Continuity of the solutions sets for parametric set optimization problems

Abstract

The current study focuses on exploring the stability of solution sets pertaining to set optimization problems, particularly with regard to the set order relation outlined by Karaman et al. 2018. Sufficient conditions are provided for the lower semicontinuity, upper semicontinuity, and compactness of m-minimal solution mappings in parametric set optimization, where the involved set-valued mapping is Lipschitz continuous.

1 Introduction

Over the past few decades, set optimization has garnered considerable attention from numerous researchers, primarily due to its widespread applications across various fields of applied mathematics, including vector optimization, interval optimization, game theory, mathematical economics, control system field, mathematical finance, and numerous others, see [2–11].

As we know, the stability of solutions holds great importance and serves as an intriguing subject in the field of set-valued optimization, see [12–20]. Xu and Li [13] introduced the notion of a lower-level mapping and established the semicontinuity of minimal solution mappings for a parametric set optimization problem. Han and Huang [7] delved into the convexity and semicontinuity of solution mappings in parametric set optimization problems, leveraging level mappings in their analysis. Karuna and Lalitha [14] primarily investigated the semicontinuity of approximate efficient solution mappings in the context of parametric set-valued optimization problems. Anh et al. [18] established the internal and external stability of the solutions sequence of perturbed problems, demonstrating their convergence towards a solution of the original problem. Li and Wei [16] conducted a thorough examination of the compactness and semicontinuity of E-minimal solution sets in parametric set optimization problems, utilizing improvement sets, and doing so under various pertinent conditions. Zhang and Huang [17] examined the semicontinuity and compactness of minimal solution mappings in parametric set optimization problems, taking into account the local Lipschitz continuity condition.

Lately, Karaman et al. [1] introduced a novel set order relations within the family of bounded sets, utilizing the Minkowski difference as a foundation. These new relations endow the family of bounded sets with a partial order, thus offering a fresh approach to the study of set optimization problems. Employing these novel set order relations, set optimization problems have been examined in [1, 20–22]. Nevertheless, to our knowledge, no research has been conducted on the continuity of solution maps for parametric set optimization problems using Karaman’s set order relations so far. Therefore, it is of significant interest and importance to delve into the semicontinuity and compactness of solution maps for the parametric set optimization problem, leveraging Karaman’s set order relations.

The rest of this paper is organized as follows. In Sect. 2, we introduce essential concepts, notations, and outcomes that are used throughout the paper. Section 3 delves into the lower semicontinuity, upper semicontinuity, and compactness of m-minimal solution mappings for parametric set optimization, specifically focusing on locally Lipschitz continuous set-valued mappings.

2 Preliminaries

Let $(X,\|\cdot\|_{X})$, $(\Lambda ,\|\cdot\|_{\Lambda})$ and $(Y,\|\cdot\|_{Y})$ be three real normed vector spaces. We denote the closed unit ball centered at origin in X (respectively, Y) by $\mathbb{B}_{X}$ (respectively, $\mathbb{B}_{Y}$), and an open ball centered at y with radius $r>0$ in Y by $\mathbb{B}_{Y}(y,r)$. Let $K\subseteq Y$ be a convex, pointed and closed cone with nonempty interior. Additionally, we use $P(Y)$ to denote the family of nonempty proper subsets of Y, and $B(Y)$ to represent the family of nonempty proper bounded subsets of Y. Then we denote by intA, clA the topological interior, and the topological closure of $A\subseteq Y$, respectively.

For $A, B \in P(Y)$, the Minkowski (Pontryagin) difference of A and B, considered in [11], is given as

$$ A\dot{-} B := \{y \in Y : y + B\subseteq A\}. $$

In [8], for any $A,B\in P(Y)$, the lower set less order relation $\leq _{K}^{l}$ and the strict lower set less order relation $\ll _{K}^{l}$ on $P(Y)$ are defined by

$$\begin{aligned}& A\leq _{K}^{l}B \quad \Leftrightarrow\quad B\subseteq A+K;\\& A\ll _{K}^{l}B \quad \Leftrightarrow\quad B\subseteq A+{ \mathrm{int}}K. \end{aligned}$$

Lately, Karaman et al. [1] introduced the following order relations on $P(Y)$.

$$ A\leq^{m} B \quad \Leftrightarrow\quad (A\dot{-} B)\cap (-K) \neq \emptyset , $$

and

$$ A\ll ^{m} B \quad \Leftrightarrow\quad (A\dot{-} B)\cap ({- \mathrm{int}}K) \neq \emptyset . $$

Obviously, $\leq^{m}$ is a partial order relation on $B(Y)$.

Lemma 2.1

[6] Let A and B be nonempty subsets of Y. If $0<\lambda <\delta$, B is convex, and $A+\delta B_{Y}\subseteq B+\lambda B_{Y}$, then $A\subseteq{\mathrm{int}}B$.

For any nonempty subset S of X, let $F:X\rightarrow 2^{Y}$ be a set-valued mapping. We consider the constrained set optimization problem:

$$ \begin{aligned} {\mathrm{{(SOP)}}}\quad & \min\quad F(x) \\ & {\mathrm{{s.t.}}} \quad x\in S. \end{aligned} $$

Throughout the paper, it is assumed that $F(x)\neq \emptyset$ for all $x\in S$, and that $F(S):=\bigcup_{x\in S}F(x)$.

We recall the concept of the optimal solutions for the problem (SOP) with regard to the set order relation $\leq _{K}^{l}$.

