Efficiency in vector quasi-equilibrium problems and applications

Abstract

In this paper, we give sufficient conditions for the existence of efficient solutions of a generalized vector quasi-equilibrium problem in topological vector spaces. The motivations for introducing this problem come from practical problems in traffic networks and the optimal control theory for discrete-time dynamical systems. The main results of the paper are proven with the help of a strongly monotonic function which can be constructed from the data of the problem under consideration. Some notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, are also used in our study. As applications, we obtain existence results in vector quasi-optimization problems, Stampacchia set-valued vector quasi-variational inequality problems and Pareto vector quasi-saddle point problems. All these results are different from the corresponding ones in the literature.

1 Introduction

Let $K$ (resp. ${\widehat{C}}$) be a subset (resp. a convex cone) of a topological vector space $X$ (resp. $Y$) and ${\widehat{f}}: K\times K\longrightarrow Y$ be a vector function. The simplest vector equilibrium problem is to find a point $x_0\in K$ such that

$$\begin{aligned} \forall \eta \in K, f(x_0,\eta ):= {\widehat{f}}(x_0,\eta )-{\widehat{f}}(x_0,x_0) \notin -\mathrm{int}\ \widehat{C}, \end{aligned}$$

(1)

or, equivalently,

$$\begin{aligned}{}[\widehat{f}(x_0,K)-\widehat{f}(x_0,x_0)] \cap -\mathrm{int}\ \widehat{C}= \emptyset , \end{aligned}$$

where $\emptyset$ stands for the empty set and int $\widehat{C}$ denotes the interior of $\widehat{C}.$ In other words, it is required to find a point $x_0\in K$ such that $\widehat{f}(x_0,x_0)$ is a weak efficient point of the set $\widehat{f}(x_0,K),$ where weak efficiency is taken from the theory of vector optimization problem (see e.g., [1, 2]). If we are interested in a point $x_0\in K$ such that $\widehat{f}(x_0,x_0)$ is an efficient point [1, 2] of the set $\widehat{f}(x_0,K),$ then we will deal with the problem of finding a point $x_0\in K$ such that

$$\begin{aligned} \forall \eta \in K, f(x_0,\eta ) \notin - \widehat{C} {\setminus } \{0\}. \end{aligned}$$

(2)

If $\widehat{f}(x,\eta )$ does not depend on $x$, then (2) is exactly the problem of finding an efficient solution $x_0$ of the problem of minimizing $\widehat{f}$ subject to $\eta \in K,$ while (1) is the problem of finding a weak efficient solution of this problem. Existence results in problem (1) and its extended versions are intensively developed (see [3] and the references therein). Recently, problem (2) has been extended as follows.

Problem $(P_1)$: Find a point $(z_0,x_0)\in E\times K$ such that $(z_0,x_0)\in B(z_0,x_0)\times A(z_0,x_0)$ and

$$\begin{aligned} \forall \eta \in A(z_0,x_0), F(z_0,x_0,\eta )\not \subset -[C(z_0,x_0,\eta ){\setminus } \{0\}], \end{aligned}$$

where $K$ (resp. $E$) is a nonempty set of a topological vector space $X$ (resp. $Z$); $A : E\times K \rightrightarrows K,\,B : E\times K\rightrightarrows E,\,F : E\times K\times K\rightrightarrows Y$ are set-valued maps with nonempty values and $C : E\times K\times K\rightrightarrows Y$ is a set-valued map taking cone values in a topological vector space $Y.$

Problem $(P_1)$ has been studied by some authors. Namely, Fang and Huang [4], and Gong [5] deal with a special case of Problem $(P_1)$ where $F$ does not depend on $z,\,A(z,x) = K$ for all $(z,x) \in E\times K,$ and $C$ is a constant convex cone (i.e., $C$ does not depend on $z,x$ and $\eta$). Fu et al. [6] consider Problem $(P_1)$ under the special assumption that both $A$ and $F$ do not depend on $z$, and $C$ does not depend on $z$ and $\eta .$ Fu [7] studies Problem $(P_1)$ where $A$ and $B$ do not depend on $z$, and $C$ is a constant convex cone. Wang and Fu [8] examine Problem $(P_1)$ where $A$ and $B$ do not depend on $z$, and $C$ does not depend on $z$ and $\eta .$

In this paper, we are interested in the following problem.

Problem $(P)$ : Find a point $(z_0,x_0)\in E\times K$ such that $(z_0,x_0)\in B(z_0,x_0)\times A(z_0,x_0)$ and

$$\begin{aligned} \forall \eta \in A(z_0,x_0), F(z_0,x_0,\eta )\cap -[C(z_0,x_0,\eta ){\setminus } \{0\}]= \emptyset . \end{aligned}$$

Problem $(P)$ is new and stronger than Problem $(P_1)$ in the sense that each solution of Problem $(P)$ is also a solution of Problem $(P_1),$ but the converse statement is no longer true. In this paper, Problem $(P)$ will be studied under the assumption that $X,Y$ and $Z$ are topological vector spaces. The main results of the present paper are established with the help of a strongly monotonic function which can be constructed under suitable assumptions on the data of Problem $(P).$ A new notion of cone-semicontinuity for set-valued maps (see [9]), more general than the usual notion of semicontinuity, will be used in our study. Observe that if $F$ is single-valued, Problems $(P)$ and $(P_1)$ are the same. However, even in this special case our results are different from the corresponding ones of [4–7]. This is because we use assumptions different from those of the just mentioned references. As applications, we give existence results in vector quasi-optimization problems (see [10]), Stampacchia set-valued vector quasi-variational inequality problems (see [6]) and Pareto vector quasi-saddle point problems. All these results are different from the corresponding ones in the literature.

Problem $(P)$ is motivated by practical problems in traffic networks [11, 12]. We can see in [12] that the well known Wardrop’s user principle in traffic networks in [11] was extended to the vector case where vector equilibrium flows can be found with the help of a problem of the kind of Problem $(P)$ (see Proposition 2.1 of [12]).

Another motivation for Problem $(P)$ comes from the optimal control theory for discrete-time dynamical systems (see, e.g., the book [13], the papers [14, 15] and the references therein). Several practical problems in economics and engineering which were described by discrete-time dynamical systems can be found in the book [13] and the references therein. We now show that Problem $(P)$ can be served as an approach to find efficient solutions in optimal control problems for discrete-time dynamical systems. To this end, we assume that $H$ (resp. $V$) is a subset of a topological vector space $\widehat{X}$ (resp. $W$) and $N$ is a fixed positive integer. Let $H_0$ and $H_1$ be nonempty subsets of $H.$ Let $\widehat{A}_z :H\times V \rightrightarrows {H\times V }$ be a family of set-valued maps depending on a parameter $z$ of a subset $E$ of a topological vector space $Z.$ For each $z\in E,$ consider the following discrete-time dynamical system $(D_z)$

$$\begin{aligned}&\widehat{x}(i) \in \widehat{A}_z (\widehat{x}(i-1), v(i-1)),\quad i=1, 2,\ldots ,N, \end{aligned}$$

(3)

$$\begin{aligned}&v(i-1)\in V,\quad i=1,2,\ldots , N, \end{aligned}$$

(4)

$$\begin{aligned}&\widehat{x}(0)\in H_0, \end{aligned}$$

(5)

$$\begin{aligned}&\widehat{x}(N)\in H_1, \end{aligned}$$

(6)

where $\widehat{x}(i)\in \widehat{X}$ (resp. $v(i)\in W$) is called the state (resp. the control) of the system at the $i$th moment. A sequence $(v(0), v(1),\ldots ,v(N-1)))$ satisfying (4) is called an admissible control (or a feasible control). Let two points $h_0$ , $h_1$ of $H$ and an admissible control $(v(0), v(1),\ldots ,v(N-1))$ be given. A sequence $(\widehat{x}(0), \widehat{x}(1),\ldots , \widehat{x}(N))$ is called a trajectory of the system $(D_z)$ starting from $h_0$ and ending at $h_1$ by admissible control $(v(0), v(1),\ldots ,v(N-1)),$ if conditions (3) and the equalities $\widehat{x}(0)=h_0$ and $\widehat{x}(N)=h_1$ are simultaneously satisfied. A sequence $(\widehat{x}(0), \widehat{x}(1),\ldots , \widehat{x}(N))$ is called an admissible trajectory of the system $(D_z)$ corresponding to the admissible control $(v(0), v(1),\ldots ,v(N-1)),$ if there exist $h_0\in H_0$ and $h_1\in H_1$ such that $(\widehat{x}(0), \widehat{x}(1),\ldots , \widehat{x}(N))$ is a trajectory starting from $h_0$ and ending at $h_1$ by admissible control $(v(0), v(1),\ldots ,v(N-1))$. For simplicity of presentation, we denote by $v$ the sequence with components $v(i), i=0,1,\ldots , N-1\!:$

$$\begin{aligned} v=(v(0), v(1),\ldots ,v(N-1))\in W^N\!, \end{aligned}$$

and similarly for $\widehat{x}\!:$

$$\begin{aligned} \widehat{x}=\left( \widehat{x}(0), \widehat{x}(1),\ldots , \widehat{x}(N)\right) \in \widehat{X}^{N+1}. \end{aligned}$$

Therefore, we can consider the set $T_z (v) \subset \widehat{X}^{N+1}$ of all admissible trajectories of the system $(D_z)$ corresponding to the admissible control $v.$ Roughly speaking, $T_z (v)$ is the set of all trajectories of the system $(D_z)$ which can reach the set $H_1$ at the $N$th moment from the initial set $H_0$ by means of the admissible control $v$. In the optimal control theory for discrete-time dynamical systems (see [13]), it is required to find a dynamical system $(D_{z_0})$, an admissible control $v_0\in V^N$ and an admissible trajectory $\widehat{x}_{0} \in T_{z_0 } (v_0)\subset \widehat{X}^{N+1}$ of dynamical system $(D_{z_0})$ such that $x_0:=(\widehat{x}_0, v_0)\in X:=\widehat{X}^{N+1} \times W^N$ satisfies some optimality criteria. In practice, for each dynamical system $(D_z)$ such criteria are often given by means of a convex cone $\widehat{C}_z$ of a topological vector space $Y$, a vector-valued function

$$\begin{aligned} (\widehat{x}, v)\in X:=\widehat{X}^{N+1} \times W^N \mapsto f_z (\widehat{x}, v)\in Y \end{aligned}$$

and a suitable notion of solutions in vector optimization. If this notion is efficiency, then we will deal with the following problem: Find a dynamical system $(D_{z_0}), z_0\in E,$ an admissible control $v_0$ and an admissible trajectory $\widehat{x}_0\in T_{z_0 } (v_0)$ of the dynamical system $(D_{z_0})$ such that, for all admissible controls $v$ and admissible trajectories $\widehat{x}\in T_{z_0 } (v)$ of the same system $(D_{z_0})$, we have

$$\begin{aligned} f_{z_0} (\widehat{x}_0, v_0)- f_{z_0} (\widehat{x}, v)\notin -\widehat{C}_{z_0}{\setminus } \{0\}. \end{aligned}$$

We will refer to this as Problem (OCP). The control $v_0$ and the corresponding trajectory $\widehat{x}_0$ mentioned in Problem (OCP) are called an optimal control and an optimal trajectory of the dynamical system $(D_{z_0}).$ Observe the difference between Problem (OCP) and the corresponding optimal control problems in [13–15]: here we deal with the existence of optimal controls and optimal trajectories, while in [13–15] it is required to find conditions which optimal controls and optimal trajectories must satisfy. So, the known approaches and results in [13–15] cannot be applied to Problem (OCP).

