parallel {u}_{n}\parallel \to +\mathrm{\infty}$ and $\varphi ({u}_{n})\le \epsilon $. This is absurd according to (b3). $A(\epsilon )$ is closed since ϕ is lower semicontinuous by Proposition 2.1 and so it is compact. Clearly, ${\mathrm{\Omega}}_{1}(\epsilon )\subset A(\epsilon )$. Thus, (b1) is satisfied by Lemma 2.1(i).
In fact, GTWPness1 or GTWPness2 for (GVQEP) can fail without the lower semicontinuity of f. The following example only states the fact under the assumption that ‘E is compact’.
Thus ${x}_{n}\in {\mathrm{\Omega}}_{2}({\epsilon}_{n})$, in other words, $\{{x}_{n}\}$ is an ASS2 of (GVQEP), but ${x}_{n}\to 0\notin \mathrm{\Omega}$. It is worth noting that (a2)(a5) are satisfied, but f is not lower semicontinuous at $(0,0)$. Indeed, let $({x}_{n},{z}_{n})=(\frac{1}{n},\frac{1}{n})\to (0,0)$. For ${y}_{0}=2\in f(0,0)$ and for any ${y}_{n}\in f({x}_{n},{z}_{n})=[\frac{2}{n},\frac{2+n}{n}]$, $\{{y}_{n}\}$ does not converge to ${y}_{0}$.
By the argument given in the proof of Theorem 3.6, it is easy to yield the following.
Theorem 2.7 If (a2) and (a4) are substituted by ‘E is closed and $P(x)=E$ for all $x\in E$ ’ and ‘W is closed’ in Theorem 2.6, respectively, then the conclusion still holds.
Corollary 2.4 If, further, assume that Ω is a singleton in Theorem 2.6 (resp., 2.7), then (GVQEP) is TWP 1 and also TWP 2.
Furthermore, if $E=F=X=Z$ is a locally convex topological space and E is a closed convex subset and if f is singlevalued and $P(x)=Q(x)=E$ for all $x\in X$ in Theorem 2.7, then Theorem 2.7 reduces to Corollary 3.1 in [39].
3 (G)HWPness for parametrically GVQEPs
where $h:E\times F\times \mathrm{\Phi}\to {2}^{Y}$, $P:E\to {2}^{E}$ and $Q:E\to {2}^{F}$ are strict setvalued mappings, and $(\mathrm{\Phi},\tilde{d})$ is a Hausdorff metric space (parametric space). ${\mathrm{\Omega}}^{p}$ denotes the solution set of (GVQEP)_{ p } for each $p\in \mathrm{\Phi}$.
Definition 3.1 (GVQEP)_{ p } is said to be generalized Hadamard wellposed (in short, GHWP) at ${p}_{0}\in \mathrm{\Phi}$ if ${\mathrm{\Omega}}^{{p}_{0}}\ne \mathrm{\varnothing}$ and for any $\{{p}_{n}\}\subset \mathrm{\Phi}$ with ${p}_{n}\to {p}_{0}$ and ${x}_{n}\in {\mathrm{\Omega}}^{{p}_{n}}$, there exists a subsequence $\{{x}_{{n}_{i}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{i}}\to \overline{x}\in {\mathrm{\Omega}}^{{p}_{0}}$; to be Hadamard wellposed (in short, HWP) at ${p}_{0}\in \mathrm{\Phi}$ if it is GHWP at ${p}_{0}\in \mathrm{\Phi}$ and ${\mathrm{\Omega}}^{{p}_{0}}$ is a singleton.
Remark 3.1 (i) Obviously, (GVQEP)_{ p } is GHWP at ${p}_{0}$ if and only if ${\mathrm{\Omega}}^{{p}_{0}}$ is nonempty compact and for any $\{{p}_{n}\}\subset \mathrm{\Phi}$ with ${p}_{n}\to {p}_{0}$ and any ${x}_{n}\in {\mathrm{\Omega}}^{{p}_{n}}$, $d({x}_{n},{\mathrm{\Omega}}^{{p}_{0}})\to 0$. (GVQEP)_{ p } is HWP at ${p}_{0}$ if and only if ${\mathrm{\Omega}}^{{p}_{0}}=\{\overline{x}\}$ and for any $\{{p}_{n}\}\subset \mathrm{\Phi}$ with ${p}_{n}\to {p}_{0}$ and any ${x}_{n}\in {\mathrm{\Omega}}^{{p}_{n}}$, ${x}_{n}\to \overline{x}$.