Definition 2.1

[8] An element $\bar{x}\in S$ is said to be

(i) a K-l-minimal solution of $(\mathit{SOP})$ if $x\in S$ such that $F(x)\leq _{K}^{l} F(\bar{x})$ imply $F(\bar{x})\leq _{K}^{l}F(x)$;

(ii) a weak K-l-minimal solution of $(\mathit{SOP})$ if $x\in S$ such that $F(x)\ll _{K}^{l} F(\bar{x})$ imply $F(\bar{x})\ll _{K}^{l}F(x)$.

Let $E_{l}(S)$ and $W_{l}(S)$ represent the K-l-minimal solution set and the weak K-l-minimal solution set of $(\mathit{SOP})$, respectively. It is evident that $E_{l}(S)\subseteq W_{l}(S)$.

We now recall the notion of the minimal and weak minimal solutions of $(\mathit{SOP})$ with respect to the relation $\leq^{m}$. To define solution concepts, we assume that $F(x)\in B(Y)$, for all $x\in S$.

Definition 2.2

[1] An element $\bar{x}\in S$ is said to be

(i) a m-minimal solution of $(\mathit{SOP})$ if there is no $x\in S$ such that $F(x)\leq^{m} F(\bar{x})$ and $F(x)\neq F(\bar{x})$, that is, either $F(x)\nleq ^{m} F(\bar{x})$ or $F(x)=F(\bar{x})$, for any $x\in S$.

(ii) a weak m-minimal solution of $(\mathit{SOP})$ if there is no $x\in S$ such that $F(x)\ll ^{m} F(\bar{x})$.

Let $E_{m}(S)$ and $W_{m}(S)$ represent the m-minimal solution set and the weak m-minimal solution set of $(\mathit{SOP})$, respectively.

Lemma 2.2

[21] If K is a closed convex pointed cone with nonempty interior, then

$$ E_{l}(S)\subseteq W_{l}(S)\subseteq W_{m}(S). $$

A subset S of a topological space is said to be arcwise connected if, for every pair of points $x,y\in S$, there exists a continuous function $\varphi : [0,1] \rightarrow S$ such that $\varphi (0)=x$ and $\varphi (1)=y$.

Definition 2.3

[22] Let S be an arcwise connected subset of X. A set-valued mapping $F:X\rightarrow 2^{Y}$ is said to be strictly m-quasiconnected on S if for any $A\in P(Y)$ and for any $x,y\in S$ with $x\neq y$, $F(x)\leq^{m}A$ and $F(y)\leq^{m}A$, there exists a continuous path $\varphi : [0,1] \rightarrow S$ with $\varphi (0)=x$ and $\varphi (1)=y$ such that

$$ F\bigl(\varphi (t)\bigr)\ll ^{m}A, \quad \forall t\in (0,1). $$

Lemma 2.3

[22] If S is an arcwise connected subset of X and $F:X\rightarrow 2^{Y}$ is strictly m-quasiconnected on S with nonempty values, then $E_{m}(S)=W_{m}(S)$.

Definition 2.4

[2] Let $F:X\rightarrow 2^{Y}$ be a set-valued mapping. Then, F is said to be

(i) upper semicontinuous (u.s.c.) at x̄ if, for any open neighborhood U of $F(\bar{x})$, there is a neighborhood $N(\bar{x})$ of x̄ such that for every $x\in N(\bar{x})$, $F(x)\subseteq U$.

(ii) lower semicontinuous (l.s.c.) at x̄ if, for any open subset U of Y with $F(\bar{x})\cap U\neq \emptyset$, there is a neighborhood $N(\bar{x})$ of x̄ such that $F(x)\cap U\neq \emptyset$ for all $x\in N(\bar{x})$.

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

Lemma 2.4

[2] A set-valued mapping $F:X\rightarrow 2^{Y}$ is l.s.c. at $\bar{x}\in X$ if and only if for any sequence $\{x_{n}\}\subseteq X$ with $x_{n}\rightarrow \bar{x}$ and for any $\bar{y}\in F(\bar{x})$, there exists $y_{n}\in F(x_{n})$ such that $y_{n}\rightarrow \bar{y}$.

Lemma 2.5

[2] Let $F:X\rightarrow 2^{Y}$ be a set-valued mapping. For any given $\bar{x}\in X$, if $F(\bar{x})$ is compact, then F is u.s.c. at $\bar{x}\in X$ if and only if for any sequence $\{x_{n}\}\subseteq X$ with $x_{n}\rightarrow \bar{x}$ and for any $y_{n}\in F(x_{n})$, there exist $\bar{y}\in F(\bar{x})$ and a subsequence $\{y_{n_{k}}\}$ of $\{y_{n}\}$ such that $y_{n_{k}}\rightarrow \bar{y}$.

Definition 2.5

[17] Let S be a nonempty subset of X. A set-valued mapping $F:X\rightarrow 2^{Y}$ is said to be locally Lipschitz continuous at $\bar{x}\in S$ if and only if there exist a constant $L>0$ and a neighborhood $N(\bar{x})$ of x̄ in X such that

$$ F(x_{1})\subseteq F(x_{2})+L \Vert x_{1}-x_{2} \Vert _{X}\mathbb{B}_{Y}, \quad \forall x_{1},x_{2}\in N(\bar{x})\cap S. $$

We say that F is locally Lipschitz continuous on S if and only if F is locally Lipschitz continuous at every $\bar{x}\in S$.