We prove that Problem (OCP) can be formulated as a special case of Problem $(P).$ For each $z\in E,$ let ${\mathbb {A}}(z)$ be the set of all points

$$\begin{aligned} x=(\widehat{x}, v)= (\widehat{x}(0),\ldots ,\widehat{x}(N), v(0),\ldots , v(N-1))\in X:=\widehat{X}^{N+1} \times W^N \end{aligned}$$

such that conditions (3)–(6) are satisfied. Clearly, ${\mathbb {A}}(z)$ is a subset of $K:=H^{N+1}\times V^N\subset X:=\widehat{X}^{N+1} \times W^N.$ Also, the fact that $v=(v(0), v(1),\ldots , v(N-1))$ is an admissible control and $\widehat{x}= (x(0),x(1),\ldots ,x(N))\in T_z (v)$ is equivalent to the fact that $x:=(\widehat{x}, v)= (\widehat{x}(0),\ldots ,\widehat{x}(N), v(0), v(1),\ldots , v(N-1))\in {\mathbb {A}}(z).$ Therefore, Problem (OCP) becomes the problem of finding a point $z_0\in E$ and a point $x_0:=(\widehat{x}_0,v_0)\in {\mathbb {A}}(z_0)$ such that, for each point $x:=(\widehat{x},v) \in {\mathbb {A}}(z_0),$ we have $F(z_0,x_0, x)\notin - \widehat{C}_{z_0}{\setminus } \{0\},$ where $F(z_0,x_0, x):=f_{z_0} ( \widehat{x},v)- f_{z_0} (\widehat{x}_0,v_0)$. Obviously, this is a special case of Problem $(P)$ with $B(z,x)=E, A(z,x)={\mathbb {A}}(z), C(z,x,\eta )=\widehat{C}_z$ for all $z\in E, x\in K,\eta \in K.$

Consider now a special case of Problem (OCP) where all the spaces $\widehat{X}, W$ and $Y$ are Euclidean finite-dimensional spaces and $\widehat{C}_{z} \equiv {\mathbb {R}}^m _+$ (the nonnegative orthant of $Y= {\mathbb {R}}^m$). Assume furthermore that

$$\begin{aligned} f_z (x)= (f^1_z (x), f^2 _z(x),\ldots , f^m _z(x)),x\in X:=\widehat{X}^{N+1} \times W^N, \end{aligned}$$

where $f^i _z : X\longrightarrow {\mathbb {R}},i=1,2,\ldots ,m,$ are convex differentiable functions. The last assumptions for $f^i _z$ allow us to write

$$\begin{aligned} f^i _z (x)-f^i _z (x_0)\ge f^{i'}_z (x_0)(x-x_0),\quad i=1,2,\ldots , m, \end{aligned}$$

for all $x\in X$ and $x_0\in X,$ where $f^{i ' }_z (x_0)$ stands for the derivative of $f^i _z$ at $x_0$. Denoting by $f' _{z_0}(x_0)(x-x_0)$ the vector with components $f^{i'}_{z_0} (x_0)(x-x_0), i=1,2,\ldots ,m,$ we can write

$$\begin{aligned} f_{z_0}(x)-f _{z_0}(x_0)\in f'_{z_0}(x_0)(x-x_0)+ {\mathbb {R}}^m _+, x \in X. \end{aligned}$$

From this it is clear that if $f_{z_0}(x)-f_{z_0} (x_0)\in -{\mathbb {R}}^m _+{\setminus } \{0\}$ for some $x \in K,$ then $f' _{z_0}(x_0)(x-x_0)\in -{\mathbb {R}}^m _+{\setminus } \{0\}.$ It follows that Problem (OCP) is solved if the following problem has a solution: find a point $z_0\in E$ and a point $x_0:=(\widehat{x}_0,v_0)\in {\mathbb {A}}(z_0)$ such that, for each point $x:=(\widehat{x},v) \in {\mathbb {A}}(z_0),$ we have $F(z_0,x_0,x):=f'_{z_0}(x_0)(x-x_0) \notin - {\mathbb {R}}^m _+{\setminus } \{0\}.$ The just formulated problem is also a special case of Problem $(P)$ whose solution set is a subset of the solution set of Problem (OCP).

The above discussion shows that Problem $(P)$ can be used as an effective tool for solving practical problems in vector traffic networks and the optimal control theory for discrete-time dynamical systems. The motivations for introducing Problem $(P)$ are thus explained.

We conclude our introduction by noticing that the notions of efficiency, weak efficiency and proper efficiency in optimization theory [1, 2] have been also exploited in duality theory for vector equilibrium problems [16] and for variational inequalities [17].

2 Preliminaries

For two subsets $X$ and $Y,$ we use the symbol $F:X\rightrightarrows Y$ to denote that $F$ is a set-valued map from $X$ to $Y.$ If $X$ (resp. $Y$) is a subset of a topological space $X'$ (resp. $Y'$), then $X$ (resp. $Y$) will be equipped with the induced topology of $X'$ (resp. $Y'$). The set dom $F:=\{x \in X: F(x) \ne \emptyset \}$ (resp. gr $F:=\{(x,y): x\in X, y\in F(x)\}$) is called the domain (resp. the graph) of $F.$ If $Y$ is a topological vector space, then the set of all the linear continuous functionals defined on $Y$ is denoted by $Y^*.$ We denote by $\langle y^*,y\rangle$ the value of $y^*\in Y^*$ at $y\in Y$. In this paper, neighborhoods of a point $x$ of a topological space $X$ are denoted by $U(x),\,U_1(x),\, U_2(x),\ldots$ The symbols “cl” and “int” are used to denote the closure and the interior, respectively. For a subset $A$ of a vector space $X,$ we denote by $\mathrm{cone}\ A$ the set of all points $x\in X$ such that $x=\lambda a$ for some nonnegative number $\lambda$ and some point $a\in A.$ This set $A$ is called a cone if $A=\mathrm{cone}\ A.$

As in [18], a set-valued map $F : X \rightrightarrows Y$ between topological spaces $X$ and $Y$ is called upper semicontinuous (shortly, usc) at $x_0\in \mathrm{dom}\ F$ if, for any open set ${\mathcal {N}}$ of $Y$ with ${\mathcal {N}} \supset F(x_0),$ there exists a neighborhood $U(x_0)$ of $x_0$ such that ${\mathcal {N}} \supset F(x)$ for all $x\in U(x_0).$ It is called lower semicontinuous (shortly, lsc) at $x_0\in \mathrm{dom }\ F$ if, for any open set ${\mathcal {N}}$ of $Y$ with ${\mathcal {N}}\cap F(x_0) \ne \emptyset ,$ there exists a neighborhood $U(x_0)$ of $x_0$ such that ${\mathcal {N}}\cap F(x)\ne \emptyset$ for all $x\in U(x_0).$ The set-valued map $F$ is called usc (resp. lsc) on a subset of $\mathrm{dom}\ F$ if it is usc (resp. lsc) at any point $x_0$ of this set. In the case where this set coincides with the whole space $X,\,F$ is called usc (resp. lsc). The set-valued map $F$ is called continuous if it is simultaneously usc and lsc.

We say that $F$ has open lower sections if the inverse set-valued map $F^{-1} : Y \rightrightarrows X,$ defined by $F^{-1}(y)=\{x\in X : y\in F(x)\},$ is open-valued, i.e., for all $y\in Y, \ F^{-1}(y)$ is open in $X.$ Clearly, a set-valued map having open lower sections is lsc (see [19]).

In this paper, the notions of semicontinuities of extended functions are understood in the usual sense (see, e.g., [20]). The following cone-semicontinuities of set-valued maps are taken from [9].

Definition 2.1

Let $F:X\rightrightarrows Y$ be a set-valued map between a topological space $X$ and a topological vector space $Y.$ Let $C:X\rightrightarrows Y$ be a set-valued map such that, for each $x\in X,\,C(x)$ is a cone. The set-valued map $F$ is called $C$-upper semicontinuous (shortly, $C$-usc) at $x_0\in \mathrm{dom}\ F$ if for any open set ${\mathcal {N}} \supset F(x_0),$ we can find a neighborhood $U(x_0)$ of $x_0$ such that ${\mathcal {N}}+C(x) \supset F(x)$ for all $x\in U(x_0).$ The set-valued map $F$ is called $C$-lower semicontinuous (shortly, $C$-lsc) at $x_0\in \mathrm{dom}\ F$ if for any open set ${\mathcal {N}}$ with $F(x_0) \cap {\mathcal {N}} \ne \emptyset ,$ we can find a neighborhood $U(x_0)$ of $x_0$ such that $F(x) \cap [{\mathcal {N}}-C(x)] \ne \emptyset$ for all $x\in U(x_0).$ The set-valued map $F$ is called $C$-usc (resp. $C$-lsc) if $X=\mathrm{dom}\ F$ and if it is $C$-usc (resp. $C$-lsc) at each point of $X.$ The set-valued map $F$ is called $C$-continuous if it is simultaneously $C$-usc and $C$-lsc.

Remark 2.1

Clearly, the usual semicontinuities are obtained with $C(x)=\{0\}.$ A set-valued map $F$ which is $C$-usc (resp. $C$-lsc) at $x_0\in \mathrm{dom}\ F$ may not be usc (resp. lsc) at this point. We can find an example illustrating this remark in [9].

Proposition 2.1

Assume that $X$ and ${\mathcal {Q}}$ are topological spaces, $Q: X\rightrightarrows {{\mathcal {Q}}}$ is a set-valued map with nonempty values and $f: \mathrm{gr}\ Q \longrightarrow {\mathbb {R}}$ is a function. Assume furthermore that $Q$ is usc and compact-valued at $x_0 \in X$ and $f$ is lsc on the set $\{(x_0,q): q\in Q(x_0)\}.$ Then the function $\varphi : X \longrightarrow {\mathbb {R}} \cup \{-\infty \}$ defined by

$$\begin{aligned} \varphi (x):=\inf _{q\in Q(x)} f(x,q),\ x\in X, \end{aligned}$$

is lsc at $x_0.$

We delete the proof of this proposition, noting that it is similar to the proof of Theorem 1 of [18, p. 67].

We now recall some concepts of generalized convexity of set-valued maps which are often used in the theory of equilibrium problems (see, e.g., [21] and the references therein).

Let $A$ be a nonempty convex subset of a vector space $X,\,C$ be a convex cone of a vector space $Y$, and $F : A \rightrightarrows Y$ be a set-valued map with nonempty values.

Set-valued map $F$ is called $C$-convex if for all $x_i\in A, \ i=1,2,$ and $\lambda \in \ ]0,1[$,

$$\begin{aligned} \lambda F(x_1)+(1-\lambda )F(x_2)\subset F(\lambda x_1+(1-\lambda )x_2)+C. \end{aligned}$$

So a function $f :A\rightarrow {\mathbb {R}}$ is convex iff it is $C$-convex with $C={\mathbb {R}}_{+}.$

The set-valued map $F$ is called proper $C$-quasiconvex if for all $x_i\in A, \ i=1,2,$ and $x\in \text {co }\{x_i,\ i=1,2\},$

$$\begin{aligned} \text {either }&\ F(x_1)\subset F(x)+C,\\ \text {or }&\ F(x_2)\subset F(x)+C. \end{aligned}$$

It is remarked from [22] that a set-valued map may be $C$-convex and not proper $C$-quasiconvex, and conversely.

