The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé–Kuratowski convergence

Abstract

In this paper, we obtain the Painlevé–Kuratowski upper convergence and the Painlevé–Kuratowski lower convergence of the approximate solution sets for set optimization problems with the continuity and convexity of objective mappings. Moreover, we discuss the extended well-posedness and the weak extended well-posedness for set optimization problems under some mild conditions. We also give some examples to illustrate our main results.

1 Introduction

It is well known that the stability of the solution sets under certain perturbations (with respect to the feasible region and the objective function) has been of great interest in optimization theory with applications. Recently, some stability results have been derived for the vector optimization and vector equilibrium problems based on a sequence of sets converging. For instance, (Huang 2000a) obtained the stability results of the set of efficient solutions of vector-valued and set-valued optimization in the sense of Painlevé–Kuratowski; (Lucchetti and Miglierina 2004) studied the convergence of the solution sets under perturbations of both the objective function and the feasible region for convex vector optimization problem; (Crespi et al. 2009) obtained the stability properties of vector optimization problems under the assumption that the objective function is cone-quasiconvex; (Lalitha and Chatterjee 2012a) established the Painlevé–Kuratowski set-convergence of the sets of minimal, weak minimal and Henig proper minimal points of the perturbed problems to the corresponding minimal set of the original problem assuming the objective functions to be (strictly) properly quasi cone-convex; (Lalitha and Chatterjee 2012b) derived the Painlevé–Kuratowski convergence of the weak efficient solution sets, efficient solution sets and Henig proper efficient sets for the perturbed vector optimization problems by using generalized quasi convexities; (Fang and Li 2012) established the Painlevé–Kuratowski convergence of the efficient solution sets, the weak efficient solution sets and various proper efficient solution sets for the perturbed vector equilibrium problems under the C-strict monotonicity. Very recently, under new assumptions, which are weaker than the assumption of C-strict monotonicity, Peng and Yang (2014) obtained sufficient conditions for the Painlevé–Kuratowski convergence of the weak efficient solution sets and efficient solution sets for the perturbed vector equilibrium problems. Zhao et al. (2016) established Painlevé–Kuratowski upper convergence of weak efficient solutions for perturbed vector optimization problems with approximate equilibrium constraints. Anh et al. (2018) discussed Painlevé–Kuratowski upper convergence and Painlevé–Kuratowski lower convergence of solution sets for the perturbed vector quasi-equilibrium problems.

It is also well known that the stability analysis of the solution sets for set optimization problems has been investigated by many authors in the literature [see, for example, Khan et al. (2015) and the references therein]. Recently, Gutiérrez et al. (2016) establish external and internal stability of the solutions of a set optimization problem in the image space using set convergence notions. Xu and Li (2014) showed the lower and upper semicontinuity of the set of minimal and weak minimal solutions to a parametric set optimization problem by using converse u-property of objective mappings. Very recently, Han and Huang (2017) discussed the upper semicontinuity and the lower semicontinuity of solution mappings to parametric set optimization problems by using the level mappings. Han and Huang (2018) established the continuity and convexity of the nonlinear scalarizing function for sets, which was introduced by Hern$\acute{a}$ndez and Rodr$\acute{{\i }}$guez-Mar$\acute{{\i }}$n Hernández and Rodríguez-Marín (2007); as applications, they derived the upper semicontinuity and the lower semicontinuity of strongly approximate solution mappings to the parametric set optimization problems. Khoshkhabar-amiranloo (2018) discussed the upper semicontinuity and lower semicontinuity and compactness of the minimal solutions of parametric set optimization problems. Karuna and Lalitha (2019) investigated external and internal stability in terms of the Hausdorff convergence and Painlevé–Kuratowski convergence of a sequence of solution sets of perturbed set optimization problems to the solution set of the original set optimization problem. However, to the best of our knowledge, the Painlevé–Kuratowski convergence of the approximate solution sets for set optimization problems has not been explored until now. Therefore, it would be quite natural and interesting to study the Painlevé–Kuratowski convergence of the approximate solution sets for set optimization problems under some mild conditions. The first aim of this paper is to make an attempt in this direction.

On the other hand, the well-posedness plays a significant role in the study of the stability theory of optimization problems. Recently, the well-posedness for set optimization problems has been studied under different conditions. Zhang et al. (2009) established the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems by using a generalized Gerstewitz’s function, respectively. Gutiérrez et al. (2012) obtained the well-posedness property in the setting of set optimization problems, which improves some results in Zhang et al. (2009) by relaxing the assumption of cone boundedness of the image of objective mappings. By using the generalized nonlinear scalarization function, Long et al. (2015) established the equivalence relations between the three kinds of pointwise well-posedness for set optimization problems and the well-posedness of three kinds of scalar optimization problems, respectively. Crespi et al. (2014) introduced a new notion of global well-posedness for set-optimization problems, which is a generalization of one of the global notion considered in Zhang et al. (2009). Very recently, Crespi et al. (2018) obtained some characterizations for pointwise and global well-posedness in set optimization.

We note that Zolezzi (1996) proposed the notion of extended well-posedness for optimization problems. In Huang (2000b, 2001), Huang generalized the notion of extended well-posedness to vector optimization problems. Crespi et al. (2009) discussed the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. However, it seems that there are no authors to study the extended well-posedness for set optimization problems. Thus, it would be important and interesting to study the extended well-posedness of set optimization problems. The second aim of this paper is to give some characterizations for the extended well-posedness of set optimization problems under suitable conditions.

The rest of the paper is organized as follows. Section 2 presents some necessary notations and lemmas. In Sect. 3, we discuss the Painlevé–Kuratowski upper convergence and the Painlevé–Kuratowski lower convergence of the approximate solution sets for set optimization problems with the continuity and convexity of objective mappings. In Sect. 4, we introduce the notions of the extended well-posedness and the weak extended well-posedness for set optimization problems and derive the extended well-posedness and the weak extended well-posedness for set optimization problems under mild conditions.

2 Preliminaries

Throughout this paper, without special statements, let $X = {{\mathbb {R}}^m}$ and $Y = {{\mathbb {R}}^l}$. Assume that $C \subseteq Y$ is a nonempty, convex, closed and pointed cones with ${\mathrm{int}} C \ne \emptyset$. We denote by ${\mathrm{int}} A$, ${\mathrm{cl}}A$, $\partial A$ and ${A^c}$ the topological interior, the topological closure, the topological boundary and the complementary set of A, respectively. Let ${{\mathbb {R}}_ + } = \left\{ {x \in {\mathbb {R}}:x \ge 0} \right\}$ and ${\mathbb {R}}_ + ^0 = \left\{ {x \in {\mathbb {R}}:x > 0} \right\}$. We denote by $B_X$ and $B_Y$ the closed unit balls in X and Y, respectively. Let A and B be two nonempty subsets of Y. The lower relation “${ \le ^l}$” and the weak lower relation “${ \ll ^l}$” are defined, respectively, by

$$\begin{aligned} A{ \le ^l}B \Leftrightarrow B \subseteq A + C \end{aligned}$$

and

$$\begin{aligned} A{ \ll ^l}B \Leftrightarrow B \subseteq A + {\mathrm{int}} C. \end{aligned}$$

Let e be a fixed point in ${\mathrm{int}} C$. For $\varepsilon \ge {\mathrm{0}}$, the $\varepsilon$-lower relation “$\le _\varepsilon ^l$” and the weak $\varepsilon$-lower relation “$\ll _\varepsilon ^l$” are defined, respectively, by

$$\begin{aligned} A \le _\varepsilon ^lB \Leftrightarrow B \subseteq A + C + \varepsilon e \end{aligned}$$

and

$$\begin{aligned} A \ll _\varepsilon ^lB \Leftrightarrow B \subseteq A + {\mathrm{int}}C + \varepsilon e. \end{aligned}$$

Let A be a nonempty subset of Y and $a \in A$. We say that a is a minimal point of A with respect to C, denoted by $a \in {\mathrm{Min}}\left( A \right)$, if $\left( {A - a} \right) \cap \left( { - C} \right) = \left\{ 0 \right\}$.

Remark 2.1

It follows from Corollary 3.8 of Luc (1989) (page 48) that, if A is compact, then ${\mathrm{Min}}\left( A \right) \ne \emptyset$.

Let $F:X \rightarrow {2^Y}$ be a set-valued mapping and $D\subseteq X$ with $D\not =\emptyset$. We consider the following set optimization problem:

$$\begin{aligned} \hbox {(SOP)} \quad \min F\left( x \right) \quad \hbox {subject to} \quad x \in D. \end{aligned}$$

Definition 2.1

For $\varepsilon \ge {\mathrm{0}}$, an element ${x_0} \in D$ is said to be

• (i) l-minimal solution of (SOP) if, for $x \in D$, $F\left( x \right) {\le ^l} F\left( x_0 \right)$ implies $F\left( x_0 \right) {\le ^l} F\left( x \right)$.

• (ii) weak l-minimal solution of (SOP) if, for $x \in D$, $F\left( x \right) {\ll ^l} F\left( x_0 \right)$ implies $F\left( x_0 \right) {\ll ^l} F\left( x \right)$.

• (iii) l-minimal approximate solution of (SOP) if, for $x \in D$, $F\left( x \right) {\le _\varepsilon ^l} F\left( x_0 \right)$ implies $F\left( x_0 \right) {\le _\varepsilon ^l} F\left( x \right)$.

• (iv) weak l-minimal approximate solution of (SOP) if, for $x \in D$, $F\left( x \right) { \ll _\varepsilon ^l} F\left( x_0 \right)$ implies $F\left( x_0 \right) { \ll _\varepsilon ^l} F\left( x \right)$.

Let ${E_l}\left( {D} \right)$, ${W_l}\left( {D} \right)$, ${E_l}\left( {\varepsilon ,D} \right)$ and ${W_l}\left( {\varepsilon ,D} \right)$ denote the l-minimal solution set of (SOP), the weak l-minimal solution set of (SOP), the l-minimal approximate solution set of (SOP) and the weak l-minimal approximate solution set of (SOP), respectively.

Remark 2.2

${E_l}\left( {\varepsilon ,D} \right)$ and ${W_l}\left( {\varepsilon ,D} \right)$ depend on the choice of $e \in {\mathrm{int}} C$.

We give an example to illustrate Remark 2.2.