Definition 2.6

[22] Let S be a nonempty subset of X. A set-valued mapping $F:X\rightarrow 2^{Y}$ is said to be weak locally K-Lipschitz continuous at $\bar{x}\in S$ if and only if there exist a constant $L>0$ and a neighborhood $N(\bar{x})$ of x̄ in X such that

$$ F(x_{1})\cap \bigl\{ z+L \Vert x_{1}-x_{2} \Vert _{X}\mathbb{B}_{Y}+K\bigr\} \neq \emptyset , \quad \forall x_{1},x_{2}\in N(\bar{x})\cap S, \forall z\in F(x_{2}). $$

We say that F is weak locally K-Lipschitz continuous on S if and only if F is weak locally K-Lipschitz continuous at every $\bar{x}\in S$.

Remark 2.1

The following examples demonstrate that the concept of continuity differs from the concept of locally Lipschitz continuity for a set-valued mapping.

Example 2.1

[17] Let $X=\mathbb{R}$, $Y=\mathbb{R}^{2}$, $S=\{x\in \mathbb{R}:x>0\}$, and $K=\mathbb{R}_{+}^{2}$. Let $F:X \rightarrow 2^{Y}$ be a set-valued mapping defined as

$$ F(x)=(\ln x, \ln x)+\mathbb{R}_{+}^{2}, \quad \forall x\in S. $$

Then it is easy to see that F is continuous on S, while it is not locally Lipschitz continous on S.

Example 2.2

[22] Let $S=X=\mathbb{R}$, and $Y=\mathbb{R}$. Consider $F:X\rightarrow 2^{Y}$ defined as

$$ F(x)= \textstyle\begin{cases} [-1,1], & {\text{if}\ x=0,} \\ \{0\}, & {\text{if}\ x\neq 0.}\end{cases} $$

Then one can easily see that F is locally Lipschitz at 0, but it is not l.s.c. at 0.

Example 2.3

[22] Let $S=X=\mathbb{R}$, and $Y=\mathbb{R}$. Consider $F:X\rightarrow 2^{Y}$ defined as

$$ F(x)= \textstyle\begin{cases} [-1,1], & {\text{if}\ x\neq 0,} \\ \{0\}, & {\text{if}\ x=0.}\end{cases} $$

Then one can easily see that F is locally Lipschitz at 0, but it is not u.s.c. at 0.

Proposition 2.1

We assume that $F:S\rightarrow 2^{Y}$ is a locally Lipschitz continuous set-valued mapping at $x_{0}\in S$ and at $y_{0}\in S$, respectively. Furthermore, $S\subseteq X$ is a nonempty, and $F(x_{0})$ is closed. If, for any sequences $\{x_{n}\}$ and $\{y_{n}\}$ satisfying the conditions $x_{n}\rightarrow x_{0}$ and $y_{n}\rightarrow y_{0}$, it holds that $F(x_{n})\leq^{m}F(y_{n})$ for sufficiently large n, then it follows that $F(x_{0})\leq^{m}F(y_{0})$.

Proof

Since F is locally Lipschitz continuous at $x_{0}\in S$, there exists a constant $L>0$ such that, for n sufficiently large, we have

$$ F(x_{n})\subseteq F(x_{0})+L \Vert x_{n}-x_{0} \Vert _{X}\mathbb{B}_{Y}. $$

(1)

Moreover, as F is locally Lipschitz continuous at $y_{0}\in S$, there exists a constant $L_{1}>0$ such that, for n sufficiently large, we have

$$ F(y_{0})\subseteq F(y_{n})+L_{1} \Vert y_{n}-y_{0} \Vert _{X}\mathbb{B}_{Y}. $$

(2)

It follows from (1), (2), and $F(x_{n})\leq^{m}F(y_{n})$ for n sufficiently large that, there exists $k_{n}\in K$ such that

$$ -k_{n}+F(y_{0})\subseteq F(x_{0})+L \Vert x_{n}-x_{0} \Vert _{X}\mathbb{B}_{Y}+L_{1} \Vert y_{n}-y_{0} \Vert _{X} \mathbb{B}_{Y}. $$

We conclude that $F(x_{0})\leq^{m}F(y_{0})$. In other words, there exists a vector $k\in K$ such that $-k+F(y_{0})\subseteq F(x_{0})$. Suppose, on the contrary, that for any $k\in K$ there exists $u_{0}\in F(y_{0})$ such that $-k+u_{0}\notin F(x_{0})$. Since $F(x_{0})$ is closed, there exists $\epsilon > 0$ such that $((-k+u_{0})+\epsilon \mathbb{B}_{Y})\cap F(x_{0})=\emptyset$, and as a result

$$ -k+u_{0}\notin F(x_{0})+\epsilon \mathbb{B}_{Y}. $$

(3)

Because $\mathbb{B}_{Y}$ is a closed unit ball and $x_{n}\rightarrow x_{0}$, $y_{n}\rightarrow y_{0}$, as $n\rightarrow \infty$, $L\|x_{n}-x_{0}\|_{X}\mathbb{B}_{Y}+L_{1}\|y_{n}-y_{0}\|_{X} \mathbb{B}_{Y}$ is reduced to the origin 0 as $n\rightarrow \infty$. Then, we get $-k_{n}+F(y_{0})\subseteq F(x_{0})+\epsilon \mathbb{B}_{Y}$ for n sufficiently large, which is a contradition to (3). Therefore, $-k+F(y_{0})\subseteq F(x_{0})$, and so $F(x_{0})\leq^{m}F(y_{0})$. □