Remark 2.2

By an induction argument we can see that $F$ is proper $C$-quasiconvex if and only if for each finite number of points $x_i\in A,\,i=1,2,\ldots ,n,$ and $x\in \mathrm{co}\{x_i,i=1,2,\ldots ,n\},$ there exists $i\in \{1,2,\ldots ,n\}$ such that $F(x_i)\subset F(x)+C$. Similarly, $F$ is $C$-convex if and only if for each $x= \sum _{i=1}^n \lambda _i x_i$ with $x_i\in A,\lambda _i \ge 0,\,i=1,2,\ldots ,n,$ and $\sum _{i=1}^n \lambda _i =1,$ we have

$$\begin{aligned} \sum _{i=1}^n \lambda _i F(x_i) \subset F(x) + C. \end{aligned}$$

Before formulating an existence result for later use, let us recall some notions. Assume that $A$ is a convex set of a vector space and $\varphi : A\times A \longrightarrow {\mathbb {R}}\cup \{-\infty , +\infty \}$ is an extended function. We say [23] that $\varphi$ is 0-diagonally quasiconvex (resp. 0-diagonally quasiconcave) if

$$\begin{aligned}&\forall x_i\in A, i=1,2,\ldots ,n, \forall x\in \mathrm{co}\{x_i,i=1,2,\ldots ,n\}, \max _{i=1,2,\ldots ,n}\varphi (x,x_i)\ge 0\\&\quad \left( \mathrm{resp.}\ \min _{i=1,2,\ldots ,n}\varphi (x,x_i)\le 0\right) . \end{aligned}$$

A set-valued map $F$ between a topological space $X$ and a topological vector space $Y$ is called acyclic if it is upper semicontinuous on $X$ and if, for all $x\in X ,\ F(x)$ is nonempty, compact and acyclic. Here a topological space is called acyclic [24] if all of its reduced $\check{\text {C}}$ech homology groups over rationals vanish. Observe that contractible spaces are acyclic; and hence, convex sets and star-shaped sets are acyclic.

The following result is established on the basis of the arguments of the proof of Lemma 4.2 of [25].

Proposition 2.2

Let $X$ and $Z$ be locally convex Hausdorff topological vector spaces; and $K \subset X$ and $E \subset Z$ be nonempty convex compact subsets. Let $\varphi : E\times K\times K \longrightarrow {\mathbb {R}} \cup \{-\infty , +\infty \}$ be an extended function. Let $A : E\times K \rightrightarrows K$ be a set-valued map with nonempty convex values and open lower sections, and let $B : E\times K \rightrightarrows E$ be an acyclic map such that the set $M=\{(z,x)\in E\times K : (z,x)\in B(z,x)\times A(z,x)\}$ is closed in $E\times K.$

Assume that $L_1$ (resp. $L_2)$ has open lower sections, where

$$\begin{aligned} L_1(z,x)&= \{\eta \in K : \varphi (z,x,\eta )< 0\},\ (z,x)\in E\times K\\ (resp.\ L_2(z,x)&= \{\eta \in K : \varphi (z,x,\eta )> 0\},\ (z,x)\in E\times K), \end{aligned}$$

and for all $\ z \in E,$ the function $\varphi (z,\cdot ,\cdot )$ is 0-diagonally quasiconvex (resp. 0-diagonally quasiconcave). Then there exists a point $(z_0,x_0)\in E\times K$ such that $(z_0,x_0)\in B(z_0,x_0)\times A(z_0,x_0)$ and

$$\begin{aligned} \varphi (z_0,x_0,\eta )&\ge 0,\ \forall \eta \in A(z_0,x_0)\\ (resp.\ \varphi (z_0,x_0,\eta )&\le 0,\ \forall \eta \in A(z_0,x_0)). \end{aligned}$$

Proof

It suffices to prove the existence of a point $( z,x) \in M$ such that $A(z,x)\cap L_1(z,x)=\emptyset$ (resp. $A(z,x)\cap L_2(z,x)=\emptyset$). Indeed, assume to the contrary that $A(z,x)\cap L_1(z,x)\ne \emptyset$ (resp. $A(z,x)\cap L_2(z,x)\ne \emptyset$) for all $(z,x) \in M.$ Then we can use the arguments of the proof of Lemma 4.2 in [25] to show that $x_0 \in A(z_0,x_0)\cap \mathrm{co} L_1(z_0,x_0) \subset \mathrm{co} L_1(z_0,x_0)$ (resp. $x_0 \in A(z_0,x_0)\cap \mathrm{co} L_2(z_0,x_0) \subset \mathrm{co} L_2(z_0,x_0)$) for some point $(z_0,x_0) \in M.$ This contradicts the 0-diagonal quasiconvexity (resp. the 0-diagonal quasiconcavity) of $\varphi$ since the condition $x_0 \in \mathrm{co} L_1(z_0,x_0)$ (resp. $x_0 \in \mathrm{co} L_2(z_0,x_0)$) implies that there exists a finite number of points $x_i\in K, i=1,2,\ldots ,n,$ such that $x_0 \in \mathrm{co} \{x_i, i=1,2,\ldots ,n\}$ and $\varphi (z_0,x_0,x_i) < 0$ (resp. $\varphi (z_0,x_0,x_i) >0$) for all $i=1,2,\ldots ,n.$ $\square$

Theorem 2.1

[19] Let $K$ be a compact Hausdorff topological space and $Y$ be a topological vector space. If $F : K \rightrightarrows Y$ is a set-valued map with nonempty convex values and open lower sections, then $f$ has a continuous selection (i.e., there exists a continuous map $g : K\longrightarrow Y$ such that $g(x)\in F(x)$ for all $x\in K)$.

Let $C$ be a cone of a vector space $Y,$ and $y$ and $\eta$ be elements of $Y.$ We write $y\le _C \eta$ (resp. $y<_C \eta$ ) if $\eta -y\in C$ (resp. $\eta -y\in C{\setminus } \{0\}$).

Definition 2.2

(see, e.g., [7]) Let $C$ be a cone of a vector space $Y.$ A function $p : Y \longrightarrow {\mathbb {R}}$ is called $C$-monotonic (resp. strongly $C$-monotonic) if $p(0)=0$ and if $p(y) \le p(\eta )$ (resp. $p(y)< p(\eta )$) for all $y$ and $\eta$ of $Y$ such that $y \le _C \eta$ (resp. $y<_C \eta$).

Clearly the strong $C$-monotonicity implies the $C$-monotonicity, but the converse is no longer true.

Remark 2.3

If $p$ is strongly $C$-monotonic, then $p(y-c) < p(y) < p(y+c)$ for all $y\in Y$ and $c \in C {\setminus } \{0\}.$ Indeed, the second of these inequalities is a consequence of Definition 2.2 with $\eta =y+c$, and the first one is derived from the second one with $y-c$ instead of $y$. Combining the above inequalities with the condition $p(0)=0$ shows that $p(-c) < 0 < p(c)$ for all $c \in C {\setminus } \{0\}.$

3 Main results

First we establish some lemmas which are needed for the proof of our main results of this paper. In Lemmas 3.1–3.4 below, we assume that $K$ and ${\mathcal {Q}}$ are topological spaces, $Y$ is a topological vector space, $C: K \rightrightarrows Y$ is a set-valued map with cone values (i.e., for each $x\in K,\,C(x)$ is a cone of $Y$), and $F: K \rightrightarrows Y$ and $Q: K \rightrightarrows {{\mathcal {Q}}}$ are set-valued maps with nonempty values.

We say that a function $p: K\times {\mathcal {Q}}\times Y \longrightarrow {\mathbb {R}}$ is $QC$-monotonic if for each $x\in K$ and each $q\in Q(x),\,p(x,q,\cdot )$ is $C(x)$-monotonic.

Let $x_0$ be a fixed point of $K.$

Lemma 3.1

Assume that $F$ is $C$-lsc at $x_0,\,Q$ is lsc at $x_0,$ and $p: K\times {\mathcal {Q}}\times Y \longrightarrow {\mathbb {R}}$ is upper semicontinuous and $QC$-monotonic. Then the extended function $\varphi : K\longrightarrow {\mathbb {R}}\cup \{-\infty \},$ defined by

$$\begin{aligned} \varphi (x):=\inf _{q\in Q(x)} \inf _{y\in F(x)} p(x,q,y),\ x\in K, \end{aligned}$$

is usc at $x_0$.

Proof

Let $\epsilon$ be an arbitrary positive number. From the definition of $\varphi (x_0)$ it follows that there exist $q_0\in Q(x_0)$ and $y_0\in F(x_0)$ such that

$$\begin{aligned}&p(x_0,q_0,y_0) < \varphi (x_0)+\epsilon /2 \ \ \ \mathrm{if}\ \varphi (x_0)> -\infty ; \end{aligned}$$

(7)

$$\begin{aligned}&p(x_0,q_0,y_0) < -3\epsilon /2 \qquad \quad \ \mathrm{if}\ \varphi (x_0)= -\infty . \end{aligned}$$

(8)

In view of the upper semicontinuity of $p,$ there exist open neighborhoods $U(x_0),\,U(q_0)$ and $U(y_0)$ such that

$$\begin{aligned} p(x,q,y)< p(x_0,q_0,y_0)+\epsilon /2 \end{aligned}$$

(9)

for all $(x,q,y)\in U(x_0)\times U(q_0)\times U(y_0).$

By the lower semicontinuity of $Q$ and the $C$-lower semicontinuity of $F$ at $x_0,$ we may assume without loss of generality that $U(x_0)$ is such that, for each $x \in U(x_0),$ there exist $q\in Q(x),\,y\in F(x)$ and $c\in C(x)$ with $q\in U(q_0)$ and $c+y\in U(y_0).$ This implies by (9) that

$$\begin{aligned} p(x,q,c+y)< p(x_0,q_0,y_0)+\epsilon /2. \end{aligned}$$

On the other hand, by the monotonicity property of $p(x,q,\cdot ),\,p(x,q,c+y) \ge p(x,q,y).$ Therefore,

$$\begin{aligned} p(x,q,y)< p(x_0,q_0,y_0)+\epsilon /2 \end{aligned}$$

which implies that $\varphi (x)< p(x_0,q_0,y_0)+\epsilon /2.$

Together with (7) (resp. (8)) this yields

$$\begin{aligned}&\varphi (x) < \varphi (x_0)+\epsilon \ \ \mathrm{if}\ \varphi (x_0)> -\infty \\&\mathrm{(resp. }\ \varphi (x) < -\epsilon \quad \mathrm{if}\ \varphi (x_0)= -\infty ). \end{aligned}$$

Since $x$ is an arbitrary point of $U(x_0),$ we can conclude that $\varphi$ is usc at $x_0,$ as desired. $\square$

Remark 3.1

It is well known that a lsc function on a compact set attains its minimum on this set. It follows that $\varphi (x)> -\infty$ for all $x\in K,$ if both $F$ and $Q$ are compact-valued and if for all $x \in K,\,p(x,\cdot ,\cdot )$ is lsc.