Example 2.1

Let $X = {\mathbb {R}}$, $Y = {{\mathbb {R}}^2}$ and $C = {\mathbb {R}}_ + ^2 = \left\{ {\left( {{x_1},{x_2}} \right) \in {{\mathbb {R}}^2}:{x_1} \ge 0,{x_2} \ge 0} \right\}$. Define a set-valued mapping $F:X \rightarrow {2^Y}$ as follows:

$$\begin{aligned} F\left( x \right) = \left( {{{\left( {x - 1} \right) }^2},{{\left( {x - 2} \right) }^2}} \right) + {B_Y},\;\;x \in X. \end{aligned}$$

Let $D = \left[ { - 5,5} \right]$. If we choose $e = \left( {2.5,0.5} \right)$, then $0 \in {E_l}\left( {1,D} \right)$ and $0 \in {W_l}\left( {1,D} \right)$. However, if we choose $e = \left( {0.5,2.5} \right)$, then $0 \notin {E_l}\left( {1,D} \right)$ and $0 \notin {W_l}\left( {1,D} \right)$.

Remark 2.3

For any $\varepsilon \ge {0}$, we have ${E_l}\left( D \right) \subseteq {E_l}\left( {\varepsilon ,D} \right)$ and ${W_l}\left( D \right) \subseteq {W_l}\left( {\varepsilon ,D} \right)$. In fact, let ${x_0} \in {E_l}\left( D \right)$. Suppose that there exists $y \in D$ such that $F\left( y \right) \le _\varepsilon ^lF\left( {{x_0}} \right)$. Then

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( y \right) + C + \varepsilon e\subseteq F\left( y \right) + C \end{aligned}$$

and so $F\left( y \right) { \le ^l}F\left( {{x_0}} \right)$. By ${x_0} \in {E_l}\left( D \right)$, one has $F\left( {{x_0}} \right) { \le ^l}F\left( y \right)$. This shows that $F\left( y \right) \subseteq F\left( {{x_0}} \right) + C$. Thus,

$$\begin{aligned} F\left( y \right) \subseteq F\left( {{x_0}} \right) + C \subseteq F\left( y \right) + C + \varepsilon e \subseteq F\left( {{x_0}} \right) + C + \varepsilon e, \end{aligned}$$

which means that $F\left( {{x_0}} \right) \le _\varepsilon ^lF\left( y \right)$. Therefore, ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$ and so ${E_l}\left( D \right) \subseteq {E_l}\left( {\varepsilon ,D} \right)$. Similarly, we can prove that ${W_l}\left( D \right) \subseteq {W_l}\left( {\varepsilon ,D} \right)$.

Remark 2.4

For any $\varepsilon \ge {\mathrm{0}}$, we have ${E_l}\left( {\varepsilon ,D} \right) \subseteq {W_l}\left( {\varepsilon ,D} \right)$. In fact, let ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$. Suppose that there exists $y \in D$ such that $F\left( y \right) \ll _\varepsilon ^lF\left( {{x_0}} \right)$. Then

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( y \right) + {\mathrm{int}} C + \varepsilon e\subseteq F\left( y \right) + C + \varepsilon e \end{aligned}$$

and so $F\left( y \right) { \le _\varepsilon ^l}F\left( {{x_0}} \right)$. By ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$, we have $F\left( {{x_0}} \right) { \le _\varepsilon ^l}F\left( y \right)$. This shows that

$$\begin{aligned} F\left( y \right)\subseteq & {} F\left( {{x_0}} \right) + C + \varepsilon e\\\subseteq & {} F\left( y \right) + {\mathrm{int}} C + C + 2\varepsilon e \\\subseteq & {} F\left( {{x_0}} \right) + C + {\mathrm{int}} C + C + 3\varepsilon e \\\subseteq & {} F\left( {{x_0}} \right) + {\mathrm{int}} C + \varepsilon e, \end{aligned}$$

which means that $F\left( {{x_0}} \right) \ll _\varepsilon ^lF\left( y \right)$. Therefore, ${x_0} \in {W_l}\left( {\varepsilon ,D} \right)$.

Remark 2.5

For any $\varepsilon > {0}$, we have ${W_l}\left( D \right) \subseteq {E_l}\left( {\varepsilon ,D} \right)$. In fact, let ${x_0} \in {W_l}\left( D \right)$. Suppose that there exists $y \in D$ such that $F\left( y \right) \le _\varepsilon ^lF\left( {{x_0}} \right)$. Then

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( y \right) + C + \varepsilon e \subseteq F\left( y \right) + C + {\mathrm{int}} C \subseteq F\left( y \right) + {\mathrm{int}} C, \end{aligned}$$

which implies $F\left( y \right) { \ll ^l}F\left( {{x_0}} \right)$. It follows from ${x_0} \in {W_l}\left( D \right)$ that $F\left( {{x_0}} \right) { \ll ^l}F\left( y \right)$, and so $F\left( y \right) \subseteq F\left( {{x_0}} \right) + {\mathrm{int}} C$. Thus,

$$\begin{aligned} F\left( y \right)\subseteq & {} F\left( {{x_0}} \right) + {\mathrm{int}} C \\\subseteq & {} F\left( y \right) + C + \varepsilon e + {\mathrm{int}} C \\\subseteq & {} F\left( {{x_0}} \right) + {\mathrm{int}} C + C + \varepsilon e + {\mathrm{int}} C \\\subseteq & {} F\left( {{x_0}} \right) + C + \varepsilon e. \end{aligned}$$

This shows that $F\left( {{x_0}} \right) \le _\varepsilon ^lF\left( y \right)$,and so ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$.

Now, let us recall the concept of the Painlevé–Kuratowski set-convergence [see, for example, Rockafellar and Wets (2004)]. Let $\{A_n\}$ be a sequence of nonempty subsets of ${{\mathbb {R}}^m}$. Set

$$\begin{aligned}&{\mathrm{Ls}}{A_n}: = \left\{ {x \in {{\mathbb {R}}^m}:x = \mathop {\lim }\limits _{k \rightarrow + \infty } {x_{{n_k}}},{x_{{n_k}}} \in {D_{{n_k}}},\left\{ {{x_{{n_k}}}} \right\} {\mathrm{is\; a \; subsequence \; of }}\left\{ {{x_n}} \right\} } \right\} , \\&{\mathrm{Li}}{A_n}: = \left\{ {x \in {{\mathbb {R}}^m}:x = \mathop {\lim }\limits _{n \rightarrow + \infty } {x_n},{x_n} \in {D_n}{\mathrm{\;\; for \; sufficiently \; large \; }} n} \right\} . \end{aligned}$$

The set ${\mathrm{Ls}}{A_n}$ is called the upper limit of the sequence $\{A_n\}$, and the set ${\mathrm{Li}}{A_n}$ is called the lower limit of the sequence $\{A_n\}$. We say that the sequence $\{A_n\}$ converges in the sense of Painlevé–Kuratowski to the set A if

$$\begin{aligned} {\mathrm{Ls}}{A_n} \subseteq A \subseteq {\mathrm{Li}}{A_n}. \end{aligned}$$

We denote the Painlevé–Kuratowski convergence by ${A_n}\mathop \rightarrow \limits ^K A$.

Definition 2.2

(Kuratowski 1968) Let $\left( {X,d} \right)$ be a metric space, A and B be two nonempty subsets of X. The Hausdorff distance between A and B is defined by

$$\begin{aligned} H\left( {A,B} \right) : = \max \left\{ {g\left( {A,B} \right) ,g\left( {B,A} \right) } \right\} , \end{aligned}$$

where

$$\begin{aligned} g\left( {A,B} \right) : = \mathop {\sup }\limits _{a \in A} d\left( {a,B} \right) with d\left( {a,B} \right) = \mathop {\inf }\limits _{b \in B} d\left( {a,b} \right) . \end{aligned}$$

Let $\{A_n\}$ be a sequence of nonempty subsets of ${{\mathbb {R}}^m}$. The sequence $\{A_n\}$ converges to $A \subseteq {{\mathbb {R}}^m}$ in the sense of Hausdorff iff $H\left( {{A_n},A} \right) \rightarrow 0$, and we denote it by ${A_n}\mathop \rightarrow \limits ^H A$. Condition $g\left( {{A_n},A} \right) \rightarrow 0$ is the upper part of Hausdorff convergence (denoted by ${A_n}\mathop {\rightharpoonup } \limits ^H A$), while condition $g\left( {{A},{A_n}} \right) \rightarrow 0$ is the lower part of Hausdorff convergence (denoted by ${A_n}\mathop {\rightharpoondown } \limits ^H A$).

Definition 2.3

Let T and $T_1$ be two topological vector spaces. A set-valued mapping $\Phi :T \rightarrow 2^{T_1}$ is said to be

• (i) upper semicontinuous (u.s.c.) at ${u_0} \in T$ if, for any neighborhood V of $\Phi \left( {{u_0}} \right)$, there exists a neighborhood $U\left( {{u_0}} \right)$ of ${u_0}$ such that for every $u \in U\left( {{u_0}} \right)$, $\Phi \left( u \right) \subseteq V$;

• (ii) lower semicontinuous (l.s.c.) at ${u_0} \in T$ if, for any $x \in \Phi \left( {{u_0}} \right)$ and any neighborhood V of x, there exists a neighborhood $U\left( {{u_0}} \right)$ of ${u_0}$ such that for every $u \in U\left( {{u_0}} \right)$, $\Phi \left( u \right) \cap V \ne \emptyset$.

We say that $\Phi$ is u.s.c. and l.s.c. on T if it is u.s.c. and l.s.c. at each point $u \in T$, respectively. We call that $\Phi$ is continuous on T if it is both u.s.c. and l.s.c. on T.