Proposition 2.2

Suppose that $F:S\rightarrow 2^{Y}$ is locally Lipschitz continuous at $x_{0}\in S$ and weak locally −K-Lipschitz continuous at $y_{0}\in S$, respectively. Furthermore, $S\subseteq X$ is a nonempty, and $F(x_{0})$ is compact. If for any sequences $\{x_{n}\}$ and $\{y_{n}\}$ satisfying $x_{n}\rightarrow x_{0}$, $y_{n}\rightarrow y_{0}$, such that $F(y_{0})\ll ^{m}F(x_{0})$ and $F(y_{n})$ are convex subsets of Y, then $F(y_{n})\ll ^{m}F(x_{n})$ for n sufficiently large.

Proof

Since $F(y_{0})\ll ^{m}F(x_{0})$, there exists $k_{0}\in{\mathrm{int}}K$ such that $-k_{0}+F(x_{0})\subseteq F(y_{0})$. Then, $-k_{0}+F(x_{0})\subseteq F(y_{0})\subseteq \bigcup_{z\in F(y_{0})} \mathbb{B}_{Y}(z,r)$, for any $r>0$. Since $F(x_{0})$ is compact, there is a finite set $\{z_{1}, z_{2}, \ldots , z_{n}\}\subseteq F(y_{0})$ such that

$$ -k_{0}+F(x_{0})\subseteq \bigcup ^{n}_{i=1} \mathbb{B}_{Y}(z_{i},r). $$

Since $-k_{0}+F(x_{0})$ is compact, we notice that there exists $\epsilon >0$ such that

$$ -k_{0}+F(x_{0})+3\epsilon \mathbb{B}_{Y} \subseteq \bigcup^{n}_{i=1} \mathbb{B}_{Y}(z_{i},r). $$

(4)

Since F is locally Lipschitz at $x_{0}\in S$, there exist a neighbourhood $N(x_{0})$ of $x_{0}$ and a constant $L>0$ such that

$$ F(x)\subseteq F(x_{0})+L \Vert x-x_{0} \Vert _{X}\mathbb{B}_{Y}, \quad \forall x \in N(x_{0}). $$

(5)

Since F is weak locally −K-Lipschitz continuous at $y_{0}\in S$, there exist a neighborhood $N(y_{0})$ of $y_{0}$ and a constant $L_{1}>0$ such that

$$ F(y)\cap \bigl\{ z+L_{1} \Vert y-y_{0} \Vert _{X}\mathbb{B}_{Y}-K\bigr\} \neq \emptyset , \quad \forall y \in N(y_{0}) \ \forall z\in F(y_{0}). $$

(6)

Since $x_{n}\rightarrow x_{0}$, $y_{n}\rightarrow y_{0}$, there are $n_{1}, n_{2}\in N$ such that $x_{n}\in N(x_{0})$ for any $n>n_{1}$ and $y_{n}\in N(y_{0})$ for any $n>n_{2}$. Let $n_{0}=\max \{n_{1}, n_{1}\}$, then, for any $n>n_{0}$, it follows from (5) that

$$ 2\epsilon \mathbb{B}_{Y}+F(x_{n})\subseteq F(x_{0})+L \Vert x_{n}-x_{0} \Vert _{X} \mathbb{B}_{Y}+2\epsilon \mathbbm{ }\mathbb{B}_{Y}. $$

(7)

Since $x_{n}\rightarrow x_{0}$ as $n\rightarrow \infty$ and $\mathbb{B}_{Y}$ is a closed unit ball, the radius of the closed ball $L\|x_{n}-x_{0}\|_{X}\mathbb{B}_{Y}$ will be close to zero as $n\rightarrow \infty$. Therefore, it follows from (4) and (7), that

$$\begin{aligned} -k_{0}+2\epsilon \mathbb{B}_{Y}+F(x_{n}) & \subseteq -k_{0}+F(x_{0})+3 \epsilon \mathbb{B}_{Y} \\ & \subseteq \bigcup^{n}_{i=1} \mathbb{B}_{Y}(z_{i},r)= \{z_{1}, z_{2}, \ldots , z_{n}\}+B_{Y}(0, r). \end{aligned}$$

(8)

For any $u\in -k_{0}+2\epsilon \mathbb{B}_{Y}+F(x_{n})$, we make a conclusion from (8) that there is $i_{0}\in \{1, 2, \ldots , n\}$ and $b_{1}\in B_{Y}(0, r)$ such that

$$ u=z_{i_{0}}+b_{1}. $$

(9)

From (6), we get $F(y_{n})\cap \{z+L_{1}\|y_{n}-y_{0}\|_{X}\mathbb{B}_{Y}-K\}\neq \emptyset$, $\forall y_{n}\in N(y_{0}) \ \forall z\in F(y_{0})$. Since $y_{n}\rightarrow y_{0}$ as $n\rightarrow \infty$, the radius of the closed ball $L_{1}\|y_{n}-y_{0}\|_{X}\mathbb{B}_{Y}$ will be close to zero as $n\rightarrow \infty$. Therefore, $F(y_{n})\cap \{z+\epsilon \mathbb{B}_{Y}-K\}\neq \emptyset$, $\forall y_{n}\in N(y_{0}) \ \forall z\in F(y_{0})$. Then there exist $t_{n}\in F(y_{n})$, $b_{2}\in \mathbb{B}_{Y}$ and $k_{1}\in K$ such that $t_{n}= z_{i_{0}}+\epsilon b_{2}-k_{1}$, and so $z_{i_{0}}=t_{n}-\epsilon b_{2}+k_{1}$. Combining it with (9), we obtain $u=z_{i_{0}}+b_{1}=t_{n}-\epsilon b_{2}+k_{1}+b_{1}$, namely,

$$ -k_{1}+u=t_{n}-\epsilon b_{2}+b_{1}\in F(y_{n})+\epsilon \mathbb{B}_{Y}+ \mathbb{B}_{Y}(0, r). $$