Corollary 3.1

Assume that $F$ is $C$-lsc at $x_0,$ and $p ': K\times Y \longrightarrow {\mathbb {R}}$ is a upper semicontinuous function such that for each $x\in K,\,p '(x,\cdot )$ is $C(x)$-monotonic. Then the extended function $f ' : K\longrightarrow {\mathbb {R}}\cup \{-\infty \},$ defined by

$$\begin{aligned} f '(x):=\inf _{y\in F(x)} p '(x,y),\ x\in K, \end{aligned}$$

is usc at $x_0.$

Proof

This is a special case of Lemma 3.1 where the function $p$ does not depend on $q$ (and hence, $Q$ plays no role in the definition of $\varphi$). $\square$

Lemma 3.2

Assume that $F$ has compact values. Assume furthermore that $F$ is $C$-lsc at $x_0,\,Q$ is usc and compact-valued at $x_0,$ and $p: K\times {\mathcal {Q}}\times Y \longrightarrow {\mathbb {R}}$ is continuous and $QC$-monotonic. Then the function $\varphi ' : K\longrightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \varphi '(x):=\inf _{q\in Q(x)}\left[ -\inf _{y\in F(x)}p(x,q,y)\right] ,\ x\in K, \end{aligned}$$

is lsc at $x_0.$

Proof

Clearly we can write

$$\begin{aligned} \varphi '(x):=\inf _{q\in Q(x)} f(x,q),\ x\in K, \end{aligned}$$

where

$$\begin{aligned} f(x,q):=-\inf _{y\in F(x)}p(x,q,y), (x,q)\in K' := \mathrm{gr}\ Q. \end{aligned}$$

Observe that $f(x,q)\in {\mathbb {R}}$ for all $(x,q)\in K':= \mathrm{gr}\ Q$, since by assumption $F(x)$ is compact and $p(x,q,\cdot )$ is continuous. By Proposition 2.1, $\varphi '$ is lsc at $x_0$ if $f$ is lsc at any point $(x_0,q)$ with $q\in Q(x_0)$. This property of $f$ can be derived from Corollary 3.1. Indeed, for each $x'=(x,q)\in K'$ and $y\in Y$ let us set $p'(x',y)= p(x,q,y), C'(x')=C(x)$ and $F'(x')=F(x).$ It remains to apply Corollary 3.1, with $K',C'$ and $F'$ instead of $K,C$ and $F,$ to obtain the desired property of $f.$ $\square$

Lemma 3.3

Assume that $F$ is $C$-usc and compact-valued at $x_0,\,Q$ is lsc at $x_0$ and $p: K\times {\mathcal {Q}}\times Y \longrightarrow {\mathbb {R}}$ is lower semicontinuous and $QC$-monotonic. Then the extended function $\varphi ': K\longrightarrow {\mathbb {R}}\cup \{-\infty ,+\infty \}$ defined in Lemma 3.2 is usc at $x_0.$

Proof

Let $\epsilon$ be an arbitrary positive number. By the definition of $\varphi '(x_0)$ we can find $q_0\in Q(x_0)$ such that

$$\begin{aligned}&-\inf _{y\in F(x_0)}p(x_0,q_0,y) < \varphi '(x_0)+\epsilon /2 \ \ \mathrm{if}\ \varphi '(x_0)> -\infty ; \end{aligned}$$

(10)

$$\begin{aligned}&-\inf _{y\in F(x_0)}p(x_0,q_0,y) < -3\epsilon /2 \qquad \quad \ \ \mathrm{if}\ \varphi '(x_0)= -\infty . \end{aligned}$$

(11)

Using a standard compactness argument and observing by assumption that $F(x_0)$ is a compact set, we can derive from the lower semicontinuity of $p$ that there exist open neighborhoods $U(x_0),\,U(q_0)$ and $U(y_i),\,i=1,2,\ldots ,n,$ such that for each $i=1,2,\ldots ,n$ the inequality

$$\begin{aligned} p(x_0,q_0,y_i) < p(x,q,y) +\epsilon /2 \end{aligned}$$

(12)

holds for all $x\in U(x_0),\,q\in U(q_0)$ and $y\in U(y_i),$ where $q_0$ is the point mentioned above, and $y_i, i=1,2,\ldots ,n,$ are some points of $F(x_0)$ such that $F(x_0)$ is contained in the union of open sets $U(y_i), i=1,2,\ldots ,n$. Since $\cup _{i=1}^n U(y_i)$ contains $F(x_0)$ and since $F$ is $C$-usc at $x_0,$ we may assume without loss of generality that $U(x_0)$ is such that, for all $x\in U(x_0),$

$$\begin{aligned} F(x)\subset \bigcup _{i=1}^n U(y_i)+ C(x). \end{aligned}$$

(13)

By the lower semicontinuity property of $Q,$ we may also assume that $U(x_0)$ is such that, for all $x\in U(x_0),$ we have

$$\begin{aligned} Q(x)\cap U(q_0)\ne \emptyset . \end{aligned}$$

Now let $x\in U(x_0)$ and let $y$ be an arbitrary point of $F(x).$ By (9), there exist $i\in \{1,2,\ldots ,n\}$ and $c\in C(x)$ such that $y-c\in U(y_i)$. Applying (12), we get

$$\begin{aligned} p(x_0,q_0,y_i) < p(x,q,y-c) +\epsilon /2, \end{aligned}$$

where $q$ is some point of $Q(x)\cap U(q_0).$ On the other hand, observing that $y= y-c+c,$ we derive from the monotonicity property of $p(x,q,\cdot )$ that $p(x,q,y-c)\le p(x,q,y).$ Therefore,

$$\begin{aligned} p(x_0,q_0,y_i) < p(x,q,y) +\epsilon /2. \end{aligned}$$

From this it follows that

$$\begin{aligned} \inf _{y\in F(x_0)} p(x_0,q_0,y) \le \inf _{y\in F(x)} p(x,q,y) +\epsilon /2. \end{aligned}$$

Together with (10) (resp. (11)), this yields

$$\begin{aligned} \begin{array}{l@{\quad }l} -\inf \limits _{y\in F(x)} p(x,q,y) <-\inf \limits _{y\in F(x_0)} p(x_0,q_0,y) < \varphi '(x_0)+\epsilon &{}\quad \mathrm{if}\ \varphi '(x_0)> -\infty \\ \quad \mathrm{(resp.}\ \! -\inf _{y\in F(x)} p(x,q,y) < -\epsilon &{} \quad \mathrm{if}\ \varphi '(x_0)= -\infty ). \end{array} \end{aligned}$$

Thus, given $\epsilon > 0,$ we can find $U(x_0)$ such that for all $x\in U(x_0),$

$$\begin{aligned} \varphi '(x)&< \varphi '(x_0)+\epsilon \ \ \mathrm{if}\ \varphi '(x_0)> -\infty ; \\ \varphi '(x)&< -\epsilon \qquad \quad \quad \,\,\, \mathrm{if}\ \varphi '(x_0)= -\infty . \end{aligned}$$

The upper semicontinuity of $\varphi '$ at $x_0$ is thus established. $\square$

Corollary 3.2

Assume that $F$ is $C$-usc and compact-valued at $x_0$, and $p ': K\times Y \longrightarrow {\mathbb {R}}$ is a lower semicontinuous function such that for each $x\in K,\,p '(x,\cdot )$ is $C(x)$-monotonic. Then the extended function $f ' : K\longrightarrow {\mathbb {R}} \cup \{-\infty \}$ defined in Corollary 3.1 is lsc at $x_0.$

Proof

This is a special case of Lemma 3.3 where the function $p$ does not depend on $q$ (and hence, $Q$ plays no role in the definition of $\varphi '$). $\square$

Lemma 3.4

Assume that $F$ has compact values. Assume furthermore that $F$ is $C$-usc at $x_0,\,Q$ is usc and compact-valued at $x_0,$ and $p: K\times {\mathcal {Q}}\times Y \longrightarrow {\mathbb {R}}$ is lower semicontinuous and $QC$-monotonic. Then the function $\varphi$ defined in Lemma 3.1 is lsc at $x_0.$

Proof

We write

$$\begin{aligned} \varphi (x):=\inf _{q\in Q(x)} f(x,q),\ x\in K, \end{aligned}$$

where

$$\begin{aligned} f(x,q):=\inf _{y\in F(x)}p(x,q,y), (x,q)\in K'=\mathrm{gr}\ Q. \end{aligned}$$

Observe that $f(x,q)\in {\mathbb {R}}$ for all $(x,q)\in K'=\mathrm{gr}\ Q,$ since by assumption $F(x)$ is compact and $p(x,q,\cdot )$ is lower semicontinuous. By Proposition 2.1, $\varphi$ is lsc at $x_0$ if $f$ is lsc at any point $(x_0,q)$ with $q\in Q(x_0).$ Arguing as in the proof of Lemma 3.2, with Corollary 3.2 instead of Corollary 3.1, we can derive this property of $f.$ $\square$

In Theorems 3.1–3.3 and Corollaries 3.3–3.5 below, we will assume that $X,Y$ and $Z$ are locally convex Hausdorff topological vector spaces; $K$ (resp. $E$) is a nonempty compact convex set of $X$ (resp. $Z$); $A : E\times K \rightrightarrows K$ is a set-valued map with nonempty convex values and open lower sections; and $B : E\times K \rightrightarrows E$ is an acyclic map such that the set

$$\begin{aligned} M=\{(z,x)\in E\times K : (z,x)\in B(z,x)\times A(z,x)\} \end{aligned}$$

is closed in $E\times K.$ We also assume that $F : E\times K\times K\rightrightarrows Y$ has nonempty values, and $C : E\times K\times K\rightrightarrows Y$ has convex cone values (i.e., each value of $C$ is a convex cone). For all $\eta \in K,$ we denote by $F_\eta$ the set-valued map defined by $F_\eta (\cdot ,\cdot ):=F(\cdot ,\cdot ,\eta )$, and similarly for $C_\eta .$

It is worth noticing that the above closedness assumption of $M$ is needed for the validity of our existence results of Problems $(P).$ An example will be given at the end of this section to illustrate this remark.

Assume that for each $\eta \in K,\,{\mathcal {Q}}_{\eta }$ is a topological space, $Q_\eta : E\times K\rightrightarrows {{\mathcal {Q}}_{\eta }}$ is a set-valued map with nonempty values, and $p_\eta : E\times K\times {\mathcal {Q}}_{\eta }\times Y\longrightarrow {\mathbb {R}}$ is a function. We say that $p_\eta$ is strongly $Q_{\eta }C_{\eta }$-monotonic if for all $(z,x)\in E\times K,\,q\in Q_\eta (z,x),\,p_\eta (z,x,q,\cdot )$ is strongly $C_\eta (z,x)$-monotonic. Evidently, a strongly $Q_{\eta }C_{\eta }$-monotonic function is also a $Q_{\eta }C_{\eta }$-monotonic function, but the converse is no longer true.