(ii) If $E=F=X=Z$, $Y=\mathbb{R}$, $P(x)=Q(x)=X$, $C(x)={\mathbb{R}}_{+}$ for all $x\in X$ and h is singlevalued, then the HWPness at ${p}_{0}$ for (GVQEP)_{ p } reduces to the HWPness at ${p}_{0}$ for (EP)_{ p }, which was investigated by Bianchi et al. [35].
where ${\mathrm{\Omega}}^{{p}_{0}}=\{\overline{x}\}$. However the converse fails to be true. See the following example.
Set ${p}_{0}=0$. $diam{\mathrm{\Omega}}^{{p}_{n}}={p}_{n}\to 0$ for any ${p}_{n}>0$ with ${p}_{n}\to {p}_{0}$, but (GVQEP)_{ p } is not HWP at ${p}_{0}$. In fact, by taking ${x}_{n}={p}_{n}+\frac{1}{{p}_{n}}\in {\mathrm{\Omega}}^{{p}_{n}}$, ${x}_{n}\to +\mathrm{\infty}$.
Theorem 3.1 Let E be compact, ${p}_{0}\in \mathrm{\Phi}$ and ${\mathrm{\Omega}}^{{p}_{0}}\ne \mathrm{\varnothing}$. If (a2)(a4) and
(a7) h is lower semicontinuous at $(x,z,{p}_{0})$ for each $(x,z)\in E\times F$
hold, then (GVQEP)_{ p } is GHWP at ${p}_{0}$. If, further, ${\mathrm{\Omega}}^{{p}_{0}}$ is a singleton, then (GVQEP)_{ p } is HWP at ${p}_{0}$.
This deduces that $\overline{x}\in {\mathrm{\Omega}}^{{p}_{0}}$ and (GVQEP)_{ p } is GHWP at ${p}_{0}$. The second conclusion follows directly from Definition 3.1. □
The conclusion in Theorem 3.1 is still true if (a2) is replaced by the assumption that P is closedvalued and upper semicontinuous on E. Theorem 3.1 can be false without (a7). See the instance as follows.
(GVQEP)_{ p } is not HWP (resp., GHWP) at ${p}_{0}=0$. It is worth noting that h is not lower semicontinuous at $(0,0,0)$. Indeed, take $({x}_{n},{z}_{n},{p}_{n})=(\frac{1}{n},\frac{1}{n},\frac{1}{n})\to (0,0,0)$. For $\overline{y}=1\in h(0,0,0)$ and for any ${y}_{n}\in h({x}_{n},{z}_{n},{p}_{n})=[n,n+1]$, it is impossible that the case ${y}_{n}\to \overline{y}$ happens.
4 P(G)TWPness for parametrical system of GVQEPs
where (GVQEP)_{ p } for each $p\in \mathrm{\Phi}$ is described at the beginning of Section 4 and $(\mathrm{\Phi},\tilde{d})$ is a Hausdorff metric space (parametric space).
Clearly, ${\mathrm{\Omega}}^{p}$, ${\mathrm{\Omega}}_{1}^{p}(\epsilon )$ and ${\mathrm{\Omega}}_{2}^{p}(\epsilon )$ are just the solution set, the type I εapproximating solution set and the type II εapproximating solution set of p(GVQEP), respectively, for each $p\in \mathrm{\Phi}$ and $\epsilon \in {\mathbb{R}}_{+}$.