(10)

Because $u\in -k_{0}+2\epsilon \mathbb{B}_{Y}+F(x_{n})$, we get $-k_{1}+u\in -k_{0}-k_{1}+2\epsilon \mathbb{B}_{Y}+F(x_{n})$. From this and (10), we have

$$ -k_{0}-k_{1}+2\epsilon \mathbb{B}_{Y}+F(x_{n}) \subseteq F(y_{n})+ \epsilon \mathbb{B}_{Y}+ \mathbb{B}_{Y}(0, r)\subseteq F(y_{n})+ \epsilon \mathbb{B}_{Y}+r\mathbb{B}_{Y}. $$

(11)

Let $k_{0}+k_{1}=k_{2}$. Since $k_{0}\in{\mathrm{int}}K$, $k_{1}\in K$, we have $k_{2}\in{\mathrm{int}}K$.

$$ -k_{2}+2\epsilon \mathbb{B}_{Y}+F(x_{n})\subseteq F(y_{n})+\epsilon \mathbb{B}_{Y}+r\mathbb{B}_{Y}. $$

(12)

Let $r\rightarrow 0$, then $r+\epsilon \rightarrow \epsilon <2\epsilon$, and hence, by Lemma 2.1, we obtain

$$ -k_{2}+F(x_{n})\subseteq {\mathrm{int}}F(y_{n}) \subseteq F(y_{n}). $$

This indicates that

$$ F(y_{n})\ll ^{m}F(x_{n}), $$

for n sufficiently large. This finishes the proof. □

For any $\lambda \in \Lambda$, we consider the following constrained parametric set-valued optimization problem (for short, PSOP):

$$ \begin{aligned} {\mathrm{{(PSOP)}}}\quad &{\min}\quad F(x) \\ & {\mathrm{{s.t.}}} \quad x\in S(\lambda ), \end{aligned} $$

where $S: \Lambda \rightarrow 2^{X}$ and $F:X\rightarrow 2^{Y}$ are two set-valued mappings.

We denote the solution mappings $E_{m}:\Lambda \rightarrow 2^{X}$, $W_{m}:\Lambda \rightarrow 2^{X}$ for (PSOP) as follows: $E_{m}(\lambda )= E_{m}(S(\lambda ))$, $W_{m}(\lambda )= W_{m}(S(\lambda ))$.

Remark 2.2

Clearly, for any $\lambda \in \Lambda$, $E_{m}(\lambda )\subseteq W_{m}(\lambda )$.

Definition 2.7

Let $S: \Lambda \rightarrow 2^{X}$ and $F:X\rightarrow 2^{Y}$ are two set-valued mappings with nonempty values. The level mapping $L:\Lambda \times X\rightarrow 2^{X}$ is defined by

$$ L(\lambda ,x)=\bigl\{ y\in S(\lambda ):F(y)\leq^{m}F(x)\bigr\} , \quad (\lambda ,x) \in \Lambda \times X. $$

Remark 2.3

Clearly, for any $\lambda \in \Lambda$, $x\in S(\lambda )$, $E_{m}(L(\lambda ,x))\subseteq E_{m}(\lambda )= E_{m}(S(\lambda ))$.

Proposition 2.3

Let $S: \Lambda \rightarrow 2^{X}$ be a nonempty closed-valued. $F:S\rightarrow 2^{Y}$ is locally Lipschitz continuous and closed-valued on $S(\lambda )$, then $L(\lambda ,x)$ is closed for all $(\lambda ,x)\in \Lambda \times X$.

Proof

For any $(\lambda ,x)\in \Lambda \times X$, let $\{y_{n}\}\subseteq L(\lambda ,x)$ be a sequence with $y_{n}\rightarrow y_{0}$, then $y_{0}\in S(\lambda )$ and $F(y_{n})\leq^{m}F(x)$. Since F is locally Lipschitz continuous and closed-valued on $S(\lambda )$, it follows from Proposition 2.1 that $F(y_{0})\leq^{m}F(x)$ and as a result $y_{0}\in L(\lambda ,x)$. □

From Theorem 5.1 in [10], Proposition 2.3, and Lemma 2.2, we can get the following Lemma.

Lemma 2.6

Let S be a nonempty and compact subset of X and $F : S \rightarrow 2^{Y}$ is locally Lipschitz continuous and closed-valued on S, then (SOP) has a m-minimal solution.

Lemma 2.7

[22] If $S(\lambda )$ is an arcwise connected subset of X and $F:X\rightarrow 2^{Y}$ is strictly m-quasiconnected on $S(\lambda )$ with nonempty values and $x_{0}\in E_{m}(\lambda )$, then $L(\lambda ,x_{0})=x_{0}$.