The existence of solutions of Problem $(P)$ is given in Theorems 3.1 and 3.2 below, with the help of a strongly monotonic function $p_\eta : E\times K\times {\mathcal {Q}}_\eta \times Y \longrightarrow {\mathbb {R}}$ which can be constructed under suitable assumptions of the data of Problems $(P)$ (see the proof of Corollaries 3.3, 3.4 and Remark 3.4). Observe that in Theorem 3.1 the values of $F$ are not required to be compact and the function $p_\eta$ is not necessarily lower semicontinuous, while Theorem 3.2 is valid under the stronger assumption that $F$ is compact-valued and $p_\eta$ is continuous. Observe also that the requirement of $Q_\eta$ in these theorems are different: the lower semicontinuity of $Q_\eta$ in the first theorem is replaced by the upper semicontinuity of $Q_\eta$ in the second one. As a consequence of Theorems 3.1 and 3.2, we give Corollaries 3.3, 3.4 and Theorem 3.3 with verifiable conditions for our existence results.

Theorem 3.1

Assume that for all $\eta \in K,\,F_\eta$ is $C_\eta$-lsc, and there exist a topological space ${\mathcal {Q}}_\eta ,$ a lsc set-valued map $Q_\eta : E\times K \rightrightarrows {{\mathcal {Q}}_\eta }$ with nonempty values, and a upper semicontinuous strongly $Q_\eta C_\eta$-monotonic function $p_\eta : E\times K\times {\mathcal {Q}}_\eta \times Y \longrightarrow {\mathbb {R}}$ such that, for each $z\in E,\,\varphi (z,\cdot ,\cdot )$ is 0-diagonally quasiconvex, where $\varphi : E\times K\times K \longrightarrow {\mathbb {R}}\cup \{-\infty \},$ is defined by

$$\begin{aligned} \varphi (z,x,\eta )= \inf _{q\in Q_\eta (z,x)}\inf _{y\in F_\eta (z,x)} p_\eta (z,x,q,y),\ (z,x,\eta )\in E\times K\times K. \end{aligned}$$

Then there exists a solution of Problem $(P).$

Proof

Consider the auxiliary problem $(\widehat{P})$ of finding a point $(z_0,x_0)\in E\times K$ such that $(z_0,x_0)\in B(z_0,x_0)\times A(z_0,x_0)$ and

$$\begin{aligned} \forall \eta \in A(z_0,x_0),\ \varphi (z_0,x_0,\eta )\ge 0. \end{aligned}$$

Making use of Lemma 3.1, we can see that the set-valued map $L_1$ defined in Proposition 2.2 has open lower sections. Therefore, by Proposition 2.2 there exists a solution $(z_0,x_0)$ of this problem. We show that $(z_0,x_0)$ is also a solution of Problem $(P).$ For this purpose, it suffices to prove that

$$\begin{aligned} \forall \eta \in A(z_0,x_0),\ F(z_0,x_0,\eta )\cap - [C(z_0,x_0,\eta ){\setminus } \{0\}] = \emptyset . \end{aligned}$$

(14)

Indeed, otherwise we can find $\eta \in A(z_0,x_0)$ and $y\in F(z_0,x_0,\eta )$ such that $y\in -C(z_0,x_0,\eta ){\setminus } \{0\}.$ Let $q$ be an arbitrary point of $Q_\eta (z_0,x_0).$ Since by assumption $p_\eta (z_0,x_0,q, \cdot )$ is strongly $C_\eta (z_0,x_0)$-monotonic, we have $p_\eta (z_0,x_0,q,y)< 0$ (see Remark 2.3). It follows that $\varphi (z_0,x_0,\eta ) < 0,$ which is impossible. $\square$

From the formulation of Theorem 3.1 and its proof, it is clear that if for each $\eta \in K$ we can construct a usc strongly $C_\eta$-monotonic function $p_\eta$ which is independent of $q,$ then the existence of a topological space ${\mathcal {Q}}_\eta$ and a lsc map $Q_\eta$ mentioned in Theorem 3.1 becomes superfluous. (A similar remark can be made for Theorem 3.2 below.) We will show in Corollaries 3.3 and 3.4 that such a function $p_\eta$ can be constructed under suitable assumptions on $C.$ For this purpose, we will use a nonlinear scalarization function introduced in [26]. Namely, assume that $\hat{C} : E\times K\times K\rightrightarrows Y$ is a set-valued map whose values are pointed closed convex cones with nonempty interior. For each $\eta \in K,$ assume that $e_\eta : E\times K\longrightarrow Y$ is a continuous selection of the map $\mathrm{int}\ \hat{C}_\eta ,$ that is, $e_\eta$ is a continuous map such that $e_\eta (z,x) \in \mathrm{int}\ \hat{C}_\eta (z,x)$ for each $(z,x) \in E\times K$. Observe from Theorem 2.1 that the map $\mathrm{int}\ \hat{C}_\eta$ has a continuous selection if, for each $y \in Y,$ the set $\{(z,x)\in E\times K : y \in \mathrm{int}\ \hat{C}_\eta (z,x)\}$ is open in $E\times K.$

Following [26], we set

$$\begin{aligned} \xi _\eta (z,x,y)=\min \{\lambda \in {\mathbb {R}}:y\in \lambda e_\eta (z,x) -C_\eta (z,x)\} \end{aligned}$$

for each $(z,x,y)\in E\times K\times Y.$

Together with the function $\xi _\eta$ we will use the set-valued map $\hat{W}_\eta (\cdot ,\cdot )=Y{\setminus } \mathrm{int}\ \hat{C}_\eta (\cdot ,\cdot ).$

Corollary 3.3

Assume that for all $\eta \in K,\,F_\eta$ is $C_\eta$-lsc, and there exists a set-valued map $\hat{C} : E\times K\times K\rightrightarrows Y$ with pointed closed convex cone values such that, for each $\eta \in K,$

• (i) The set-valued map $\mathrm{int}\ \hat{C}_\eta$ has a continuous selection $e_\eta ;$

• (ii) $C_\eta (z,x){\setminus } \{0\} \subset \mathrm{int}\ \hat{C}_\eta (z,x), \forall (z,x)\in E\times K;$

• (iii) $\hat{W}_\eta$ has graph closed in $E\times K\times Y$ where each of the sets $E$ and $K$ is equipped with the induced topology.

Assume furthermore that for each $z\in E,\,\hat{\varphi }(z,\cdot ,\cdot )$ is 0-diagonally quasiconvex, where $\hat{\varphi }: E\times K\times K \longrightarrow {\mathbb {R}}\cup \{-\infty \},$ is defined by

$$\begin{aligned} \hat{\varphi }(z,x,\eta )= \inf _{y\in F_\eta (z,x)} \xi _\eta (z,x,y),\ (z,x,\eta )\in E\times K\times K. \end{aligned}$$

Then there exists a solution of Problem $(P).$

Proof

As we have said above, this corollary is a consequence of Theorem 3.1 if for each $\eta \in K$ there exists a usc strongly $C_\eta$ -monotonic function $p_\eta$ which is independent of $q$. Such a function can be constructed by setting for each $\eta \in K$

$$\begin{aligned} p_\eta (z,x,y) = \xi _\eta (z,x,y), (z,x,y)\in E\times K\times Y. \end{aligned}$$

Indeed, it is known from Proposition 2.1 of [27] that, for each $(z,x)\in E\times K,\,\xi _\eta (z,x,\cdot )$ is $\hat{C}_\eta (z,x)$-monotonic, that is, if $y_1\in y_2 + \mathrm{int}\ \hat{C}_\eta (z,x)$, then $\xi _ \eta (z,x,y_2) < \xi _ \eta (z,x,y_1).$ Together with (ii), this shows that $\xi _\eta (z,x,\cdot )$ is strongly $C_\eta (z,x)$-monotonic. To complete our proof, it suffices to observe that the upper semicontinuity of $\xi _\eta$ can be derived by the arguments similar to those used in the proof of Theorem 2.1 of [26]. $\square$

Theorem 3.2

Assume that for all $\eta \in K,\,F_\eta$ is $C_\eta$-lsc and compact-valued, and there exist a topological space ${\mathcal {Q}}_\eta ,$ a usc set-valued map $Q_\eta : E\times K \rightrightarrows {{\mathcal {Q}}_\eta }$ with nonempty compact values, and a continuous strongly $Q_\eta C_\eta$-monotonic function $p_\eta : E\times K\times {\mathcal {Q}}_\eta \times Y \longrightarrow {\mathbb {R}}$ such that, for each $z\in E,\,\varphi '(z,\cdot ,\cdot )$ is 0-diagonally quasiconcave where $\varphi ' : E\times K\times K \longrightarrow {\mathbb {R}}$ is defined by

$$\begin{aligned} \varphi '(z,x,\eta )= \inf _{q\in Q_\eta (z,x)}\left[ -\inf _{y\in F_\eta (z,x)} p_\eta (z,x,q,y)\right] ,\ (z,x,\eta )\in E\times K\times K. \end{aligned}$$

Then there exists a solution of Problem $(P).$

Proof

Consider Problem $(\widehat{P})$ of finding a point $(z_0,x_0)\in E\times K$ such that $(z_0,x_0)\in B(z_0,x_0)\times A(z_0,x_0)$ and

$$\begin{aligned} \forall \eta \in A(z_0,x_0),\ \varphi '(z_0,x_0,\eta ) \le 0. \end{aligned}$$

Making use of Lemma 3.2, we can see that the set-valued map $L_2$ defined in Proposition 2.2, with $\varphi '$ instead of $\varphi ,$ has open lower sections. Therefore, by Proposition 2.2 there exists a solution $(z_0,x_0)$ of this problem. We show that it is also a solution of Problem $(P).$ For this purpose, it suffices to prove that (14) holds. Indeed, as we have seen in the proof of Theorem 3.1, if (14) does not hold, then there exist $\eta \in A(z_0,x_0)$ and $y\in F(z_0,x_0,\eta )$ such that

$$\begin{aligned} p_\eta (z_0,x_0,q,y) < 0,\ \forall q\in Q_\eta (z_0,x_0). \end{aligned}$$

This implies that

$$\begin{aligned} -\inf _{y\in F_\eta (z_0,x_0)} p_\eta (z_0,x_0,q,y) > 0,\ \forall q\in Q_\eta (z_0,x_0). \end{aligned}$$

By the compactness of $Q_\eta (z_0,x_0),$ we obtain

$$\begin{aligned} \varphi '(z_0,x_0,\eta ) > 0, \end{aligned}$$

which is impossible. $\square$

Remark 3.2

Since $\varphi \le -\varphi '$ (in the sense that each value of $\varphi$ is less than or equal to the corresponding value of $-\varphi '$), the 0-diagonal quasiconvexity of $\varphi$ implies the 0-diagonal quasiconcavity of $\varphi '.$

Corollary 3.4

Assume that for all $\eta \in K,\,F_\eta$ is $C_\eta$-lsc and compact-valued, and there exists a set-valued map $\hat{C} : E\times K\times K\rightrightarrows Y$ with pointed closed convex cone values such that, for each $\eta \in K,$

• (i) The set-valued map $\mathrm{int}\ \hat{C}_\eta$ has a continuous selection $e_\eta ;$

• (ii) $C_\eta (z,x){\setminus } \{0\} \subset \mathrm{int}\ \hat{C}_\eta (z,x), \forall (z,x)\in E\times K;$

• (iii) $\hat{C}_\eta$ and $\hat{W}_\eta$ have graphs closed in $E\times K\times Y$ where each of the sets $E$ and $K$ is equipped with the induced topology.