A sequence $\{{x}_{n}\}$ is called a type I approximating solution sequence related to $p\in \mathrm{\Phi}$, $\text{ASS1}(p)$ for brevity (resp., type II approximating solution sequence related to $p\in \mathrm{\Phi}$, $\text{ASS2}(p)$ for brevity) of $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ if there exist $\{{\epsilon}_{n}\},\{{\delta}_{n}\}\subset {\mathbb{R}}_{+}$ with ${\epsilon}_{n},{\delta}_{n}\to 0$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}^{p}({\epsilon}_{n},{\delta}_{n})$ (resp., ${x}_{n}\in {\mathrm{\Omega}}_{2}^{p}({\epsilon}_{n},{\delta}_{n})$).
Remark 4.1 (i) An $\text{ASS2}(p)$ of $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ is its $\text{ASS1}(p)$ for each $p\in \mathrm{\Phi}$.
on the assumption of
(a8) $0\in h(x,Q(x),p)$ for all $x\in E$ and $p\in \mathrm{\Phi}$.
Definition 4.2 $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ is said to be GTWP1 (resp., GTWP2) if for each $p\in \mathrm{\Phi}$, ${\mathrm{\Omega}}^{p}\ne \mathrm{\varnothing}$ and for any $\text{ASS1}(p)$ (resp., $\text{ASS2}(p)$) $\{{x}_{n}\}$ of $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$, there exists a subsequence $\{{x}_{{n}_{i}}\}$ such that ${x}_{{n}_{i}}\to \overline{x}\in {\mathrm{\Omega}}^{p}$; to be TWP1 (resp., TWP2) if it is GTWP1 (resp., GTWP2) and ${\mathrm{\Omega}}^{p}$ is a singleton for each $p\in \mathrm{\Phi}$.
Remark 4.2 (i) The GTWPness1 for $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ implies its GTWPness2 according to Remark 4.1(i).
(ii) $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP1 if and only if for each $p\in \mathrm{\Phi}$, ${\mathrm{\Omega}}^{p}$ is nonempty compact and for any $\text{ASS1}(p)$ $\{{x}_{n}\}$ for $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$, $d({x}_{n},{\mathrm{\Omega}}^{p})\to 0$. When ${\mathrm{\Omega}}^{p}$ is compact for each $p\in \mathrm{\Phi}$, $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP2 if and only if for any $p\in \mathrm{\Phi}$, ${\mathrm{\Omega}}^{p}\ne \mathrm{\varnothing}$ and for any $\text{ASS2}(p)$ $\{{x}_{n}\}$ for $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$, $d({x}_{n},{\mathrm{\Omega}}^{p})\to 0$. In addition, $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ is TWP1 (resp., TWP2) if and only if for each $p\in \mathrm{\Phi}$, ${\mathrm{\Omega}}^{p}=\{{\overline{x}}^{p}\}$ and for any $\text{ASS1}(p)$ (resp., $\text{ASS2}(p)$) $\{{x}_{n}\}$ for $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$, $d({x}_{n},{\overline{x}}^{p})\to 0$.
Lemma 4.1 Assume that (a2)(a5) hold and
(a9) h is lower semicontinuous on $E\times F\times \mathrm{\Phi}$;
(a10) Φ is compact.
 (i)
${\mathrm{\Omega}}_{1}^{p}(\epsilon ,\delta )$ is closed for each $p\in \mathrm{\Phi}$ and $\epsilon ,\delta >0$.
 (ii)
${\mathrm{\Omega}}^{p}=\bigcap \{{\mathrm{\Omega}}_{1}^{p}(\epsilon ,\delta ):\epsilon ,\delta >0\}$ for each $p\in \mathrm{\Phi}$.
Accordingly, $\overline{x}\in {\mathrm{\Omega}}_{1}^{\overline{p}}(\epsilon )$ and so $\overline{x}\in {\mathrm{\Omega}}_{1}^{p}(\epsilon ,\delta )$ by $\tilde{d}(\overline{p},p)\le \delta $.