Definition 2.4

(Han and Huang 2018) Let D be a nonempty convex subset of X. A set-valued mapping $\Phi :X \rightarrow {2^{Y}}$ is said to be

• (i) natural quasi C-convex on D if, for any ${x_1},{x_2} \in D$ and for any $t \in \left[ {0,1} \right]$, there exists $\lambda \in \left[ {0,1} \right]$ such that

  $$\begin{aligned} \lambda \Phi \left( {{x_1}} \right) + \left( {1 - \lambda } \right) \Phi \left( {{x_2}} \right) \subseteq \Phi \left( {t{x_1} + \left( {1 - t} \right) {x_2}} \right) + C. \end{aligned}$$

• (iii) strictly natural quasi C-convex on D if, for any ${x_1},{x_2} \in D$ with ${x_1} \ne {x_2}$ and for any $t \in \left( {0,1} \right)$, there exists $\lambda \in \left[ {0,1} \right]$ such that

  $$\begin{aligned} \lambda \Phi \left( {{x_1}} \right) + \left( {1 - \lambda } \right) \Phi \left( {{x_2}} \right) \subseteq \Phi \left( {t{x_1} + \left( {1 - t} \right) {x_2}} \right) + {\mathrm{int}}C. \end{aligned}$$

Definition 2.5

Let $F:X \rightarrow {2^Y}$ be a set-valued mapping and D be a nonempty subset of X. The $\varepsilon$-level set ${Q_l}\left( {\varepsilon ,x,D} \right)$ is defined as follows:

$$\begin{aligned} {Q_l}\left( {\varepsilon ,x,D} \right) = \left\{ {u \in D:F\left( u \right) \le _\varepsilon ^lF\left( x \right) } \right\} \bigcup \left\{ x \right\} . \end{aligned}$$

Remark 2.6

Crespi et al. (2017) defined general level sets for any map g from a domain D to a range R with a binary relation $\precsim$ on R, as

$$\begin{aligned} {\mathrm{Lev}}\left( {g, \precsim ,r} \right) = \left\{ {d \in D:g\left( d \right) \precsim r} \right\} , \end{aligned}$$

for any $r \in R$. However, in Definition 2.5, since the $\varepsilon$-lower relation “$\le _\varepsilon ^l$” is not reflexive for $\varepsilon > 0$, $x \in \left\{ {u \in D:F\left( u \right) \le _\varepsilon ^lF\left( x \right) } \right\}$ may not be true. Thus, we define the $\varepsilon$-level set ${Q_l}\left( {\varepsilon ,x,D} \right)$ by

$$\begin{aligned} {Q_l}\left( {\varepsilon ,x,D} \right) = \left\{ {u \in D:F\left( u \right) \le _\varepsilon ^lF\left( x \right) } \right\} \bigcup \left\{ x \right\} . \end{aligned}$$

In the following two lemmas, let T and $T_1$ be two normed vector spaces.

Lemma 2.1

(Aubin and Ekeland 1984) A set-valued mapping $G:T \rightarrow {2^{{T_1}}}$ is l.s.c. at ${u_0} \in T$ if and only if for any sequence $\left\{ {{u_n}} \right\} \subseteq T$ with ${u_n} \rightarrow {u_0}$ and for any ${x_0} \in G\left( {{u_0}} \right)$, there exists ${x_n} \in G\left( {{u_n}} \right)$ such that ${x_n} \rightarrow {x_0}$.

Lemma 2.2

(Göpfert et al. 2003) Let $G:T \rightarrow {2^{{T_1}}}$ be a set-valued mapping. For any given $u_0\in T$, if $G\left( {{u_0}} \right)$ is compact, then G is u.s.c. at ${u_0} \in T$ if and only if for any sequence $\left\{ {{u_n}} \right\} \subseteq T$ with ${u_n} \rightarrow {u_0}$ and for any ${x_n} \in G\left( {{u_n}} \right)$, there exist ${x_0} \in G\left( {{u_0}} \right)$ and a subsequence $\left\{ {{x_{{n_k}}}} \right\}$ of $\left\{ {{x_n}} \right\}$ such that ${x_{{n_k}}} \rightarrow {x_0}$.

Lemma 2.3

(Rockafellar and Wets 2004) Let ${A_n} \subseteq {{\mathbb {R}}^m}$ with $n=1,2,\ldots$ and ${A} \subseteq {{\mathbb {R}}^m}$. Then $A \subseteq {\mathrm{Li}}{A_n}$ if and only if for any open set W with $W \cap A \ne \emptyset$, there exists ${n_0} \in {\mathbb {N}}$ such that $W \cap {A_n} \ne \emptyset$ for any $n \ge {n_0}$.

Lemma 2.4

(Karuna and Lalitha 2019) Let A be a nonempty subset of X and ${A_n}$ be a sequence of nonempty subsets of X. Then the following assertions hold:

• (i) If ${A_n}\mathop \rightharpoonup \limits ^H A$ and A is closed, then ${\mathrm{Ls}}{A_n} \subseteq A$.

• (ii) ${A_n}\mathop \rightharpoonup \limits ^H A$ if and only if for any $\varepsilon > 0$, there exists ${n_\varepsilon } \in {\mathbb {N}}$ such that ${A_n} \subseteq A + \varepsilon {B_X}$ for all $n \ge {n_\varepsilon }$.

Lemma 2.5

(Han and Huang 2017; Alonso and Rodríguez-Marín 2005) Assume that D is nonempty compact and F is u.s.c. on D. Then ${E_l}\left( {D} \right) \ne \emptyset$.

Lemma 2.6

(Han et al. 2019) Assume that D is convex and F is strictly natural quasi C-convex on D with nonempty compact values. Then ${E_l}\left( D \right) = {W_l}\left( D \right)$.

Lemma 2.7

Assume that D is closed and F is u.s.c. on D with nonempty compact values. Then ${Q_l}\left( {\varepsilon ,x,D} \right)$ is closed.

Proof

It suffices to prove that $\left\{ {u \in D:F\left( u \right) \le _\varepsilon ^lF\left( x \right) } \right\}$ is closed. Assume that

$$\begin{aligned} \left\{ {{u_n}} \right\} \subseteq \left\{ {u \in D:F\left( u \right) \le _\varepsilon ^lF\left( x \right) } \right\} \end{aligned}$$

with ${u_n} \rightarrow {u_0}$. Then ${u_0} \in D$ and ${F\left( {{u_n}} \right) \le _\varepsilon ^lF\left( x \right) }$, which means that $F\left( x \right) \subseteq F\left( {{u_n}} \right) + C + \varepsilon e$. For any $z \in F\left( x \right)$, there exists ${v _n} \in F\left( {{u_n}} \right)$ such that

$$\begin{aligned} z - {v _n} \in C + \varepsilon e. \end{aligned}$$

(1)

Since F is u.s.c. at ${u _0}$, it follows from Lemma 2.2 that there exist ${v _0} \in F\left( {{u _0}} \right)$ and a subsequence $\left\{ {v _{n_k }} \right\}$ of $\left\{ {{v_n }} \right\}$ such that ${v _{n_k}} \rightarrow {v_0}$. This together with (1) implies that $z - {v _0} \in C + \varepsilon e$ and so $F\left( x \right) \subseteq F\left( {{u_0}} \right) + C + \varepsilon e$. Therefore, ${u_0} \in \left\{ {u \in D:F\left( {{u}} \right) \le _\varepsilon ^lF\left( x \right) } \right\}$. $\square$

Lemma 2.8

Assume that ${x_0} \in D$ and $F(x_0)$ is compact.

• (i) If $\varepsilon > 0$, then ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$ if and only if there does not exist $y \in D$ satisfying $F\left( y \right) \le _\varepsilon ^lF\left( {{x_0}} \right)$;

• (ii) Then ${x_0} \in {W_l}\left( {\varepsilon ,D} \right)$ if and only if there does not exist $y \in D$ satisfying $F(x)\le _\varepsilon ^u F(x_0)$.

Proof

(i). It suffices to prove the necessity. Suppose that there exists ${y_0} \in D$ such that $F\left( {y_0} \right) \le _\varepsilon ^lF\left( {{x_0}} \right)$, and so

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( {{y_0}} \right) + C + \varepsilon e. \end{aligned}$$

(2)

It follows from ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$ that $F\left( {x_0} \right) \le _\varepsilon ^lF\left( {{y_0}} \right)$. Then,

$$\begin{aligned} F\left( {{y_0}} \right) \subseteq F\left( {{x_0}} \right) + C + \varepsilon e. \end{aligned}$$

(3)

Due to (2) and (3), we have

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( {{y_0}} \right) + C + \varepsilon e \subseteq F\left( {{x_0}} \right) + C + 2\varepsilon e \subseteq F\left( {{x_0}} \right) + C\backslash \left\{ 0 \right\} . \end{aligned}$$

(4)

In view of Remark 2.1, we have ${\mathrm{Min}}\left( {F\left( {{x_0}} \right) } \right) \ne \emptyset$. Let ${{\mathrm{z}}_0} \in {\mathrm{Min}}\left( {F\left( {{x_0}} \right) } \right)$. Consequently,

$$\begin{aligned} \left( {F\left( {{x_0}} \right) - {z_0}} \right) \cap \left( { - C} \right) = \left\{ 0 \right\} . \end{aligned}$$

(5)

It follows from (4) that there exists ${u _0} \in F\left( {{x_0}} \right)$ and ${c_0} \in C\backslash \left\{ 0 \right\}$ such that ${z_0} = {u _0} + {c_0}$. Thus,

$$\begin{aligned} 0 \ne -{c_0} = {u _0} - {z_0} \in \left( {F\left( {{x_0}} \right) - {z_0}} \right) \cap \left( { - C} \right) , \end{aligned}$$

which contradicts (5).

The proof of (ii) is similar to the proof of (i) and so we omit it here. $\square$

Lemma 2.9

Assume that $\varepsilon > 0$, ${x_0} \in D$ and $F(x_0)$ is compact. Then ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$ if and only if ${Q_l}\left( {\varepsilon ,{x_0},D} \right) = \left\{ {{x_0}} \right\}$.

Proof

In view of Lemma 2.8, it is easy to see that the necessity is true. Next, we prove the sufficiency. We claim that

$$\begin{aligned} \left\{ {u \in D:F\left( u \right) \le _\varepsilon ^lF\left( {{x_0}} \right) } \right\} = \emptyset . \end{aligned}$$

(6)

In fact, if not, due to ${Q_l}\left( {\varepsilon ,{x_0},D} \right) = \left\{ {{x_0}} \right\}$, we have ${F\left( {{x_0}} \right) \le _\varepsilon ^lF\left( {{x_0}} \right) }$ and so

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( {{x_0}} \right) + C + \varepsilon e \subseteq F\left( {{x_0}} \right) + C\backslash \left\{ 0 \right\} . \end{aligned}$$

(7)

In view of Remark 2.1, similar to the proof of Lemma 2.8, we can see that (7) is not true. This shows that (6) holds. Thus, it follows from (6) and Lemma 2.8 that ${x_0} \in {E_l}\left( {\varepsilon ,D} \right)$. $\square$

Lemma 2.10

Assume that $x \in D$ and $F\left( y \right)$ is compact for any $y \in D$. Then

$$\begin{aligned} {E_l}\left( {\varepsilon ,{Q_l}\left( {\varepsilon ,x,D} \right) } \right) \subseteq {E_l}\left( {\varepsilon ,D} \right) . \end{aligned}$$

Proof

Suppose to the contrary that there exists

$$\begin{aligned} {v _0} \in {E_l}\left( {\varepsilon ,{Q_l}\left( {\varepsilon ,x,D} \right) } \right) \end{aligned}$$

(8)

such that ${v _0} \notin {E_l}\left( {\varepsilon ,D} \right)$. It follows from ${v _0} \notin {E_l}\left( {\varepsilon ,D} \right)$ and Lemma 2.8 that there exists ${z_0} \in D$ such that $F\left( {{z_0}} \right) \le _\varepsilon ^lF\left( {{v _0}} \right)$ and so

$$\begin{aligned} F\left( {{v _0}} \right) \subseteq F\left( {{z_0}} \right) + C + \varepsilon e. \end{aligned}$$

(9)

We claim that ${z_0} \in {Q_l}\left( {\varepsilon ,x,D} \right)$. In fact, in view of ${v _0} \in {Q_l}\left( {\varepsilon ,x,D} \right)$, there are two cases to be considered.