Assume furthermore that for each $z\in E,\,\hat{\varphi }'(z,\cdot ,\cdot )$ is 0-diagonally quasiconvex, where $\hat{\varphi }' : E\times K\times K \longrightarrow {\mathbb {R}}\cup \{-\infty \},$ is defined by

$$\begin{aligned} \hat{\varphi }'(z,x,\eta )=-\inf _{y\in F_\eta (z,x)} \xi _\eta (z,x,y) ,\ (z,x,\eta )\in E\times K\times K. \end{aligned}$$

Then there exists a solution of Problem $(P).$

Proof

For each $\eta \in K$, consider the function $p_\eta$ defined in the proof of Corollary 3.3:

$$\begin{aligned} p_\eta (z,x,y) = \xi _\eta (z,x,y) , (z,x,y)\in E\times K\times Y. \end{aligned}$$

We have seen in this proof that, for each $(z,x)\in E\times K,\,\xi _\eta (z,x,\cdot )$ is strongly $C_\eta (z,x)$-monotonic. Also, making use of (iii) and arguing as in the proof of Theorem 2.1 of [26], we can show that $\xi _\eta$ is continuous. To complete our proof it remains to apply Theorem 3.2 (and to observe that $\xi _\eta$ is independent of $q$). $\square$

Remark 3.3

To apply Theorem 3.1 (resp. Theorem 3.2), we need to verify the 0-diagonal quasiconvexity (resp. the 0-diagonal quasiconcavity) of $\varphi (z,\cdot ,\cdot )$ (resp. $\varphi ' (z,\cdot ,\cdot )$). Sufficient conditions for the validity of this property of $\varphi (z,\cdot ,\cdot )$ (resp. $\varphi ' (z,\cdot ,\cdot )$) are given in Proposition 3.1. In particular, we can combine this proposition with Corollaries 3.3 and 3.4 to obtain existence results in Problem $(P),$ with verifiable conditions imposed directly on $F$ and $C.$

Before going further, let us introduce the set-valued map $C^{+i}: E\times K\times K \rightrightarrows {Y^*}$ defined by

$$\begin{aligned} C^{+i}(z,x,\eta )= \{y^*\in Y^* : \langle y^*,y \rangle > 0, \forall c\in C(z,x,\eta ){\setminus } \{0\}\}, (z,x,\eta )\in E\!\times \! K\!\times \! K. \end{aligned}$$

Recall that $Y^*$ is the set of all linear continuous functionals defined on $Y.$ From now on, when dealing with $Y^*$ we always assume that $Y^*$ is equipped with a topology for which $Y^*$ becomes a topological vector space with the bilinear form $\bar{p}(y^*,y):= \langle y^*,y \rangle$ being continuous in $(y^*,y)\in Y^*\times Y.$ We will refer to this as Assumption (I).

Assumption (I) holds, if $Y$ is a normed space and if $Y^*$ is taken as the normed space of the bounded linear functionals on $Y.$ For another example, see Lemma 1 of [28].

Remark 3.4

Let Assumption (I) be satisfied. For each $\eta \in K,$ we set ${\mathcal {Q}}_\eta = Y^*.$ Then the function

$$\begin{aligned} p_\eta (z,x,q,y)= \bar{p}(q,y), (z,x)\in E\times K, q\in {\mathcal {Q}}_\eta = Y^*, y\in Y, \end{aligned}$$

(15)

is continuous (and does not depend on $z,x$ and $\eta$). It is strongly $Q_\eta C_\eta$-monotonic for any set-valued map $Q_\eta \subset C^{+i}_\eta$ where $Q_\eta \subset C^{+i}_\eta$ means that each value of $Q_\eta$ is a subset of the corresponding value of $C^{+i}_\eta .$

Now, if we assume that, for each $\eta \in K,\,C^{+i}_\eta$ has a continuous selection $e_\eta$, then the function

$$\begin{aligned} p_\eta (z,x,q,y)=\langle e_\eta (z,x),y \rangle , (z,x)\in E\times K, q\in {\mathcal {Q}}_\eta = Y^*, y\in Y, \end{aligned}$$

(16)

is continuous (and does not depend on $q$). It is strongly $Q_\eta C_\eta$-monotonic for any set-valued map $Q_\eta$ taking values in ${\mathcal {Q}}_\eta =Y^*$.

Observe from Theorem 2.1 that the map $C_\eta ^{+i}$ has a continuous selection if for all $(z,x)\in E\times K,\,C_\eta ^{+i}(z,x)\ne \emptyset$, and if for all $q\in {\mathcal {Q}}_\eta = Y^*,$ the set $\{(z,x)\in E\times K : q\in C_\eta ^{+i}(z,x)\}$ is open in $E\times K.$

In the theory of equilibrium problems, we often deal with the case where $C(z,x,\eta )$ does not depend on the variable $\eta \in K.$ In this case, for the sake of simplicity of presentation, we will write $C(z,x)$ and $C^{+i}(z,x)$ instead of $C(z,x,\eta )$ and $C^{+i}(z,x,\eta ).$

Sufficient conditions for the 0-diagonal quasiconvexity of $\varphi$ and the 0-diagonal quasiconcavity of $\varphi '$ can be found in the following proposition.

Proposition 3.1

Assume that, for each $\eta \in K,\,{\mathcal {Q}}_\eta$ is a topological space, $Q_\eta : E\times K \rightrightarrows {{\mathcal {Q}}_\eta }$ is a set-valued map with nonempty values and $p_\eta : E\times K\times {\mathcal {Q}}_\eta \times Y \longrightarrow {\mathbb {R}}$ is a $Q_\eta C_\eta$-monotonic function. Consider the following conditions:

(a$_1)$ :

For all $z\in E,\,x_i\in K,i=1,2,\ldots ,n,$ and $x\in \mathrm{co}\{x_i,i=1,2,\ldots ,n\}$ there exists $i\in \{1,2,\ldots ,n\}$ such that $F(z,x,x_i)\subset C(z,x,x_i);$

(a$_2)$ :

$C$ does not depend on $\eta \in K,$ and $C$ is such that for all $(z,x)\in E\times K,\,F(z,x,x)\subset C(z,x)$ and $F(z,x,\cdot )$ is proper $C(z,x)$-quasiconvex;

(a$_3)$ :

$C$ does not depend on $\eta \in K,$ and $C$ is such that for all $(z,x)\in E\times K,\,F(z,x,x)\subset C(z,x)$ and $F(z,x,\cdot )$ is $C(z,x)$-convex.

Then the following conclusions are true:

• (i) If (a $_1)$ or (a $_2)$ holds, then $\varphi$ (resp. $\varphi ')$ is 0-diagonally quasiconvex (resp. 0- diagonally quasiconcave).

• (ii) If (a $_3)$ holds and if for all $(z,x,\eta )\in E\times K\times K,$ the function $p_\eta (z,x,q,y)$ is convex in $y\in Y$ and either $p_\eta (z,x,q,y)$ does not depend on $q\in {\mathcal {Q}}_\eta$, or $p_\eta (z,x,q,y)$ does not depend on $\eta \in K$, then $\varphi$ (resp. $\varphi ')$ is 0-diagonally quasiconvex (resp. 0-diagonally quasiconcave).

Proof

By Remark 3.2, the 0-diagonal quasiconvexity of $\varphi$ implies the 0-diagonal quasiconcavity of $\varphi '.$ So, we need to show only the 0-diagonal quasiconvexity of $\varphi$.

Let $z\in E,x_i\in K,i=1,2,\ldots ,n,$ and $x\in \mathrm{co}\{x_i,i=1,2,\ldots ,n\}.$ Then $x= \sum _{i=1}^n \lambda _i x_i$ with $\lambda _i \ge 0,\,i=1,2,\ldots ,n,$ and $\sum _{i=1}^n \lambda _i =1.$ If condition (a$_1$) holds and if $i\in \{1,2,\ldots ,n\}$ is the index mentioned in this condition, then we have $p_{x_i}(z,x,q,y)\ge 0$ for all $q\in Q_{x_i}(z,x)$ and $y\in F_{x_i}(z,x).$ Thus, $\varphi (z,x,x_i)\ge 0$ and the 0-diagonal quasiconvexity of $\varphi$ is established.

To complete the proof of the conclusion (i) of our proposition it remains to show that (a$_2$) implies (a$_1$). Indeed, making use of the proper $C(z,x)$-quasiconvexity of $F(z,x,\cdot )$ and an induction argument, we can show (see Remark 2.2) that $F(z,x,x_i)\subset F(z,x,x)+C(z,x)$ for some $i\in \{1,2,\ldots ,n\}.$ Since by assumption $F(z,x,x)\subset C(z,x),$ we can conclude that $F(z,x,x_i)\subset C(z,x),$ showing that (a$_2$) $\Rightarrow$ (a$_1$), as desired.

To prove conclusion (ii), we take $z\in E,x_i\in K,i=1,2,\ldots ,n,$ and $x\in \mathrm{co}\{x_i,i=1,2,\ldots ,n\}.$ Then $x= \sum _{i=1}^n \lambda _i x_i$ with $\lambda _i \ge 0,\,i=1,2,\ldots ,n,$ and $\sum _{i=1}^n \lambda _i =1.$ By the $C(z,x)$-convexity property of $F(z,x,\cdot )$ we obtain from Remark 2.2 that

$$\begin{aligned} \sum _{i=1}^n \lambda _i F(z,x,x_i) \subset F(z,x,x) + C(z,x) \subset C(z,x). \end{aligned}$$

(The last inclusion holds since by assumption $F(z,x,x)\subset C(z,x).$) Together with the $Q_\eta C_\eta$ monotonicity property of $p_\eta$, this yields

$$\begin{aligned} p_\eta \left( z,x,q,\sum _{i=1}^n \lambda _i y_i\right) \ge 0 \end{aligned}$$

for all $y_i \in F(z,x,x_i),i=1,2,\ldots ,n,$ and $(\eta ,q)\in K\times Q_\eta (z,x)$. In view of the convexity of the function $p_\eta (z,x,q,\cdot )$, we can write

$$\begin{aligned} \sum _{i=1}^n \lambda _i p_\eta (z,x,q,y_i) \ge 0. \end{aligned}$$

Since this inequality is valid for all $(\eta ,q)\in K\times Q_\eta (z,x)$ and $y_i\in F(z,x,x_i), i=1,2,\ldots ,n,$ we can conclude that

$$\begin{aligned} \sum _{i=1}^n \lambda _i \left[ \inf _{y'\in F(z,x,x_i)} p_\eta (z,x,q,y')\right] \ge 0, \end{aligned}$$

or, equivalently,

$$\begin{aligned} \sum _{i=1}^n \lambda _i \varphi (z,x,x_i) \ge 0 \end{aligned}$$

since by assumption either $p_\eta (z,x,q,y)$ does not depend on $q\in {\mathcal {Q}}_\eta$, or $p_\eta (z,x,q,y)$ does not depend on $\eta \in K$. Hence, $\varphi (z,x,x_i) \ge 0$ for some $i,$ and the 0-diagonal quasiconvexity of $\varphi$ is thus established. $\square$

Theorem 3.3

Let Assumption (I) be satisfied. Assume furthermore that one of the following conditions is fulfilled:

• (i) For all $\eta \in K,\,F_\eta$ is $C_\eta$-lsc, and $C_\eta ^{+i}$ has nonempty values and is lsc;

• (ii) For all $\eta \in K,\,F_\eta$ is $C_\eta$-lsc and compact-valued, and the map $C_\eta ^{+i}$ has a continuous selection.