Therefore, there exists ${p}_{n}\in \mathrm{\Phi}$ such that $\tilde{d}({p}_{n},p)\le {\delta}_{n}$ and $\overline{x}\in {\mathrm{\Omega}}_{1}^{{p}_{n}}({\epsilon}_{n})$. As a result, ${p}_{n}\to p$, $\overline{x}\in P(\overline{x})$ and $h(\overline{x},z,{p}_{n})\cap ({\epsilon}_{n}e(\overline{x})intC(\overline{x}))=\mathrm{\varnothing}$ for all $z\in Q(\overline{x})$. Then $h(\overline{x},z,p)\cap (intC(\overline{x}))=\mathrm{\varnothing}$ for all $z\in Q(\overline{x})$ proceeds from (a2), (a3) and (a10) and so $\overline{x}\in {\mathrm{\Omega}}^{p}$. □
By a resemblant argument given in the proof of Lemma 2.2, we have the following.
 (i)
${\mathrm{\Omega}}_{2}^{p}(\epsilon ,\delta )$ is closed for each $p\in \mathrm{\Phi}$ and $\epsilon ,\delta >0$.
 (ii)
${\mathrm{\Omega}}^{p}=\bigcap \{{\mathrm{\Omega}}_{2}^{p}(\epsilon ,\delta ):\epsilon ,\delta >0\}$ for each $p\in \mathrm{\Phi}$.
 (i)If $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP 1, then for each $p\in \mathrm{\Phi}$,${\mathrm{\Omega}}_{1}^{p}(\epsilon ,\delta )\ne \mathrm{\varnothing}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\epsilon ,\delta 0\mathit{\text{and}}\underset{(\epsilon ,\delta )\to (0,0)}{lim}\alpha ({\mathrm{\Omega}}_{1}^{p}(\epsilon ,\delta ))=0.$(4.6)
 (ii)
Suppose that (a2)(a5) and (a9)(a10) hold. If (4.6) holds for each $p\in \mathrm{\Phi}$, then $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP 1.
for any $p\in \mathrm{\Phi}$ and $\epsilon ,\delta >0$. It is enough to testify $\tilde{e}({\mathrm{\Omega}}_{1}^{p}(\epsilon ,\delta ),{\mathrm{\Omega}}^{p})\to 0$ as $(\epsilon ,\delta )\to (0,0)$ for each $p\in \mathrm{\Phi}$. Otherwise, there exist $p,{p}_{n}\in \mathrm{\Phi}$, $r>0,{\epsilon}_{n}\downarrow 0$ and ${\delta}_{n}\downarrow 0$ with $\tilde{d}({p}_{n},p)\le {\delta}_{n}$ and ${x}_{n}\in {\mathrm{\Omega}}_{1}^{{p}_{n}}({\epsilon}_{n})$ such that $d({x}_{n},{\mathrm{\Omega}}^{p})\ge r$. This says that $\{{x}_{n}\}$ is an $\text{ASS1}(p)$ of $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$. By Remark 4.2(ii), $d({x}_{n},{\mathrm{\Omega}}^{p})\to 0$, which contradicts $d({x}_{n},{\mathrm{\Omega}}^{p})\ge r$.
(ii) For any $\text{ASS1}(p)$ $\{{x}_{n}\}$ of $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$, there exist $\{{\epsilon}_{n}\},\{{\delta}_{n}\}\subset {\mathbb{R}}_{+}$ with $({\epsilon}_{n},{\delta}_{n})\to (0,0)$ such that ${x}_{n}\in {\mathrm{\Omega}}_{1}^{p}({\epsilon}_{n},{\delta}_{n})$ by Remark 4.2(ii). In view of Lemma 4.1 and Kuratowski theorem [51], ${\mathrm{\Omega}}^{p}$ is nonempty compact and $H({\mathrm{\Omega}}_{1}^{p}({\epsilon}_{n},{\delta}_{n}),{\mathrm{\Omega}}^{p})\to 0$. Thus $d({x}_{n},{\mathrm{\Omega}}^{p})\to 0$. It follows that $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP1. □
 (i)Suppose that for each $p\in \mathrm{\Phi}$, ${\mathrm{\Omega}}^{p}$ is compact. If $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP 2, then for $p\in \mathrm{\Phi}$,${\mathrm{\Omega}}_{2}^{p}(\epsilon ,\delta )\ne \mathrm{\varnothing}\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}\epsilon ,\delta 0,\mathit{\text{and}}\underset{(\epsilon ,\delta )\to (0,0)}{lim}\alpha ({\mathrm{\Omega}}_{2}^{p}(\epsilon ,\delta ))=0.$(4.7)
 (ii)
Assume that (a2)(a5) and (a8)(a10) hold. If (4.7) holds for each $p\in \mathrm{\Phi}$, then $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP 2.