Case 1. ${v _0} = x$. It is clear that $F\left( {{z_0}} \right) \le _\varepsilon ^lF\left( {{x}} \right)$ and so ${z_0} \in {Q_l}\left( {\varepsilon ,x,D} \right)$.

Case 2. $F\left( {{v _0}} \right) \le _\varepsilon ^lF\left( x \right)$. Then

$$\begin{aligned} F\left( {{x}} \right) \subseteq F\left( {{v _0}} \right) + C + \varepsilon e. \end{aligned}$$

(10)

Due to (9) and (10), we have

$$\begin{aligned} F\left( x \right) \subseteq F\left( {{v _0}} \right) + C + \varepsilon e \subseteq F\left( {{z_0}} \right) + C + C + 2\varepsilon e \subseteq F\left( {{z_0}} \right) + C + \varepsilon e, \end{aligned}$$

which means that $F\left( {{z_0}} \right) \le _\varepsilon ^lF\left( {{x}} \right)$. Thus, ${z_0} \in {Q_l}\left( {\varepsilon ,x,D} \right)$.

In view of Lemma 2.8, ${z_0} \in {Q_l}\left( {\varepsilon ,x,D} \right)$ and $F\left( {{z_0}} \right) \le _\varepsilon ^lF\left( {{v _0}} \right)$ show that ${v _0} \notin {E_l}\left( {\varepsilon ,{Q_l}\left( {\varepsilon ,x,D} \right) } \right)$, which contradicts (8). $\square$

3 Painlevé–Kuratowski convergence

In this section, we discuss the Painlevé–Kuratowski upper convergence and the Painlevé–Kuratowski lower convergence of the approximate solution sets for set optimization problems.

Lemma 3.1

Let $\{D_n\}$ be a sequence of subsets of X, D be a bounded subset of X, ${x_n} \in {D_n}$ with ${x_n} \rightarrow x \in D$ and $\left\{ {{\varepsilon _n}} \right\} \subseteq {{\mathbb {R}}_ + }$ with ${\varepsilon _n} \rightarrow {\varepsilon _0}$. Assume that F is continuous on D with nonempty compact values and any of the following conditions is satisfied:

• (a) ${\mathrm{Ls}}{D_n} \subseteq D$ and there exist $\delta > 0$ and ${n_0} \in {\mathbb {N}}$ such that ${D_n} \subseteq D + \delta {B_X}$ for any $n \ge {n_0}$;

• (b) ${D_n}\mathop {\rightharpoonup } \limits ^H D$ and D is closed;

• (c) ${\varepsilon _0} = 0$, ${\mathrm{Ls}}{D_n} \subseteq D$, ${D_n}$ is convex and and F is naturally quasi C-convex on $D_n$.

Then, for any $\alpha > 0$, there exists ${\bar{n}} \in {\mathbb {N}}$ such that

$$\begin{aligned} {Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right) \subseteq {Q_l}\left( {{\varepsilon _0},x,D} \right) + \alpha {B_X},\;\;\forall n \ge {\bar{n}}. \end{aligned}$$

Proof

Suppose to the contrary that there exists ${\alpha _0} > 0$ such that, for any $n \in {\mathbb {N}}$, there exists ${m_n} \ge n$ satisfying

$$\begin{aligned} {Q_l}\left( {{\varepsilon _{m_n}},{x_{m_n}},{D_{m_n}}} \right) \not \subset {Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}. \end{aligned}$$

Without loss of generality, we assume that

$$\begin{aligned} {Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right) \not \subset {Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X},\;\;\forall n \in {\mathbb {N}}. \end{aligned}$$

Then there exists

$$\begin{aligned} {v _n} \in {Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right) \end{aligned}$$

(11)

such that

$$\begin{aligned} {v _n} \notin {Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}. \end{aligned}$$

(12)

It is clear that ${v _n} \in {D_n}$. It follows from ${x_n} \rightarrow x$ and $x \in {Q_l}\left( {{\varepsilon _0},x,D} \right)$ that

$$\begin{aligned} {x_n} \in {Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X} \end{aligned}$$

(13)

for n large enough. This together with (12) implies that ${x_n} \ne {v _n}$ for n large enough. Thus, by (11), we have $F\left( {{v _n}} \right) \le _{\varepsilon _n} ^lF\left( {{x_n}} \right)$ and so

$$\begin{aligned} F\left( {{x_n}} \right) \subseteq F\left( {{v _n}} \right) + C + {\varepsilon _n} e. \end{aligned}$$

(14)

• (a) By virtue of condition (a), we can see that $\left\{ {{v_n}} \right\} \subseteq X$ is bounded. Without loss of generality, we assume that ${v_n} \rightarrow {v_0} \in X$. In view of ${\mathrm{Ls}}{D_n} \subseteq D$, we have ${v_0} \in D$. We claim that

  $$\begin{aligned} F\left( x \right) \subseteq F\left( {{v _0}} \right) + C + {\varepsilon _0} e. \end{aligned}$$

  (15)

  In fact, if not, then there exists ${z_0} \in F\left( x \right)$ such that

  $$\begin{aligned} {z_0} \notin F\left( {{v _0}} \right) + C + {\varepsilon _0} e. \end{aligned}$$

  (16)

  Since F is l.s.c. at x, by Lemma 2.1, there exists ${z_n} \in F\left( {{x_n}} \right)$ such that ${z_n} \rightarrow {z_0}$. Due to (14), there exists ${u_n} \in F\left( {{v _n}} \right)$ such that

  $$\begin{aligned} {z_n} - {u_n} \in C + {\varepsilon _n} e. \end{aligned}$$

  (17)

  Since F is u.s.c. at ${v _0}$, it follows from Lemma 2.2 that there exist ${u_0} \in F\left( {{v _0}} \right)$ and a subsequence $\left\{ {u_{n_k }} \right\}$ of $\left\{ {{u_n }} \right\}$ such that ${u_{n_k}} \rightarrow {u_0}$. In view of (17), we have ${z_0} - {u_0} \in C + {\varepsilon _0} e$. which contradicts (16). Therefore, (15) holds, which means that ${v _0} \in {Q_l}\left( {{\varepsilon _0},x,D} \right)$. Noting that

  $$\begin{aligned} {v _n} \rightarrow {v _0} \in {Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}, \end{aligned}$$

  we can see that ${v _n} \in {Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}$ for n large enough, which contradicts (12).

• (b) In view of Lemma 2.4, it is easy to see that condition (b) implies condition (a).

• (c) Since $F\left( {x} \right)$ is compact and F is l.s.c. at x. we can see that F is H-l.s.c. at x. Noting that $C - {1 \over n}e$ is a neighborhood of $0 \in X$, then there exists ${m_n} \ge n$ such that $F\left( x \right) \subseteq F\left( {{{x}_{{m_n}}}} \right) + C - {1 \over n}e$. Without loss of generality, we assume that

  $$\begin{aligned} F\left( x \right) \subseteq F\left( {{{x}_{{n}}}} \right) + C - {1 \over n}e,\;\; \forall n \in {\mathbb {N}}. \end{aligned}$$

  (18)

  Due to (14) and (18), we have

  $$\begin{aligned} F\left( x \right) \subseteq F\left( {{v _n}} \right) + C + \left( {{\varepsilon _n} - {1 \over n}} \right) e. \end{aligned}$$

  (19)

  Let ${x_n}\left( t \right) = t{v _n} + \left( {1 - t} \right) {{x}_n}$ for $t \in \left[ {0,1} \right]$. By the convexity of ${D_n}$, we have ${x_n}\left( t \right) \in {D_n}$. Since F is naturally quasi C-convex on $D_n$, for the above $t \in \left[ {0,1} \right]$, there exists $\lambda \in \left[ {0,1} \right]$ such that

  $$\begin{aligned} \lambda F\left( {{v _n}} \right) + \left( {1 - \lambda } \right) F\left( {{{x}_n}} \right) \subseteq F\left( {{x_n}\left( t \right) } \right) + C. \end{aligned}$$

  (20)

  Applying (18)–(20), we have

  $$\begin{aligned}&F\left( x \right) \subseteq \lambda F\left( x \right) + \left( {1 - \lambda } \right) F\left( x \right) \nonumber \\\subseteq & {} \lambda F\left( {{v _n}} \right) + \lambda C + \lambda \left( {{\varepsilon _n} - {1 \over n}} \right) e \nonumber \\&+\, \left( {1 - \lambda } \right) F\left( {{x_n}} \right) + \left( {1 - \lambda } \right) C - \left( {1 - \lambda } \right) {1 \over n}e \nonumber \\\subseteq & {} F\left( {{x_n}\left( t \right) } \right) + \lambda C - {1 \over n}e + \left( {1 - \lambda } \right) C + C + \lambda {\varepsilon _n}e \nonumber \\\subseteq & {} F\left( {{x_n}\left( t \right) } \right) + C - {1 \over n}e. \end{aligned}$$

  (21)

  It follows from (12) and (13) that there exists ${t_n} \in \left[ {0,1} \right]$ such that ${x_n}\left( {{t_n}} \right) \in \partial \left[ {{Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}} \right]$. Since ${{Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}}$ is bounded and $X = {{\mathbb {R}}^m}$, it is easy to see that $\partial \left[ {{Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}} \right]$ is compact. Without loss of generality, we assume that

  $$\begin{aligned} {x_n}\left( {{t_n}} \right) \rightarrow w \in \partial \left[ {{Q_l}\left( {{\varepsilon _0},x,D} \right) + {\alpha _0}{B_X}} \right] \end{aligned}$$