3 Stability of solutions of (PSOP)

In this section, we undertake an analysis of the continuity properties of solution mappings for (PSOP). Initially, we establish the upper semi-continuity and compactness of the weak m-minimal solution mapping associated with (PSOP).

Theorem 3.1

Let $\lambda _{0}\in \Lambda$. Suppose that

(i) S is continuous at $\lambda _{0}$, and $S(\lambda _{0})$ is compact;

(ii) F is locally Lipschitz continuous and weak locally −K-Lipschitz continuous on $S(\lambda _{0})$ with nonempty and convex compact values.

Then, $W_{m}(\lambda )$ is u.s.c. at $\lambda _{0}$ and $W_{m}(\lambda _{0})$ is compact.

Proof

Since $E_{m}(S)\subseteq W_{m}(S)$. It follows from Lemma 2.6 that $W_{m}(\lambda _{0})\neq \emptyset$. Now, we claim that $W_{m}(\lambda )$ is u.s.c. at $\lambda _{0}$. By the contrary, we suppose that $W_{m}(\lambda )$ is not u.s.c. at $\lambda _{0}$, then there is at least a neighborhood U with $W_{m}(\lambda _{0})\subseteq U$, a sequence $\{\lambda _{n}\}$ with $\lambda _{n}\rightarrow \lambda _{0}$, and $x_{n}\in W_{m}(\lambda _{n})$ such that

$$ x_{n}\notin U, \quad \forall n\in N. $$

(13)

Since S is u.s.c. at $\lambda _{0}$ and $S(\lambda _{0})$ is compact, by Lemma 2.5, there exists $x_{0}\in S(\lambda _{0})$ such that $x_{n}\rightarrow x_{0}$. (Of course, if necessary, we can extract a subsequence).

We now assert that $x_{0}\in W_{m}(\lambda _{0})$. If $x_{0}\notin W_{m}(\lambda _{0})$, then there is $y_{0}\in S(\lambda _{0})$ such that $F(y_{0})\ll ^{m}F(x_{0})$. It follows from the l.s.c. of S at $\lambda _{0}$ and Lemma 2.4, there exists a sequence $\{y_{n}\}$ with $y_{n}\in S(\lambda _{n})$ such that $y_{n}\rightarrow y_{0}$. Since F is locally Lipschitz continuous and weak locally −K-Lipschitz continuous on $S(\lambda _{0})$ with nonempty and convex compact values. By Proposition 2.2 we get $F(y_{n})\ll ^{m}F(x_{n})$ for large enough n, which is a contradiction to $x_{n}\in W_{m}(\lambda _{n})$, therefore $x_{0}\in W_{m}(\lambda _{0})$. Based on the assumption that $x_{n}\rightarrow x_{0}$, it follows that $x_{n}\in U$ for n sufficiently large. However, this contradicts (13). Consequently, $W_{m}(\lambda )$ is u.s.c. at $\lambda _{0}$.

Next, we claim that $W_{m}(\lambda _{0})$ is compact. Given that $W_{m}(\lambda _{0})\subseteq S(\lambda _{0})$ and $S(\lambda _{0})$ is compact, it suffices to demonstrate that $W_{m}(\lambda _{0})$ is closed. Let $\{z_{n}\}\subseteq W_{m}(\lambda _{0})$ be a sequence with $z_{n}\rightarrow z_{0}$. Assuming $z_{0}\notin W_{m}(\lambda _{0})$, there exists an element $z^{*}\in S(\lambda _{0})$ satisfying $F(z^{*})\ll ^{m}F(z_{0})$. By employing the same reasoning as previously demonstrated, we can conclude that for large enough n, $F(z^{*})\ll ^{m}F(z_{n})$ holds. However, this contradicts $\{z_{n}\}\subseteq W_{m}(\lambda _{0})$, therefore $z_{0}\in W_{m}(\lambda _{0})$. We can deduce that $W_{m}(\lambda _{0})$ is also compact. □

Now, we present an example to demonstrate Theorem 3.1.

Example 3.1

Let $X=Y=\mathbb{R}^{2}$, $\Lambda =[0,1]$, and $K=\mathbb{R}_{+}^{2}$. Assume that $S(\lambda )=[0,\lambda ]\times [0,\lambda ]$ for all $\lambda \in \Lambda$. Define a set-valued mapping $F:X\rightarrow 2^{Y}$ as $F(x,y)=(x^{2}y^{2}, x^{2}y^{2})+\mathbb{B}_{Y}$ for all $(x,y)\in \mathbb{R}^{2}$. Let $\lambda _{0}=1$. It is straightforward to verify that all the conditions of Theorem 3.1 are satisfied. Through a simple computation, we can determine that $W_{m}(\lambda )=[0,\lambda ]\times \{0\}\cup \{0\}\times [0,\lambda ]$ for all $\lambda \in \Lambda$. Evidently, we observe that $W_{m}(\lambda )$ is u.s.c. at 1.

Theorem 3.2

Let $\lambda _{0}\in \Lambda$. Suppose that

(i) S is continuous at $\lambda _{0}$, and $S(\lambda _{0})$ is compact;

(ii) F is locally Lipschitz continuous and weak locally −K-Lipschitz continuous on $S(\lambda _{0})$ with nonempty and convex compact values;

(iii) F is strictly m-quasiconnected on $S(\lambda _{0})$.

Then, $E_{m}(\lambda )$ is u.s.c. at $\lambda _{0}$ and $E_{m}(\lambda _{0})$ is compact.