If at least one of the conditions (a$_1)$, (a$_2)$ and (a$_3)$ holds, then there exists a solution of Problem $(P)$.

Proof

Let condition (i) be satisfied. For each $\eta \in K$, define $p_\eta$ by (15) and take $Q_\eta =C_\eta ^{+i}$. Using Proposition 3.1, we can see that all assumptions of Theorem 3.1 are fulfilled. The existence of a solution of Problem $(P)$ is thus established.

Now let condition (ii) be satisfied. For each $\eta \in K$, define $p_\eta$ by (16) (where $e_\eta$ is a continuous selection of $C_\eta ^{+i}$) and take $Q_\eta \equiv \{0\}$. Using Proposition 3.1, we can see that all assumptions of Theorem 3.2 are fulfilled. Hence there exists a solution of Problem $(P).$ $\square$

The following corollary gives an existence result for Problem $(P)$ in the case where $C$ is a constant convex cone.

Corollary 3.5

Let Assumption (I) be satisfied. Let $C(z,x,\eta ) = C_1$ for all $( z,x,\eta ) \in E\times K\times K,$ where $C_1$ is a convex cone of $Y.$ Assume that the set

$$\begin{aligned} C_1^{+i}= \{y^*\in Y^* : \langle y^*,y \rangle > 0, \forall c\in C_1{\setminus } \{0\}\} \end{aligned}$$

is nonempty. Assume furthermore that for each $\eta \in K,\,F_\eta$ is $C_1$-lsc, and at least one of the following conditions is satisfied:

(a$'_1)$ :

For all $z\in E,\,x_i\in K,i=1,2,\ldots ,n,$ and $x\in \mathrm{co}\{x_i,i=1,2,\ldots ,n\}$, there exists $i\in \{1,2,\ldots ,n\}$ such that $F(z,x,x_i)\subset C_1.$

(a$'_2)$ :

$C_1$ is such that for all $(z,x)\in E\times K,\,F(z,x,x)\subset C_1$ and $F(z,x,\cdot )$ is proper $C_1$-quasiconvex.

(a$'_3)$ :

$C_1$ is such that for all $(z,x)\in E\times K,\,F(z,x,x)\subset C_1$ and $F(z,x,\cdot )$ is $C_1$-convex.

Then there exists a solution of Problem $(P)$.

Proof

This is immediate from Theorem 3.3. $\square$

As we remarked above, our existence results may fail to hold if the assumption that the set $M$ is closed in $E\times K$ is violated. This conclusion can be derived from the following example taken from [21, 25].

Example 3.1

Consider Problem $(P)$ where $X=Y=Z={\mathbb {R}},\,E=K=[0,1]\subset {\mathbb {R}},\,C_1={\mathbb {R}}_+,$ and for all $z, x, \eta \in [0,1], \,\widehat{B}(z,x)\equiv \{1\},\,F(z,x,\eta )=\{ z(x-\eta )\}\subset {\mathbb {R}}$ and

$$\begin{aligned} A(z,x)= \left\{ \begin{array}{l@{\quad }l} [0,1], &{} \text {if }\ x\in [0,1[\\ \{0\},&{} \text {if }\ x=1.\\ \end{array}\right. \end{aligned}$$

It is easy to see that Problem $(P)$ in this example has no solutions. Observe that, in our case, $F$ is continuous and satisfies condition (a$'_3$) of Corollary 3.5, but $M$ is not closed.

4 Applications

In this section, we apply the main results of the paper to study the existence of solutions of the vector quasi-optimization problems $(GVQOP)_p$ and $(GVQOP)_{Pr}$ (see [10]), the Stampacchia set-valued vector quasi-variational inequality problem (see [7]) and the Pareto vector quasi-saddle point problem. Existence results are formulated in Theorems 4.1–4.3 and Corollary 4.1. We will see that they are new, and are established under assumptions different from the corresponding ones in the literature.

We will denote by $\mathrm{Min}[D/C]$ (resp. $\mathrm{Max}[D/C]$) the set of minimal (resp. maximal) points of a subset $D$ with respect to a convex cone $C$ of a vector space $Y.$ Recall that

$$\begin{aligned} \mathrm{Min}[D/C]&= \{d\in D : (D - d)\cap (-C{\setminus } \{0\})= \emptyset \},\\ \mathrm{Max}[D/C]&= \{d\in D : (D - d)\cap (C{\setminus } \{0\})= \emptyset \} \end{aligned}$$

and $\mathrm{Max}[D/C]=\mathrm{Min}[D/-C],$ where 0 is the origin of $Y.$

We denote by $\mathrm{PrMin}[D/C]$ the set of (global) proper efficient points of $D$ with respect to $C.$ Recall that $y\in \mathrm{PrMin}[D/C]$ if and only if $y\in D$ and there exists a pointed convex cone $C_1$ with $C{\setminus } \{0\}\subset \mathrm{int}\ C_1$ and $y\in \mathrm{Min}[D/C_1].$

Assume that $F: A\rightrightarrows {{\mathbb {R}}}$ is a set-valued map from a convex set $A$ to the real line ${\mathbb {R}}.$ We say that $F$ is quasiconvex if, for each $r\in {\mathbb {R}},$ the generalized level set

$$\begin{aligned} \{x\in A: \exists r'\in F(x)\ \mathrm{with}\ r'\le r\} \end{aligned}$$

is convex. This definition is taken from [29], and it reduces to the usual notion of quasiconvexity if $F$ is single-valued. In this paper, we do not use extensions of the above notion of quasiconvexity to set-valued maps taking values in partially ordered vector spaces. The reader who is interested in the corresponding definition in this general case is referred to [29] and [30], where some relationships with other generalized cone-convexity notions and applications of this definition to the theory of loose saddle points of set-valued maps can be found.

Now we give some examples of quasiconvex maps $F: A\rightrightarrows {{\mathbb {R}}}.$

Example 4.1

Let $F$ be compact-valued. Then $F$ is quasiconvex if and only if the single-valued function $x\mapsto \inf F(x)$ is quasiconvex in the usual sense.

Example 4.2

Let $C$ be a convex cone of a vector space $Y,\,F: A \rightrightarrows Y$ be a proper $C$-quasiconvex map with nonempty compact values, and $\sigma : Y\longrightarrow {\mathbb {R}}$ be a quasiconvex $C$-monotonic function. Then the set-valued map $F'=\sigma F$ (the composite of $\sigma$ and $F$) is quasiconvex. This is because $x\mapsto \inf F'(x)$ is quasiconvex.

Example 4.3

(see [29, 30]) Let $C$ be a convex cone of a vector space $Y,\,F: A \rightrightarrows Y$ be a $C$-convex map with nonempty compact values, and $\sigma : Y\longrightarrow {\mathbb {R}}$ be a convex $C$-monotonic function. Then $F'=\sigma F$ is quasiconvex, since $x\mapsto \inf F'(x)$ is convex.

Lemma 4.1

Let $E$ and $K$ be nonempty convex sets and $F: E\times K\times K \rightrightarrows {{\mathbb {R}}}$ be a set-valued map such that, for each $(z,x)\in E\times K,\,F(z,x,\cdot )$ is quasiconvex and compact-valued. Let $f: E\times K\times K \longrightarrow {\mathbb {R}}$ be defined by

$$\begin{aligned} f(z,x,\eta )= \inf F(z,x,\eta ) -\inf F(z,x,x),\ (z,x,\eta )\in E\times K\times K. \end{aligned}$$

Then, for each $z\in E,\,f(z,\cdot ,\cdot )$ is 0-diagonally quasiconvex.

Proof

Let $z\in E, x_i\in K,i=1,2,\ldots ,n,$ and $x\in \mathrm{co}\{x_i,i=1,2,\ldots ,n\}.$ We need to prove that $f(z,x,x_i)\ge 0$ for some $i\in \{1,2,\ldots ,n\}.$ Indeed, otherwise we can find $r\in {\mathbb {R}}$ such that

$$\begin{aligned} \inf F(z,x,x_i) < r < \inf F(z,x,x),\quad i=1,2,\ldots ,n. \end{aligned}$$

Making use of the quasiconvexity of $F(z,x,\cdot ),$ we can write $\inf F(z,x,x) \le r < \inf F(z,x,x).$ This is impossible. $\square$

Theorem 4.1

Assume that $X,Y$ and $Z$ are locally convex Hausdorff topological vector spaces; $K\subset X$ and $E\subset Z$ are nonempty compact convex sets; $A: E\times K \rightrightarrows K$ is a set-valued map with nonempty convex values and open lower sections; and $B: E\times K \rightrightarrows E$ is an acyclic map such that the set $M,$ defined in Sect. 3, is closed in $E\times K.$ Assume that $C\subset Y$ is a convex cone, and $F: E\times K\times K \rightrightarrows Y$ is a $C$-continuous map with nonempty compact values. Assume that there exists a (single-valued) continuous strongly $C$-monotonic map $\sigma : Y\longrightarrow {\mathbb {R}}$ such that $\sigma F(z,x,\cdot )$ is quasiconvex for each $(z,x)\in E\times K.$ Then there exists $(z,x)\in M$ such that

• (i) $\min \sigma F(z,x,A(z,x))= \min \sigma F(z,x,x);$

• (ii) $F(z,x,x)\cap \mathrm{Min}[F(z,x,A(z,x))/C]\ne \emptyset .$

Proof

Consider Problem $(P)$ where ${\mathbb {R}}$ (resp. ${\mathbb {R}}_+$) plays the role of $Y$ (resp. $C$) and $f:E\times K\times K\longrightarrow Y={\mathbb {R}}$ is defined by

$$\begin{aligned} f(z,x,\eta )= \inf \sigma F(z,x,\eta ) -\inf \sigma F(z,x,x),\ (z,x,\eta )\in E\times K\times K. \end{aligned}$$

We set ${\mathcal {Q}}_\eta = {\mathbb {R}},\,Q_\eta (z,x)\equiv \{0\},\,p_\eta (z,x,q,y)\equiv 1 \in {\mathcal {Q}}_\eta ={\mathbb {R}}$ for each $\eta \in K$ and $(z,x,q,y) \in E\times K\times {\mathcal {Q}}_\eta \times Y.$ Then clearly $\varphi (z,x,\eta )=f(z,x,\eta )$ for all $(z,x,\eta )\in E\times K\times K.$ By Lemma 4.1 $\varphi (z,\cdot ,\cdot )$ is 0-diagonally quasiconvex, for each $z\in K.$ We now prove that $F$ is continuous. For this purpose, we write

$$\begin{aligned} f(z,x,\eta )= f_1(z,x,\eta ) + f_2(z,x) \end{aligned}$$

where

$$\begin{aligned} f_1(z,x,\eta )&=\quad \inf _{y\in F(z,x,\eta )} \sigma (y),\\ f_2(z,x)&= -\inf _{y\in F(z,x,x)} \sigma (y),\ (z,x,\eta )\in E\times K\times K. \end{aligned}$$

Applying Lemmas 3.1 and 3.4, with $\sigma (y)$ (resp. $F(z,x,\eta )$) instead of $p(z,x,q,y)$ (resp. $F(x)$), we see that $f_1$ is continuous. Similarly, applying Lemmas 3.2 and 3.3 proves the continuity of $f_2.$ Basing on these discussions we can conclude that $f$ is continuous, as desired.