  (22)

  Due to (22), ${x_n}\left( {{t_n}} \right) \in {D_n}$ and ${\mathrm{Ls}}{D_n} \subseteq D$, we have $w \in D$. It follows from (21) that

  $$\begin{aligned} F\left( x \right) \subseteq F\left( {{x_n}\left( {{t_n}} \right) } \right) + C - {1 \over n}e. \end{aligned}$$

  (23)

  We claim that

  $$\begin{aligned} F\left( x \right) \subseteq F\left( w \right) + C = F\left( w \right) + C + {\varepsilon _0}e. \end{aligned}$$

  (24)

  In fact, for any $v \in F\left( x \right)$, it follows from (23) that there exists ${z_n} \in F\left( {{x_n}\left( {{t_n}} \right) } \right)$ such that

  $$\begin{aligned} v - {z_n} + {1 \over n}e \in C. \end{aligned}$$

  (25)

  Since F is u.s.c. at $w \in D$, by Lemma 2.2, there exist ${z_0} \in F\left( {{w}} \right)$ and a subsequence $\left\{ {z_{n_k }} \right\}$ of $\left\{ {{z_n }} \right\}$ such that ${z_{n_k}} \rightarrow {z_0}$. Due to (25) and the closedness of C, we have $v - {z_0} \in C$, and so $v \in {z_0} + C \subseteq F\left( w \right) + C$. Thus, $F\left( x \right) \subseteq F\left( w \right) + C = F\left( w \right) + C + {\varepsilon _0}e$. This together with $w \in D$ implies that $w \in {Q_l}\left( {{\varepsilon _0},x,D} \right)$, which contradicts (22). This completes the proof.

$\square$

Theorem 3.1

Let $\{D_n\}$ be a sequence of subsets of X and $\left\{ {{\varepsilon _n}} \right\} \subseteq {{\mathbb {R}}_ + }$ with ${\varepsilon _n} \rightarrow {\varepsilon _0}$. Assume that F is continuous on D with nonempty compact values and ${D_n}\mathop \rightarrow \limits ^K D$. Then ${\mathrm{Ls}}{W_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {W_l}\left( {{\varepsilon _0},D} \right)$.

Proof

Let ${x_0} \in {\mathrm{Ls}}{W_l}\left( {{\varepsilon _n},{D_n}} \right)$. Then there exist a subsequence $\left\{ {{n_k}} \right\}$ of the integers and ${x_{{n_k}}} \in {W_l}\left( {{\varepsilon _{{n_k}}},{D_{{n_k}}}} \right)$ such that ${x_{{n_k}}} \rightarrow {x_0}$. Due to ${D_n}\mathop \rightarrow \limits ^K D$, we have ${x_0} \in D$.

We now show that ${x_0} \in {W_l}\left( {{\varepsilon _0},D} \right)$. Suppose to the contrary that ${x_0} \notin {W_l}\left( {{\varepsilon _0},D} \right)$. Then, in view of Lemma 2.8, there exists ${y_0} \in D$ such that $F\left( {{y_0}} \right) \ll _{\varepsilon _0} ^lF\left( {{x_0}} \right)$ and so

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( {{y_0}} \right) + {\mathrm{int}} C + {\varepsilon _0}e. \end{aligned}$$

(26)

Due to ${y_0} \in D$ and ${D_n}\mathop \rightarrow \limits ^K D$, there exists ${y_n} \in D$ such that ${y_n} \rightarrow {y_0}$. We claim that there exists ${k_0} \in {\mathbb {N}}$ such that

$$\begin{aligned} F\left( {{x_{{n_k}}}} \right) \subseteq F\left( {{y_{{n_k}}}} \right) + {\mathrm{int}} C + {\varepsilon _{{n_k}}}e,\;\;\forall k \ge {k_0}. \end{aligned}$$

(27)

In fact, if not, then there exist a subsequence $\left\{ {{x_{{n_{{k_j}}}}}} \right\}$ of $\left\{ {x_{n_k }} \right\}$ and a subsequence $\left\{ {{y_{{n_{{k_j}}}}}} \right\}$ of $\left\{ {y_{n_k }} \right\}$ such that

$$\begin{aligned} F\left( {{x_{{n_{{k_j}}}}}} \right) \not \subset F\left( {{y_{{n_{{k_j}}}}}} \right) + {\mathrm{int}} C + {\varepsilon _{{n_{{k_j}}}}}e. \end{aligned}$$

Without loss of generality, we assume that

$$\begin{aligned} F\left( {{x_{{n_k}}}} \right) \not \subset F\left( {{y_{{n_k}}}} \right) + {\mathrm{int}} C + {\varepsilon _{{n_k}}}e,\;\;\forall k \in {\mathbb {N}}. \end{aligned}$$

Then there exists ${v_{{n_k}}} \in F\left( {{x_{{n_k}}}} \right)$ such that

$$\begin{aligned} {v_{{n_k}}} \notin F\left( {{y_{{n_k}}}} \right) + {\mathrm{int}} C + {\varepsilon _{{n_k}}}e. \end{aligned}$$

(28)

Since F is u.s.c. at ${x _0}$, by Lemma 2.2, without loss of generality, we assume that ${v_{{n_k}}} \rightarrow {v_0} \in F\left( {{x_0}} \right)$. It follows from (26) that there exists ${u_0} \in F\left( {{y_0}} \right)$ such that

$$\begin{aligned} {v_0} - {u_0} \in {\mathrm{int}} C + {\varepsilon _0}e. \end{aligned}$$

(29)

Noting that F is l.s.c. at ${y _0}$, in view of Lemma 2.1, there exists ${u_{{n_k}}} \in F\left( {{y_{{n_k}}}} \right)$ such that ${u_{{n_k}}} \rightarrow {u_0}$. Due to (29), we have ${v_{{n_k}}} - {u_{{n_k}}} \in {\mathrm{int}} C + {\varepsilon _{{n_k}}}e$ for k large enough, which contradicts (28). This shows that (27) holds and so $F\left( {{y_{{n_k}}}} \right) \ll _{{\varepsilon _{{n_k}}}}^lF\left( {{x_{{n_k}}}} \right)$. It follows from Lemma 2.8 that ${x_{{n_k}}} \notin {W_l}\left( {{\varepsilon _{{n_k}}},{D_{{n_k}}}} \right)$, which contradicts ${x_{{n_k}}} \in {W_l}\left( {{\varepsilon _{{n_k}}},{D_{{n_k}}}} \right)$. This shows that ${x_0} \in {W_l}\left( {{\varepsilon _0},D} \right)$. $\square$

Theorem 3.2

Let $\{D_n\}$ be a sequence of subsets of X, D be a bounded subset of X and $\left\{ {{\varepsilon _n}} \right\} \subseteq {{\mathbb {R}}_ + }$ with ${\varepsilon _n} \rightarrow {\varepsilon _0} \in {\mathbb {R}}_ +$. Assume that F is continuous on D with nonempty compact values, $D \subseteq {\mathrm{Li}}{D_n}$, $D_n$ is closed and F is u.s.c. on $D_n$, and suppose that any of the following conditions is satisfied:

• (a) ${\varepsilon _0} > 0$, ${\mathrm{Ls}}{D_n} \subseteq D$ and there exist $\delta > 0$ and ${n_0} \in {\mathbb {N}}$ such that ${D_n} \subseteq D + \delta {B_X}$ for any $n \ge {n_0}$;

• (b) ${\varepsilon _0} > 0$, ${D_n}\mathop {\rightharpoonup } \limits ^H D$ and D is closed;

• (c) ${\varepsilon _0} = 0$, ${\mathrm{Ls}}{D_n} \subseteq D$, ${D_n}$ and D are convex, F is naturally quasi C-convex on $D_n$ and F is strictly naturally quasi C-convex on D.

Then, ${E_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{E_l}\left( {{\varepsilon _n},{D_n}} \right)$. Moreover, if D is convex and F is strictly naturally quasi C-convex on D, then ${W_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{W_l}\left( {{\varepsilon _n},{D_n}} \right)$.

Proof

Let ${x_0} \in {E_l}\left( {{\varepsilon _0},D} \right)$. Due to ${x_0} \in D$ and $D \subseteq {\mathrm{Li}}{D_n}$, there exists ${x_n} \in {D_n}$ such that ${x_n} \rightarrow {x_0}$. If ${\varepsilon _0} > 0$, in view of Lemma 2.9 and ${x_0} \in {E_l}\left( {{\varepsilon _0},D} \right)$, we have ${Q_l}\left( {{\varepsilon _0} ,{x_0},D} \right) = \left\{ {{x_0}} \right\}$. If ${\varepsilon _0} = 0$, it follows from Lemma 3.1 of Han et al. (2019) that ${Q_l}\left( {{\varepsilon _0} ,{x_0},D} \right) = \left\{ {{x_0}} \right\}$. By Lemma 3.1 , we can see that for any $\alpha > 0$, there exists ${n_0} \in {\mathbb {N}}$ such that

$$\begin{aligned} {Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right) \subseteq {Q_l}\left( {{\varepsilon _0},{x_0},{D}} \right) + \alpha {B_X} = \left\{ {{x_0}} \right\} + \alpha {B_X},\quad \forall n \ge {n_0}. \end{aligned}$$

(30)

It follows from Lemma 2.7 that ${Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right)$ is closed. Due to (30), we know that ${Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right)$ is bounded, and consequently, ${Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right)$ is compact. In view of Remark 2.3 and Lemma 2.5, we have

$$\begin{aligned} {E_l}\left( {{\varepsilon _n},{Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right) } \right) \ne \emptyset . \end{aligned}$$

Let ${v _n} \in {E_l}\left( {{\varepsilon _n},{Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right) } \right)$. From Lemma 2.10, one has

$$\begin{aligned} {v _n} \in {E_l}\left( {{\varepsilon _n},{Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right) } \right) \subseteq {E_l}\left( {{\varepsilon _n},{D_n}} \right) . \end{aligned}$$

Due to ${v _n} \in {Q_l}\left( {{\varepsilon _n},{x_n},{D_n}} \right)$ and (30), we have ${v _n} \rightarrow {x_0}$. Therefore, ${x_0} \in {\mathrm{Li}}{E_l}\left( {{\varepsilon _n},{D_n}} \right)$ and so ${E_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{E_l}\left( {{\varepsilon _n},{D_n}} \right)$.