Proof

It follows from Lemma 2.6 that $E_{m}(\lambda _{0})\neq \emptyset$. Now, we assert that $E_{m}(\lambda )$ is u.s.c. at $\lambda _{0}$. By the contrary, we suppose that $E_{m}(\lambda )$ is not u.s.c. at $\lambda _{0}$, then there exists at least a neighborhood U such that $E_{m}(\lambda _{0})\subseteq U$ and a sequence $\{\lambda _{n}\}$ with $\lambda _{n}\rightarrow \lambda _{0}$ such that

$$ E_{m}(\lambda _{n})\nsubseteq U. $$

(14)

Since F is strictly m-quasiconnected on $S(\lambda _{0})$, based on Lemma 2.3, we deduce that $E_{m}(\lambda _{0})=W_{m}(\lambda _{0})\subseteq U$. Leveraging Theorem 3.1, we can infer that $W_{m}(\lambda )$ is u.s.c. at $\lambda _{0}$. Consequently, there exists a neighborhood V of $\lambda _{0}$ such that $W_{m}(\lambda )\subseteq U$, for all $\lambda \in V$. Because $\lambda _{n}\rightarrow \lambda _{0}$, it follows that $\lambda _{n}\in V$ for n large enough. Therefore, $E_{m}(\lambda _{n})\subseteq W_{m}(\lambda _{n})\subseteq U$ for n large enough. However, this conclusion contradicts (14). Consequently, $E_{m}(\lambda )$ is u.s.c. at $\lambda _{0}$. Since $E_{m}(\lambda _{0})=W_{m}(\lambda _{0})$, as a result, by Theorem 3.1$E_{m}(\lambda _{0})$ is compact. □

In the theorem that follows, we explore the lower semicontinuity of the m-minimal solution for the (PSOP) problem.

Lemma 3.1

Let $\lambda _{0}\in \Lambda$. Assume that

(i) S is continuous at $\lambda _{0}$, S is compact-valued on Λ and $S(\lambda _{0})$ is an arcwise connected subset of X;

(ii) F is locally Lipschitz continuous on $S(\lambda _{0})$ with nonempty and closed values;

(iii) F is strictly m-quasiconnected on $S(\lambda _{0})$.

Then, $L(\lambda , x)$ is u.s.c. at $\lambda _{0}\times S(\lambda _{0})$.

Proof

Assume to the contrary that there exists $x_{0}\in S(\lambda _{0})$ such that $L(\lambda ,x)$ is not u.s.c. at $(\lambda _{0}, x_{0})$. Then, there exist a neighborhood U with $L(\lambda _{0}, x_{0})\subseteq U$ and a sequence $\{(\lambda _{n},x_{n})\}$ with $(\lambda _{n}, x_{n})\rightarrow (\lambda _{0}, x_{0})$ such that $L(\lambda _{n}, x_{n})\nsubseteq U$. Therefore, there exists a sequence $\{y_{n}\}$ with $y_{n}\in L(\lambda _{n},x_{n})$ and

$$ y_{n}\notin U, \quad \forall n\in \mathbb{N}. $$

(15)

Since S is u.s.c. and compact-valued at $\lambda _{0}$, by Lemma 2.5, there exist $y_{0}\in S(\lambda _{0})$ and a subsequence $\{y_{n_{k}}\}$ of $\{y_{n}\}$ such that $y_{n_{k}}\rightarrow y_{0}$. Without loss of generality, let $y_{n}\rightarrow y_{0}$. It follows from $y_{n}\in L(\lambda _{n},x_{n})$ that

$$ F(y_{n})\leq^{m}F(x_{n}), \quad \forall ( \lambda _{n},x_{n}) \in \Lambda \times X. $$

By Proposition 2.1 we have $F(y_{0})\leq^{m}F(x_{0})$. and so $y_{0}\in L(\lambda _{0}, x_{0})$. Therefore, from $y_{n}\rightarrow y_{0}$, we have $y_{n}\in U$ for large enough n, which contradicts (15). Therefore, $L(\lambda )$ is u.s.c. at $\lambda _{0}\times S(\lambda _{0})$. □

Theorem 3.3

Let $\lambda _{0}\in \Lambda$. Assume that

(i) S is continuous at $\lambda _{0}$, S is compact-valued on Λ and $S(\lambda _{0})$ is an arcwise connected subset of X;

(ii) F is locally Lipschitz continuous on $S(\lambda _{0})$ with nonempty and closed values;

(iii) F is strictly m-quasiconnected on $S(\lambda _{0})$.

Then, $E_{m}(\lambda )$ is l.s.c. at $\lambda _{0}$.

Proof

We conclude from Lemma 2.6 that $E_{m}(\lambda _{0})\neq \emptyset$. Now, we assert that $E_{m}(\lambda )$ is l.s.c. at $\lambda _{0}$. By the contrary, we suppose that $E_{m}(\lambda )$ is not l.s.c. at $\lambda _{0}$, then there exists a point $y\in E_{m}(\lambda _{0})$, along with a neighborhood U of 0 in X. Additionally, there is a sequence $\{\lambda _{n}\}$ with $\lambda _{n}\rightarrow \lambda _{0}$ such that

$$ (y+U)\cap E_{m}(\lambda _{n})=\emptyset , \quad \forall n\in N. $$

(16)