Proof This proof is completed by using Lemma 4.2 and a similar argument proposed in the proof of Theorem 2.2 and is omitted. □
 (i)
If $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is TWP 1 (resp., TWP 2), then (4.6) (resp., (4.7)) holds for each $p\in \mathrm{\Phi}$.
 (ii)
Assume that (a2)(a5) and (a9)(a10) (resp., (a2)(a5) and (a8)(a10)) hold. If (4.6) (resp., (4.7)) holds for each $p\in \mathrm{\Phi}$, then $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is TWP 1 (resp., TWP 2).
By a similar method of the proof in Theorem 2.6, we have the following.
Theorem 4.3 Let (a2)(a5) and (a9)(a10) hold and ${\mathrm{\Omega}}^{p}\ne \mathrm{\varnothing}$ for each $p\in \mathrm{\Phi}$. If
(b4) For each $p\in \mathrm{\Phi}$, ${\mathrm{\Omega}}_{1}^{p}({\epsilon}_{0},{\delta}_{0})$ is compact for some ${\epsilon}_{0},{\delta}_{0}>0$; or
(b5) X is a finitedimensional normed linear space and for each $p\in \mathrm{\Phi}$, ${\mathrm{\Omega}}_{1}^{p}({\epsilon}_{0},{\delta}_{0})$ is bounded for some ${\epsilon}_{0},{\delta}_{0}>0$
holds, then $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is GTWP 1 and also GTWP 2.
Corollary 4.2 Further suppose that ${\mathrm{\Omega}}^{p}$ is a singleton for each $p\in \mathrm{\Phi}$ in Theorem 4.3. Then $\{{(\mathit{\text{GVQEP}})}_{p}:p\in \mathrm{\Phi}\}$ is TWP 1 and also TWP 2.
5 Relations among the types of proposed wellposedness
In this section we are interested in the comparison among the types of proposed wellposedness defined in previous sections.
It seems on the surface to have no relations between the (G)HWPness for (GVQEP)_{ p } and (G)TWPness for (GVQEP). However, if there are some connections between their objective mappings, we may discuss the relations.
if (a6) holds. If, further, $P(x)=E$ for all $x\in E$, then both (5.2) and (5.3) are indeed equalities.
It follows from (4.4) and (4.5) that GTWPness1 and GTWPness2 for $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ imply that GTWPness1 and GTWPness2 for p(GVQEP) for each $p\in \mathrm{\Phi}$, respectively, and GTWPness1 for $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ also implies GHWPness for (GVQEP)_{ p } at each $p\in \mathrm{\Phi}$, while GTWPness2 implies GHWPness for (GVQEP)_{ p } at each $p\in \mathrm{\Phi}$ if (a10) holds. But these converses fail to hold. See the following example.
for each $p\in \mathrm{\Phi}$ and $0<\epsilon ,\delta <1$. It is clear that (GVQEP)_{ p } is GHWP at each $p\in \mathrm{\Phi}$, and p(GVQEP) is GTWP1 and GTWP2 for each $p\in \mathrm{\Phi}$, while $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$ is neither GTWP1 nor GTWP2. In fact, take ${p}_{n}=\frac{1}{{n}^{2}}$, ${\epsilon}_{n}=\frac{1}{n}$, ${\delta}_{n}=\frac{1}{{n}^{2}}$ and ${x}_{n}=n$. It is easy to see that $\{{x}_{n}\}$ is an ASS1(0) (resp., ASS2(0)) of $\{{(\text{GVQEP})}_{p}:p\in \mathrm{\Phi}\}$, but it has no convergent subsequence.
Declarations
Acknowledgements
This research is supported by the Doctoral Fund of innovation of Beijing University of Technology.
Authors’ Affiliations
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