We next show that ${W_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{W_l}\left( {{\varepsilon _n},{D_n}} \right)$. For any open set V with $V \cap {W_l}\left( {{\varepsilon _0},D} \right) \ne \emptyset$, we claim that

$$\begin{aligned} V \cap {E_l}\left( {{\varepsilon _0},D} \right) \ne \emptyset . \end{aligned}$$

(31)

In fact, let ${z_0} \in V \cap {W_l}\left( {{\varepsilon _0},D} \right)$. Suppose that $V \cap {E_l}\left( {{\varepsilon _0},D} \right) = \emptyset$. Then ${z_0} \notin {E_l}\left( {{\varepsilon _0},D} \right)$. It follows from Lemma 2.8 that there exists ${y_0} \in D$ such that $F\left( {{y_0}} \right) \le _{{\varepsilon _0}}^lF\left( {{z_0}} \right)$ and so

$$\begin{aligned} F\left( {{z_0}} \right) \subseteq F\left( {{y_0}} \right) + C + {\varepsilon _0}e \subseteq F\left( {{y_0}} \right) + C\backslash \left\{ 0 \right\} . \end{aligned}$$

(32)

Noting Remark 2.1 and (4), from the proof of Lemma 2.8, we can see that ${z_0} \ne {y_0}$. Since F is strictly naturally quasi C-convex on D, for any $t \in \left( {0,1} \right)$, there exists $\lambda \in \left[ {0,1} \right]$ such that

$$\begin{aligned} \lambda F\left( {{z_0}} \right) + \left( {1 - \lambda } \right) F\left( {{y_0}} \right) \subseteq F\left( {t{z_0} + \left( {1 - t} \right) {y_0}} \right) + {\mathrm{int}} C. \end{aligned}$$

(33)

By (32) and (33), we have

$$\begin{aligned} F\left( {{z_0}} \right) \subseteq F\left( {t{z_0} + \left( {1 - t} \right) {y_0}} \right) + {\mathrm{int}} C, \quad \forall t \in \left( {0,1} \right) . \end{aligned}$$

(34)

In fact, if (34) does not hold, then there exist ${t_0} \in \left( {0,1} \right)$ and ${u_0} \in F\left( {{z_0}} \right)$ such that

$$\begin{aligned} {u_0} \notin F\left( {{t_0}{z_0} + \left( {1 - {t_0}} \right) {y_0}} \right) + {\mathrm{int}} C. \end{aligned}$$

(35)

It follows from (32) that there exist ${s _0} \in F\left( {{y_0}} \right)$ and ${c_0} \in C$ such that ${u_0} = {s _0} + {c_0}$. Due to (33), one has

$$\begin{aligned} \lambda {u_0} + \left( {1 - \lambda } \right) {s _0} \in F\left( {{t_0}{z_0} + \left( {1 - {t_0}} \right) {y_0}} \right) + {\mathrm{int}} C. \end{aligned}$$

This together with ${u_0} = {s _0} + {c_0}$ implies that ${u_0} \in F\left( {{t_0}{z_0} + \left( {1 - {t_0}} \right) {y_0}} \right) + {\mathrm{int}} C$, which contradicts (35). Therefore, (34) is true. Let $z\left( t \right) = t{z_0} + \left( {1 - t} \right) {y_0}$ for $t \in \left( {0,1} \right)$. It is clear that there exists ${\hat{t}} \in \left( {0,1} \right)$ such that $z ( {\hat{t}} ) \in V$. This together with $V \cap {E_l}\left( {{\varepsilon _0},D} \right) = \emptyset$ implies that $z\left( {\hat{t}} \right) \notin {E_l}\left( {{\varepsilon _0},D} \right)$. In view of Lemma 2.8, there exists ${w _0} \in D$ such that $F\left( {{w _0}} \right) \le _{{\varepsilon _0}}^lF\left( {z\left( {\hat{t}} \right) } \right)$ and so

$$\begin{aligned} F\left( {z\left( {\hat{t}} \right) } \right) \subseteq F\left( {{w _0}} \right) + C + {\varepsilon _0}e. \end{aligned}$$

(36)

Due to (34) and (36), we have

$$\begin{aligned} F\left( {{z_0}} \right) \subseteq F\left( {z\left( {\hat{t}} \right) } \right) + {\mathrm{int}} C \subseteq F\left( {{w _0}} \right) + {\mathrm{int}} C + C + {\varepsilon _0}e \subseteq F\left( {{w _0}} \right) + {\mathrm{int}} C + {\varepsilon _0}e, \end{aligned}$$

which means that $F\left( {{w _0}} \right) \ll _{{\varepsilon _0}}^lF\left( {z _0} \right)$. From Lemma 2.8, we have ${z_0} \notin {W_l}\left( {{\varepsilon _0},D} \right)$, which contradicts ${z_0} \in {W_l}\left( {{\varepsilon _0},D} \right)$. Hence, (31) holds. Applying (31), ${E_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{E_l}\left( {{\varepsilon _n},{D_n}} \right)$ and Lemma 2.3, we know that there exists $\bar{n} \in {\mathbb {N}}$ such that

$$\begin{aligned} V \cap {E_l}\left( {{\varepsilon _n},{D_n}} \right) \ne \emptyset , \quad \forall n \ge \bar{n}. \end{aligned}$$

(37)

It follows from Remark 2.4 that ${E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {W_l}\left( {{\varepsilon _n},{D_n}} \right)$. This together with (37) implies that

$$\begin{aligned} V \cap {W_l}\left( {{\varepsilon _n},{D_n}} \right) \ne \emptyset , \quad \forall n \ge \bar{n}. \end{aligned}$$

Thus, from Lemma 2.3, we have ${W_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{W_l}\left( {{\varepsilon _n},{D_n}} \right)$. $\square$

Now, we give an example to illustrate Theorems 3.1 and 3.2.

Example 3.1

Let $X = {\mathbb {R}}$, $Y = {{\mathbb {R}}^2}$ and $C = {\mathbb {R}}_ + ^2 = \left\{ {\left( {{x_1},{x_2}} \right) \in {{\mathbb {R}}^2}:{x_1} \ge 0,{x_2} \ge 0} \right\}$. Define a set-valued mapping $F:X \rightarrow {2^Y}$ as follows:

$$\begin{aligned} F\left( x \right) = \left( { - 5\cos {1 \over 6}x,{x^2} - x + 3} \right) + {B_Y},\;\;x \in X. \end{aligned}$$

Let $D = \left[ { - 9,9} \right]$ and ${D_n} = \left[ { - 9 + \sin {1 \over n},8 + \cos {1 \over n}} \right]$. Then it is easy to see that ${D_n}\mathop \rightarrow \limits ^K D$ and F is strictly naturally quasi C-convex on D. Let ${\varepsilon _0} = 1$, ${\varepsilon _n} = {{n + 1} \over n}$ and $e = \left( {1,1} \right)$. Then,

$$\begin{aligned} 0 \in {E_l}\left( {{\varepsilon _0},D} \right) \subseteq {W_l}\left( {{\varepsilon _0},D} \right) , \quad 0 \in {E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {W_l}\left( {{\varepsilon _n},{D_n}} \right) . \end{aligned}$$

It is easy to check that all conditions of Theorems 3.1 and 3.2 are satisfied. Thus, Theorem 3.1 shows that ${\mathrm{Ls}}W\left( {{\varepsilon _n},{D_n}} \right) \subseteq W\left( {{\varepsilon _0},D} \right)$ and Theorem 3.2 implies that ${E_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{E_l}\left( {{\varepsilon _n},{D_n}} \right)$ and ${W_l}\left( {{\varepsilon _0},D} \right) \subseteq {\mathrm{Li}}{W_l}\left( {{\varepsilon _n},{D_n}} \right)$.

Theorem 3.3

Let $\{D_n\}$ be a sequence of subsets of X, D be a convex subset of X and $\left\{ {{\varepsilon _n}} \right\} \subseteq {{\mathbb {R}}_ + }$ with ${\varepsilon _n} \rightarrow {\varepsilon _0} = 0$. Assume that

• (i) F is continuous on D with nonempty compact values and ${D_n}\mathop \rightarrow \limits ^K D$.

• (ii) F is strictly naturally quasi C-convex on D.

Then, ${\mathrm{Ls}}{E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {E_l}\left( {{\varepsilon _0},D} \right)$.

Proof

Noting that ${\varepsilon _0} = 0$ and Lemma 2.6, we have ${E_l}\left( {{\varepsilon _0},D} \right) = {E_l}\left( D \right) = {W_l}\left( D \right) = {W_l}\left( {{\varepsilon _0},D} \right)$. From Remark 2.4, we have ${E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {W_l}\left( {{\varepsilon _n},{D_n}} \right)$ and so ${\mathrm{Ls}}{E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {\mathrm{Ls}}{W_l}\left( {{\varepsilon _n},{D_n}} \right)$. It follows from Theorem 3.1 that ${\mathrm{Ls}}W\left( {{\varepsilon _n},{D_n}} \right) \subseteq W\left( {{\varepsilon _0},D} \right)$. Consequently,

$$\begin{aligned} {\mathrm{Ls}}{E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {\mathrm{Ls}}{W_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {W_l}\left( {{\varepsilon _0},D} \right) {\mathrm{= }}{E_l}\left( {{\varepsilon _0},D} \right) . \end{aligned}$$

This completes the proof. $\square$

If ${\varepsilon _0} > 0$ and F is strictly naturally quasi C-convex on D, then ${E_l}\left( {{\varepsilon _0},D} \right) = {W_l}\left( {{\varepsilon _0},D} \right)$ may not be true. We give the following counterexample to illustrate it.

Example 3.2

Let $X = {\mathbb {R}}$, $Y = {{\mathbb {R}}^2}$ and $C = {\mathbb {R}}_ + ^2 = \left\{ {\left( {{x_1},{x_2}} \right) \in {{\mathbb {R}}^2}:{x_1} \ge 0,{x_2} \ge 0} \right\}$. Define a set-valued mapping $F:X \rightarrow {2^Y}$ as follows:

$$\begin{aligned} F\left( x \right) = \left( { - 2\cos {1 \over 3}x,{x^2}} \right) + {B_Y},\;\;x \in X. \end{aligned}$$

Let $D = \left[ { - 4,4} \right]$, ${\varepsilon _0} = 1$ and $e = \left( {1,1} \right)$. Then it is easy to see that F is strictly naturally quasi C-convex on D. Moreover, from Lemma 2.8, we can see that ${E_l}\left( {{\varepsilon _0},D} \right) = \left( { - \pi ,\pi } \right)$ and ${W_l}\left( {{\varepsilon _0},D} \right) = \left[ { - \pi ,\pi } \right]$. Thus, ${E_l}\left( {{\varepsilon _0},D} \right) \ne {W_l}\left( {{\varepsilon _0},D} \right)$.