In view of Theorem 3.1, there exists $(z,x)\in M$ such that

$$\begin{aligned} \forall \eta \in A(z,x), f(z,x,\eta ) \ge 0. \end{aligned}$$

It follows that

$$\begin{aligned} \inf \sigma F(z,x,A(z,x))\ge \inf \sigma F(z,x,x). \end{aligned}$$

Observe that the converse of this inequality is obvious since the condition $(z,x)\in M$ implies that $x\in A(z,x)$ and hence $\sigma F(z,x,x)\subset \sigma F(z,x,A(z,x)).$ Thus,

$$\begin{aligned} \inf \sigma F(z,x,A(z,x))= \inf \sigma F(z,x,x). \end{aligned}$$

To complete the proof of conclusion (i) of Theorem 4.1 it remains to observe that $\sigma F(z,x,x)$ is a compact set and hence

$$\begin{aligned} \inf \sigma F(z,x,x)= \min \sigma F(z,x,x). \end{aligned}$$

To prove the second conclusion of Theorem 4.1, we denote by $y$ the element of $F(z,x,x)$ such that

$$\begin{aligned} \sigma (y)= \min \sigma F(z,x,x). \end{aligned}$$

We claim that $y\in \mathrm{Min}[F(z,x,A(z,x))/C].$ Indeed, otherwise we can find $\eta \in A(z,x)$ and $y'\in F(z,x,\eta )$ with $y'-y\in -C{\setminus } \{0\}.$ By the strong $C$-monotonicity of $\sigma ,$ this yields $\sigma (y) > \sigma (y'),$ a contradiction to conclusion (i). $\square$

Remark 4.1

Examples 4.2 and 4.3 show that the quasiconvexity of the composite map $\sigma F(z,x,\cdot )$ used in Theorem 4.1 can be assured by verifiable conditions imposed separately on $\sigma$ and $F$.

Remark 4.2

For the special case where $\sigma (y)\equiv 1,$ the first conclusion of Theorem 4.1 is obtained in Theorem 3.1 of [7], under the assumption that $A$ does not depend on $z,$ and $A$ is continuous and has compact convex values. Even for this special case, the first inclusion of Theorem 4.1 cannot be derived from Theorem 3.1 of [7], since in our case the open lower openness of $A$ is used instead of the continuity of $A$ in [7], and the $C$-continuity of $F$ is used instead of the continuity of $F$ in [7]. Recall that the $C$-continuity property of $F$ is weaker than the usual continuity of $F$ (see Remark 2.1).

Remark 4.3

Theorem 4.1 gives an existence result for a solution of the general vector quasi-optimization problem $(GVQOP)_p$ in [10]. The solution existence of $(GVQOP)_p$ in Corollary 3.17 of [10] is proven by an approach different of that of our paper. Also, this corollary uses the continuity of $A,$ while in our result it is replaced by the openness of the lower sections of $A.$

Corollary 4.1

Assume that $X,Y,Z,E,K,A$ and $B$ are as in Theorem 4.1. Assume that $C\subset Y$ is a convex cone, and $F: E\times K\times K \rightrightarrows Y$ is a $C$-continuous map with nonempty compact values. Assume furthermore that there exist a convex cone $C_1\subset Y$ with $C{\setminus } \{0\}\subset C_1$ and a (single-valued) continuous strongly $C_1$-monotonic map $\sigma : Y\longrightarrow {\mathbb {R}}$ such that $\sigma F(z,x,\cdot )$ is quasiconvex for each $(z,x)\in E\times K.$ Then there exists $(z,x)\in M$ such that

$$\begin{aligned} F(z,x,x)\cap \mathrm{PrMin}[F(z,x,A(z,x))/C]\ne \emptyset . \end{aligned}$$

Proof

Since $C\subset C_1,$ the $C$-continuity of $F$ implies the $C_1$-continuity of $F.$ Applying Theorem 4.1 with $C_1$ instead of $C,$ we find $(z,x)\in M$ such that the conclusion (ii) of Theorem 4.1 holds, with $C_1$ instead of $C.$ This proves Corollary 4.1. $\square$

Remark 4.4

Corollary 4.1 gives an existence result for a solution of the general vector quasi-optimization problem $(GVQOP)_{Pr}$ in [10]. This result is different from the solution existence in Corollary 3.16 of [10], by the same reason shown at the end of Remark 4.3.

Theorem 4.2

Assume that $X,Y$ and $Z$ are Banach spaces, $K\subset X$ and $E\subset Z$ are nonempty compact convex sets; $A: E\times K \rightrightarrows K$ is a set-valued map with nonempty convex values and open lower sections; and $B: E\times K \rightrightarrows E$ is an acyclic map such that the set $M,$ defined in the previous section, is closed in $E\times K;$ and $C: E\times K \rightrightarrows Y$ is a map with convex cone values such that $C^{+i}$ is a lsc map with nonempty values. Assume furthermore that $T: E\times K \rightrightarrows {L(X,Y)}$ is a continuous set-valued map with nonempty compact values, where the family $L(X,Y)$ of linear continuous maps from $X$ into $Y$ is equipped with the norm-topology. Then there exists a solution of the Stampacchia set-valued vector quasi-variational inequality problem of finding a point $(z,x)\in M$ such that

$$\begin{aligned} \forall \eta \in A(z,x), T(z,x)(x-\eta )\cap -[C(z,x)){\setminus } \{0\}]= \emptyset , \end{aligned}$$

where $T(z,x)(x-\eta )$ denotes the value of $T(z,x)$ at $x-\eta .$

Proof

It suffices to show that Problem $(P)$ has a solution where $F: E\times K\times K \rightrightarrows Y$ is defined by $F(z,x,\eta )= T(z,x)(x-\eta ).$ Observe that $F$ is continuous and, for each $(z,x)\in E\times K,\,F(z,x,\cdot )$ is $C(z,x)$-convex and $F(z,x,x)\equiv \{0\}\subset C(z,x).$ To complete our proof it remains to apply Theorem 3.3. $\square$

Remark 4.5

Theorem 4.2 differs from Theorem 4.1 of [6] in that in Theorem 4.1 of [6] it is assumed that $T$ and $C$ do not depend on $z$, and $T$ is single-valued, while all these requirements are not used in our Theorem 4.2. Also, in [6] an assumption guaranteeing the existence of a continuous selector of $C^{+i}$ is introduced, while in our Theorem 4.2 this assumption is replaced by the lower semicontinuity of $C^{+i}.$

Theorem 4.3

Assume that $X_i,i=1,2,\,Y$ and $Z$ are locally convex Hausdorff topological vector spaces; $E,K_1$ and $K_2$ are nonempty compact convex subsets of $Z,X_1$ and $X_2$ respectively; $A_i: E\times K_1\times K_2 \rightrightarrows {K_i},\,i=1,2,$ are set-valued maps with nonempty convex values and open lower sections; and $b: E\times K_1\times K_2 \rightrightarrows E$ is an acyclic map such that the set

$$\begin{aligned} M_1&= \{(z,x_1,x_2)\in E\times K_1\times K_2 : (z,x_1,x_2)\in b(z,x_1,x_2)\times A_1(z,x_1,x_2)\\&\times A_2(z,x_1,x_2)\} \end{aligned}$$

is closed in $E\times K_1\times K_2.$ Assume that $f: E\times K_1\times K_2\longrightarrow Y$ is a single-valued continuous map, and $C_i: E\times K_i \rightrightarrows Y,\,i=1,2,$ are set-valued maps with convex cone values. Assume that Assumption (I) is satisfied, and

• (i) Either both $C_1^{+i}$ and $C_2^{+i}$ are lsc maps with nonempty values, or both $C_1^{+i}$ and $C_2^{+i}$ have continuous selections;

• (ii) For each $z\in E$ and $(x_1,x_2)\in K_1\times K_2,\,f(z,\cdot ,x_2)$ is $C_2(z,x_2)$-convex and $f(z,x_1,\cdot )$ is $[-C_1(z,x_1)]$-convex.

Then there exists a solution of the Pareto vector quasi-saddle point problem of finding a point $(z,x_1,x_2)\in M_1$ such that

$$\begin{aligned} f(z,x_1,x_2)&\in \mathrm{Min}[f(z,A_1(z,x_1,x_2),x_2)/C_2(z,x_2)],\end{aligned}$$

(17)

$$\begin{aligned} f(z,x_1,x_2)&\in \mathrm{Max}[f(z,x_1,A_2(z,x_1,x_2))/C_1(z,x_1)]. \end{aligned}$$

(18)

Proof

Consider Problem $(P)$ where $K:= K_1\times K_2 \subset X:=X_1\times X_2,$ and for all $z\in E, x=(x_1,x_2)\in K:= K_1\times K_2$ and $\eta =(\eta _1,\eta _2)\in K:= K_1\times K_2,$ define $B(z,x)= b(z,x_1,x_2),\,A(z,x)= A_1(z,x_1,x_2)\times A_2(z,x_1,x_2),\,F(z,x,\eta )= (f(z,\eta _1,x_2)-f(z,x_1,x_2),f(z,x_1,x_2)-f(z,x_1,\eta _2))\in Y\times Y,\,C(z,x)= C_2(z,x_2)\times C_1(z,x_1)\subset Y\times Y.$ Observe from (ii) that condition $(a_3)$ holds.

Making use of Theorem 3.3, we can claim that Problem $(P)$ has at least a solution, denoted by $(z,x)= (z,x_1,x_2)\in E\times K= E\times K_1\times K_2.$ Therefore, $(z,x)= (z,x_1,x_2)\in M_1$ and

$$\begin{aligned} F(z,x,\eta )\notin -[C(z,x){\setminus } \{(0,0)\}], \end{aligned}$$

(19)

for all $\eta =(\eta _1,\eta _2)\in A_1(z,x_1,x_2)\times A_2(z,x_1,x_2).$ Here $(0,0)$ is the origin of $Y\times Y.$

Observing that $x_2\in A_2(z,x_1,x_2)$ since $(z,x_1,x_2)\in M_1.$ Therefore, for an arbitrary point $\eta _1\in A_1(z,x_1,x_2),$ we get $\eta = (\eta _1,x_2)\in A_1(z,x_1,x_2)\times A_2(z,x_1,x_2)$ and by (19)

$$\begin{aligned} (f(z,\eta _1,x_2)-f(z,x_1,x_2),0)\notin -[C_2(z,x_2)\times C_1(z,x_1){\setminus } \{(0,0)\}]. \end{aligned}$$

From this it follows the validity of (13). Similarly, we can see that (18) holds. $\square$

Remark 4.6

Notions of loose saddle points are introduced for the case where $f$ is set-valued (see [7, 29–31] and the references therein). Existence results for loose saddle points in [7, 29–31] are proven under the assumption that $C_i, i=1,2,$ are constant convex cones, and the maps $A$ and $f$ do not depend on $z,$ while our Theorem 4.3 may be valid without these restrictions. So, Theorem 4.3 is different from the corresponding ones of [7, 29–31], applied to single-valued maps.