It follows from $y\in E_{m}(\lambda _{0})$ that $y\in S(\lambda _{0})$. Because S is l.s.c. at $\lambda _{0}$, according to Lemma 2.4, there exists a sequence $\{y_{n}\}$ with $y_{n}\in S(\lambda _{n})$ such that $y_{n}\rightarrow y$. By Lemma 2.7, we have that $L(\lambda _{0}, y)=\{y\}$. Then $L(\lambda _{0}, y)\subseteq y+U$. By Lemma 3.1 we know that $L(\cdot,\cdot)$ u.s.c. at $(\lambda _{0}, y)$. Then for large enough n,

$$ L(\lambda _{n}, y_{n})\subseteq y+U. $$

(17)

It is clear that $y_{n}\in L(\lambda _{n}, y_{n})$ and so $L(\lambda _{n}, y_{n})\neq \emptyset$. By Proposition 2.3 we know that $L(\lambda _{n}, y_{n})$ is closed. Noting that $S(\lambda _{n})$ is compact and $L(\lambda _{n}, y_{n})\subseteq S(\lambda _{n})$, we get that $L(\lambda _{n}, y_{n})$ is compact. It follows from Lemma 2.6 that $E_{m}(L(\lambda _{n}, y_{n}))\neq \emptyset$. Let $z_{n}\in E_{m}(L(\lambda _{n}, y_{n}))$, it follows from (17) that

$$ z_{n}\in E_{m}\bigl(L(\lambda _{n}, y_{n})\bigr)\subseteq L(\lambda _{n}, y_{n}) \subseteq y+U, $$

(18)

for large enough n. By Remark 2.2, we get

$$ z_{n}\in E_{m}\bigl(L(\lambda _{n}, y_{n})\bigr)\subseteq E_{m}(\lambda _{n}), $$

(19)

for large enough n. It follows from (18) and (19) that $z_{n}\in (y+U)\cap E_{m}(\lambda _{n})$ for large enough n, which contradicts (16). Therefore, $E_{m}(\lambda )$ is l.s.c. at $\lambda _{0}$. □

Lastly, we demonstrate the lower semicontinuity of the weak m-minimal solution for the (PSOP) problem, as outlined below:

Theorem 3.4

Let $\lambda _{0}\in \Lambda$. Assume that

(i) S is continuous at $\lambda _{0}$, S is compact-valued on Λ and $S(\lambda _{0})$ is an arcwise connected subset of X;

(ii) F is locally Lipschitz continuous on $S(\lambda _{0})$ with nonempty and closed values;

(iii) F is strictly m-quasiconnected on $S(\lambda _{0})$.

Then, $W_{m}(\lambda )$ is l.s.c. at $\lambda _{0}$.

Proof

Since $E_{m}(S)\subseteq W_{m}(S)$. It follows from Lemma 2.6 that $W_{m}(\lambda _{0})\neq \emptyset$. Now, we assert that $W_{m}(\lambda )$ is l.s.c. at $\lambda _{0}$. By Lemma 2.3, it follows that $W_{m}(\lambda _{0})=E_{m}(\lambda _{0})$. For any $y_{0}\in W_{m}(\lambda _{0})$, we get $y_{0}\in E_{m}(\lambda _{0})$. By Theorem 3.3, we know $E_{m}(\lambda )$ is l.s.c. at $\lambda _{0}$. For any neighborhood U of $y_{0}$, there exists a neighborhood V of $\lambda _{0}$ such that

$$ U\cap E_{m}(\lambda )\neq \emptyset , \quad \forall \lambda \in V. $$

Using Remark 2.1, we have $E_{m}(\lambda )\subseteq W_{m}(\lambda )$.

$$ U\cap W_{m}(\lambda )\neq \emptyset , \quad \forall \lambda \in V. $$

This shows that $W_{m}(\lambda )$ is l.s.c. at $\lambda _{0}$. □

Now, we present an example to exemplify Theorem 3.4.

Example 3.2

Let $X=\mathbb{R}$, $\Lambda =[0,1]$, $Y=\mathbb{R}^{2}$, and $K=\mathbb{R}_{+}^{2}$. Assume that $S(\lambda )=[1,2]$ for all $\lambda \in \Lambda$. A set-valued mapping $F:X\rightarrow 2^{Y}$ can be defined as follows:

$$ F(x)=[1,x]\times [1,x], \quad \forall x\in S. $$

Let $\lambda _{0}=0$, then it is straightforward to verify that all the conditions stated in Theorem 3.4 are met. Through a straightforward computation, we deduce that $W_{m}(\lambda )=[1,2]$ for all $\lambda \in \Lambda$. Evidently, we observe that $W_{m}(\lambda )$ is l.s.c. at 0.

Remark 3.1

According to Remark 2.1, it is evident that the semicontinuity exhibited by objective set-valued mappings in [7, 12–16, 21] is distinct from the local Lipschitz continuity employed for objective set-valued mappings in Theorems 3.1 to 3.4.

Remark 3.2

The objective set-valued mappings with nonempty and closed values used in Theorems 3.3-3.4 are weaker than the objective set-valued mappings with nonempty and compact values used in [7, 12–16, 21]. That $S(\lambda _{0})$ is an arcwise connected subset, which is also used in Theorems 3.3–3.4 is weaker than $S(\lambda _{0})$, a convex subset appearing in [7, 12–16, 21].

Remark 3.3

By Lemma 2.2, we can know that $E_{l}(S)\subseteq W_{l}(S)\subseteq W_{m}(S)$. Hence, we have studied Theorems 3.1 and Theorems 3.4 for a larger set than any in [7, 12–16, 21].