Inspired by Example 3.2, we give the following proposition.

Proposition 3.1

Let ${\varepsilon _0} > 0$, D be a nonempty and compact subset of X and F be continuous on D with nonempty and compact values. Then ${E_l}\left( {{\varepsilon _0},D} \right)$ is open in D and ${W_l}\left( {{\varepsilon _0},D} \right)$ is closed in D.

Proof

We show that ${E_l}\left( {{\varepsilon _0},D} \right)$ is open in D. For any $\left\{ {{x_n}} \right\} \subseteq D\backslash {E_l}\left( {{\varepsilon _0},D} \right)$ with ${x_n} \rightarrow {x_0} \in D$, it follows from Lemma 2.8 that there exists ${y_n} \in D$ such that $F\left( {{y_n}} \right) \le _{{\varepsilon _0}}^lF\left( {{x_n}} \right)$ and so

$$\begin{aligned} F\left( {{x_n}} \right) \subseteq F\left( {{y_n}} \right) + C + {\varepsilon _0}e. \end{aligned}$$

(38)

Since D is compact, without loss of generality, we assume that ${y_n} \rightarrow {y_0} \in D$. Similar to the proof of (15), by (38), it is easy to prove that $F\left( {{x_0}} \right) \subseteq F\left( {{y_0}} \right) + C + {\varepsilon _0}e$. This together with Lemma 2.8 implies that ${x_0} \notin {E_l}\left( {{\varepsilon _0},D} \right)$. Therefore, ${E_l}\left( {{\varepsilon _0},D} \right)$ is open in D.

Next, we prove that ${W_l}\left( {{\varepsilon _0},D} \right)$ is closed in D. For any $\left\{ {{x_n}} \right\} \subseteq {W_l}\left( {{\varepsilon _0},D} \right)$ with $x_n \rightarrow x_0 \in D$, it suffices to show that ${x_0} \in {W_l}\left( {{\varepsilon _0},D} \right)$. Suppose that ${x_0} \notin {W_l}\left( {{\varepsilon _0},D} \right)$. Then it follows from Lemma 2.8 that there exists $v _0\in D$ such that $F\left( {{v _0}} \right) \ll _{\varepsilon _0} ^lF\left( {{x_0}} \right)$, i.e.,

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( {{v _0}} \right) + {\mathrm{int}} C + {\varepsilon _0}e. \end{aligned}$$

It is clear that $F\left( {{v _0}} \right) + {\mathrm{int}} C + {\varepsilon _0}e$ is a neighborhood of $F(x_0)$. Since F is u.s.c. at ${x _0}$, there exists $n_0\in {\mathbb {N}}$ such that

$$\begin{aligned} F(x_n)\subset F\left( {{v _0}} \right) + {\mathrm{int}} C + {\varepsilon _0}e,\;\; \forall n\ge n_0, \end{aligned}$$

which means that $F\left( {{v _0}} \right) \ll _\varepsilon ^lF\left( {{x_n}} \right)$. This together with Lemma 2.8 implies that ${x_n} \notin {W_l}\left( {{\varepsilon _0},D} \right)$, which contradicts $\left\{ {{x_n}} \right\} \subseteq {W_l}\left( {{\varepsilon _0},D} \right)$. This shows that ${x_0} \in {W_l}\left( {{\varepsilon _0},D} \right)$ and so ${W_l}\left( {{\varepsilon _0},D} \right)$ is closed in D. $\square$

Remark 3.1

In Theorem 3.3, we show that ${\mathrm{Ls}}{E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {E_l}\left( {{\varepsilon _0},D} \right)$ under the assumption that ${\varepsilon _0} = 0$. If ${\varepsilon _0} > 0$, then it follows from Proposition 3.1 that ${E_l}\left( {{\varepsilon _0},D} \right) \ne {W_l}\left( {{\varepsilon _0},D} \right)$ in the general case. Therefore, it is interesting to obtain some suitable conditions to ensure ${\mathrm{Ls}}{E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {E_l}\left( {{\varepsilon _0},D} \right)$ holds when ${\varepsilon _0} > 0$.

4 Extended well-posedness

The concept of the extended well-posedness for vector optimization problems is due to Huang (2000b, 2001). In this section, we introduce the notions of the extended well-posedness and the weak extended well-posedness for set optimization problems. We denote by $d\left( {x,A} \right) = \inf \left\{ {\left\| {y - a} \right\| ,a \in A} \right\}$ the distance of a point $x \in X$ to a set $A \subseteq X$.

Definition 4.1

Let $\{D_n\}$ be a sequence of subsets of X and D be a subset of X. We say that (SOP) is extended well-posed (resp., weak extended well-posed) with respect to the perturbation defined by the sequence $\{D_n\}$ if ${{E_l}\left( D \right) \ne \emptyset }$ (resp., ${{W_l}\left( D \right) \ne \emptyset }$) and for every sequence $\{x_n\} \subset {D_n}$ with ${x_n} \in {E_l}\left( {{\varepsilon _n},{D_n}} \right)$ (resp., ${x_n} \in {W_l}\left( {{\varepsilon _n},{D_n}} \right)$) for some sequence $\{\varepsilon _n\}$ with ${\varepsilon _n} \rightarrow {0^ + }$, there exists a subsequence $\left\{ {{x_{{n_k}}}} \right\}$ of $\left\{ {{x_n}} \right\}$ such that $d\left( {{x_{{n_k}}},{E_l}\left( D \right) } \right) \rightarrow 0$ (resp., $d\left( {{x_{{n_k}}},{W_l}\left( D \right) } \right) \rightarrow 0$) as $k \rightarrow + \infty$.

Theorem 4.1

Let $\{D_n\}$ be a sequence of subsets of X and D be a compact subset of X. Assume that

• (i) F is continuous on D with nonempty compact values and for any $x \in {D_n}$, $F\left( x \right)$ is compact;

• (ii) ${D_n}\mathop \rightarrow \limits ^K D$ and there exist $\delta > 0$ and ${n_0} \in {\mathbb {N}}$ such that ${D_n} \subseteq D + \delta {B_X}$ for any $n \ge {n_0}$.

Then (SOP) is weak extended well-posed. Moreover, if D is convex and F is strictly naturally quasi C-convex on D, then (SOP) is extended well-posed.

Proof

In view of Proposition 2.7 of Hernández and Rodríguez-Marín (2007), we have ${E_l}\left( D \right) \subseteq {W_l}\left( D \right)$. This together with Lemma 2.5 implies that ${W_l}\left( D \right) \ne \emptyset$. Suppose that (SOP) is not weak extended well-posed. Then we can find a sequence $\{\varepsilon _n\}$ with ${\varepsilon _n} \rightarrow {0^ + }$ and ${x_n} \in {W_l}\left( {{\varepsilon _n},{D_n}} \right)$ such that $d\left( {{x_n},{W_l}\left( D \right) } \right) \nrightarrow 0$ as $n \rightarrow + \infty$. Thus, there exists $\delta > 0$ such that, for any $n \in {\mathbb {N}}$, there exists ${m_n} \ge n$ satisfying ${x_{{m_n}}} \notin {W_l}\left( D \right) + \delta {B_X}$. Without loss of generality, we assume that

$$\begin{aligned} {x_n} \notin {W_l}\left( D \right) + \delta {B_X}, \quad \forall n \in {\mathbb {N}}. \end{aligned}$$

(39)

Then it is easy to see that ${x _n} \in {D_n}$. By condition (ii), we can see that $\left\{ {{x_n}} \right\} \subseteq X$ is bounded. Without loss of generality, we assume that ${x_n} \rightarrow {x_0} \in X$. It follows from ${D_n}\mathop \rightarrow \limits ^K D$ that ${x_0} \in D$. Suppose that ${x_0} \notin {W_l}\left( D \right)$. In view of Lemma 2.8, there exists ${y_0} \in D$ such that $F\left( {{y_0}} \right) { \ll ^l}F\left( {{x_0}} \right)$ and so

$$\begin{aligned} F\left( {{x_0}} \right) \subseteq F\left( {{y_0}} \right) + {\mathrm{int}} C. \end{aligned}$$

(40)

Due to $D \subseteq {\mathrm{Li}}{D_n}$, there exists ${y_n} \in {D_n}$ such that ${y_n} \rightarrow {y_0}$.

We claim that there exists ${n_0} \in {\mathbb {N}}$ such that

$$\begin{aligned} F\left( {{x_{{n}}}} \right) \subseteq F\left( {{y_{{n}}}} \right) + {\mathrm{int}} C + {\varepsilon _{{n}}}e,\;\;\forall n \ge {n_0}. \end{aligned}$$

(41)

In fact, if not, then there exist a subsequence $\left\{ {{x_{{n_{k}}}}} \right\}$ of $\left\{ {x_{n}} \right\}$ and a subsequence $\left\{ {{y_{{n_{k}}}}} \right\}$ of $\left\{ {y_{n}} \right\}$ such that

$$\begin{aligned} F\left( {{x_{{n_{k}}}}} \right) \not \subset F\left( {{y_{{n_{k}}}}} \right) + {\mathrm{int}} C + {\varepsilon _{{n_{k}}}}e. \end{aligned}$$

Without loss of generality, we assume that

$$\begin{aligned} F\left( {{x_{n}}} \right) \not \subset F\left( {{y_{n}}} \right) + {\mathrm{int}} C + {\varepsilon _{n}}e,\;\;\forall n \in {\mathbb {N}}. \end{aligned}$$

Then there exists ${v_{n}} \in F\left( {{x_{n}}} \right)$ such that

$$\begin{aligned} {v_{n}} \notin F\left( {{y_{n}}} \right) + {\mathrm{int}} C + {\varepsilon _{n}}e. \end{aligned}$$

(42)

Since F is u.s.c. at ${x _0}$, by Lemma 2.2, there exist ${v_0} \in F\left( {{x _0}} \right)$ and a subsequence $\left\{ {v_{n_k }} \right\}$ of $\left\{ {{v_n }} \right\}$ such that ${v_{n_k}} \rightarrow {v_0}$. It follows from (40) that there exists ${u_0} \in F\left( {{y_0}} \right)$ such that

$$\begin{aligned} {v_0} - {u_0} \in {\mathrm{int}} C. \end{aligned}$$

(43)

Noting that F is l.s.c. at ${y _0}$, in view of Lemma 2.1, there exists ${u_{n}} \in F\left( {{y_{n}}} \right)$ such that ${u_{n}} \rightarrow {u_0}$. By (43), we have ${v_{{n_k}}} - {u_{{n_k}}} \in {\mathrm{int}} C + {\varepsilon _{{n_k}}}e$ for k large enough, which contradicts (42). This shows that (41) holds and so $F\left( {{y_n}} \right) \ll _{{\varepsilon _n}}^lF\left( {{x_n}} \right)$. It follows from Lemma 2.8 that ${x_n} \notin {W_l}\left( {{\varepsilon _n},{D_n}} \right)$, which contradicts ${x_n} \in {W_l}\left( {{\varepsilon _n},{D_n}} \right)$. Therefore, ${x_0} \in {W_l}\left( D \right)$. Noting that ${x_n} \rightarrow {x_0} \in {W_l}\left( D \right) + \delta {B_X}$, we can see that ${x_n} \in {W_l}\left( D \right) + \delta {B_X}$ for n large enough, which contradicts (39). This shows that (SOP) is weak extended well-posed.

Next, we show that (SOP) is extended well-posed. Since F is strictly naturally quasi C-convex on D, it follows from Lemma 2.6 that ${E_l}\left( {D} \right) = {W_l}\left( {D} \right)$. For every sequence $\{x_n\} \subset {D_n}$ with ${x_n} \in {E_l}\left( {{\varepsilon _n},{D_n}} \right)$ for some sequence $\{\varepsilon _n\}$ with ${\varepsilon _n} \rightarrow {0^ + }$, it follows from Remark 2.4 that ${x_n} \in {E_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {W_l}\left( {{\varepsilon _n},{D_n}} \right)$. Noting that (SOP) is weak extended well-posed, then there exists a subsequence $\left\{ {{x_{{n_k}}}} \right\}$ of $\left\{ {{x_n}} \right\}$ such that $d\left( {{x_{{n_k}}},{W_l}\left( D \right) } \right) \rightarrow 0$ as $k \rightarrow + \infty$. This together with ${E_l}\left( {D} \right) = {W_l}\left( {D} \right)$ implies that $d\left( {{x_{{n_k}}},{E_l}\left( D \right) } \right) \rightarrow 0$ as $k \rightarrow + \infty$. This show that (SOP) is extended well-posed. $\square$

From Lemma 2.4 and Theorem 4.1, we can get the following corollary.

Corollary 4.1

Let $\{D_n\}$ be a sequence of subsets of X and D be a compact subset of X. Assume that

• (i) F is continuous on D with nonempty compact values and for any $x \in {D_n}$, $F\left( x \right)$ is compact;

• (ii) ${D_n}\mathop \rightharpoonup \limits ^H D$ and $D \subseteq {\mathrm{Li}}{D_n}$.

Then (SOP) is weak extended well-posed. Moreover, if D is convex and F is strictly naturally quasi C-convex on D, then (SOP) is extended well-posed.

Now, we give an example to illustrate Theorem 4.1.

Example 4.1

Let $X = {\mathbb {R}}$, $Y = {{\mathbb {R}}^2}$ and $C = {\mathbb {R}}_ + ^2 = \left\{ {\left( {{x_1},{x_2}} \right) \in {{\mathbb {R}}^2}:{x_1} \ge 0,{x_2} \ge 0} \right\}$. Define a set-valued mapping $F:X \rightarrow {2^Y}$ as follows:

$$\begin{aligned} F\left( x \right) = \left( { - 2\sin {1 \over 3}x,{{\left( {x - 4} \right) }^2} + 1} \right) + {B_Y},\>\>x \in X. \end{aligned}$$

Let $D = \left[ {0,3\pi } \right]$ and ${D_n} = \left[ { \sin {1 \over n},3\pi - 1 + \cos {1 \over n}} \right]$. Then it is easy to see that ${D_n}\mathop \rightarrow \limits ^K D$ and F is strictly naturally quasi C-convex on D. Let ${\varepsilon _0} = 1$, ${\varepsilon _n} = {{n + 1} \over n}$ and $e = \left( {1,1} \right)$. Then we can see that $4 \in {E_l}\left( {{\varepsilon _0},D} \right) \subseteq {W_l}\left( {{\varepsilon _0},D} \right)$. Moreover, we can check that all conditions of Theorem 4.1 are satisfied. Thus, Theorem 4.1 shows that (SOP) is weak extended well-posed and extended well-posed.

Theorem 4.2

Let $\{D_n\}$ be a sequence of convex subsets of X and D be a convex and compact subset of X. Assume that

• (i) F is continuous and strictly naturally quasi C-convex on D with nonempty compact values;

• (ii) ${D_n}\mathop \rightarrow \limits ^K D$ and F is naturally quasi C-convex on $D_n$ with nonempty compact values;

• (iii) for any $\varepsilon > 0$ and for any $n \in {\mathbb {N}}$, ${W_l}\left( {\varepsilon ,{D_n}} \right)$ is connected.

Then (SOP) is weak extended well-posed. Moreover, (SOP) is extended well-posed.

Proof

Suppose that (SOP) is not weak extended well-posed. Then we can find a sequence $\{\varepsilon _n\}$ with ${\varepsilon _n} \rightarrow {0^ + }$ and ${x_n} \in {W_l}\left( {{\varepsilon _n},{D_n}} \right)$ such that $d\left( {{x_n},{W_l}\left( D \right) } \right) \nrightarrow 0$ as $n \rightarrow + \infty$. Thus, there is a constant $\delta > 0$ such that, for any $n \in {\mathbb {N}}$, there exists ${m_n} \ge n$ satisfying ${x_{{m_n}}} \notin {W_l}\left( D \right) + \delta {B_X}$. Without loss of generality, we assume that

$$\begin{aligned} {x_n} \notin {W_l}\left( D \right) + \delta {B_X}, \quad \forall n \in {\mathbb {N}}. \end{aligned}$$

(44)

Now we claim that there exists

$$\begin{aligned} {z_n} \in \partial \left[ {{W_l}\left( D \right) + \delta {B_X}} \right] \cap {W_l}\left( {{\varepsilon _n},{D_n}} \right) . \end{aligned}$$

(45)

In fact, if $\partial \left[ {{W_l}\left( D \right) + \delta {B_X}} \right] \cap {W_l}\left( {{\varepsilon _n},{D_n}} \right) = \emptyset$, then

$$\begin{aligned} {W_l}\left( {{\varepsilon _n},{D_n}} \right) \subseteq {\mathrm{int}} \left( {{W_l}\left( D \right) + \delta {B_X}} \right) \cup {\left( {{W_l}\left( D \right) + \delta {B_X}} \right) ^c}. \end{aligned}$$

(46)

It follows from (44) that

$$\begin{aligned} {W_l}\left( {{\varepsilon _n},{D_n}} \right) \cap {\left( {{W_l}\left( D \right) + \delta {B_X}} \right) ^c} \ne \emptyset . \end{aligned}$$

(47)

Now we show that

$$\begin{aligned} {W_l}\left( {{\varepsilon _n},{D_n}} \right) \cap {\mathrm{int}} \left( {{W_l}\left( D \right) + \delta {B_X}} \right) \ne \emptyset . \end{aligned}$$

(48)

Let ${v _0} \in {W_l}\left( D \right)$. In view of Theorem 3.2, we have ${W_l}\left( D \right) = {W_l}\left( {0,D} \right) \subseteq {\mathrm{Li}}{W_l}\left( {{\varepsilon _n},{D_n}} \right)$ and so there exists ${v _n} \in {W_l}\left( {{\varepsilon _n},{D_n}} \right)$ such that ${v _n} \rightarrow {v _0} \in {\mathrm{int}} \left( {{W_l}\left( D \right) + \delta {B_X}} \right)$. This means that ${v _n} \in {\mathrm{int}} \left( {{W_l}\left( D \right) + \delta {B_X}} \right)$ for n large enough, which implies that (48) holds. Due to Proposition 2.3 of Han and Huang (2017), we can see that ${{W_l}\left( D \right) }$ is closed. Since ${{B_X}}$ is compact, we obtain that ${{W_l}\left( D \right) + \delta {B_X}}$ is closed. Thus, it follows from (46)–(48) that ${W_l}\left( {{\varepsilon _n},{D_n}} \right)$ is not connected, which contradicts condition (iii). Therefore, we know that (45) holds.

Noting that $\partial \left[ {{W_l}\left( D \right) + \delta {B_X}} \right]$ is compact, without loss of generality, we assume that

$$\begin{aligned} {z_n} \rightarrow {z_0} \in \partial \left[ {{W_l}\left( D \right) + \delta {B_X}} \right] . \end{aligned}$$

(49)

Due to ${\mathrm{Ls}}{D_n} \subseteq D$ and ${{z_n} \in {D_n}}$, we have ${z_0} \in D$.

Next we claim that ${z_0} \in {W_l}\left( D \right)$. In fact, if not, it follows from Lemma 2.8 that there exists ${y_0} \in D$ such that $F\left( {{y_0}} \right) { \ll ^l}F\left( {{z_0}} \right)$ and so $F\left( {{z_0}} \right) \subseteq F\left( {{y_0}} \right) + {\mathrm{int}} C$. Due to $D \subseteq {\mathrm{Li}}{D_n}$, there exists ${y_n} \in {D_n}$ such that ${y_n} \rightarrow {y_0}$. Similar to the proof of (41), we can prove that there exists ${n_0} \in {\mathbb {N}}$ such that

$$\begin{aligned} F\left( {{z_{{n}}}} \right) \subseteq F\left( {{y_{{n}}}} \right) + {\mathrm{int}} C + {\varepsilon _{{n}}}e,\;\;\forall n \ge {n_0}. \end{aligned}$$

(50)

(50) yields $F\left( {{y_n}} \right) \ll _{{\varepsilon _n}}^lF\left( {{z_n}} \right)$. This together with 2.8 implies that ${z_n} \notin {W_l}\left( {{\varepsilon _n},{D_n}} \right)$, which contradicts (45). Therefore, ${z_0} \in {W_l}\left( D \right)$, which contradicts (49).

Similar to the proof of Theorem 4.1, we can show that (SOP) is extended well-posed. This completes the proof. $\square$