CONTINUITY OF GENERALIZED CONVEX AND GENERALIZED CONCAVE SET-VALUED FUNCTIONS

Criteria for the continuity of convex and concave functions play am important role in functional analysis and optimization theory, no matter whether these functions are single- or set-valued. But in the case of single-valued functions it suffices to state such criteria for one of these two classes of functions, because a real- or vector-valued function $f$$f$ff$f$ is convex (resp. concave) if and only if $-f$$-f$-f-f$-f$ is concave (resp. convex). Unfortunately, this advantage is lost in the case of set-valued functions. Although the continuity properties of convex set-valued functions are quite similar to those of concave set-valued functions, they have to be proved for each of these classes of functions separately. The present paper will show that the same phenomenon arises in the case of a pair of more general set-valued functions. We call these two new types of setvalued functions $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-convex and $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-concave, respectively. They are defined as follows.

Assume that $A$$A$AA$A$ is a subset of the open interval $]0,1$$]0,1$]0,1] 0,1$]0,1$ [.having zero as a cluster point, and that $s$$s$ss$s$ is a positive real number. Throughout the paper $A$$A$AA$A$ and $s$$s$ss$s$ will always have this meaning. Let $X$$X$XX$X$ and $Y$$Y$YY$Y$ be real topological linear spaces, let ${\mathcal{P}}_0(Y)$${\mathcal{P}}_0(Y)$P_(0)(Y)\mathscr{P}_{0}(Y)${\mathcal{P}}_0(Y)$ denote the set consisting of all nonempty subsets of $Y$$Y$YY$Y$, and let $M$$M$MM$M$ be a nonempty convex subset of $X$$X$XX$X$. A function $F^1:M\to {\mathcal{P}}_0(Y)$$F^1:M\to {\mathcal{P}}_0(Y)$F^(1):M rarrP_(0)(Y)F^{1}: M \rightarrow \mathscr{P}_{0}(Y)$F^1:M\to {\mathcal{P}}_0(Y)$ is said to be :

whenever $a\in A$$a\in A$a in Aa \in A$a\in A$ and $x,y\in M$$x,y\in M$x,y in Mx, y \in M$x,y\in M$;
(ii) $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-concave if

whenever $a\in A$$a\in A$a in Aa \in A$a\in A$ and $x,y\in M$$x,y\in M$x,y in Mx, y \in M$x,y\in M$.
Obviously, for special choices of $A$$A$AA$A$ and $s$$s$ss$s$ in these definitions, we obtain certain kinds of convexity and concavity for set-valued functions which have already been investigated. So, when $A=]0,1[$$A=]0,1[$A=]0,1[A=] 0,1[$A=]0,1[$ and $s=1$$s=1$s=1s=1$s=1$, we get the convex and concave set-valued functions; when $A=]0,1[\cap \mathbf{Q}$$A=]0,1[\cap \mathbf{Q}$A=]0,1[nnQA=] 0,1[\cap \mathbf{Q}$A=]0,1[\cap \mathbf{Q}$ and $s=1$$s=1$s=1s=1$s=1$, we get the rationally convex and rationally concave set-valued functions; finally, when $A=\{2^{-n}:n\in \mathbf{N}\}$$A=\left\{2^{-n}:n\in \mathbf{N}\right\}$A={2^(-n):n inN}A=\left\{2^{-n}: n \in \mathbf{N}\right\}$A=\{2^{-n}:n\in \mathbf{N}\}$ and $s=1$$s=1$s=1s=1$s=1$, we get the sequentially midpoint conves and sequentially midpoint concave set-valued functions.

A relationship between our generalized convexity concept introduced for set-valued functions and some convexity concepts known hitherto merely for single-valued functions has also to be mentioned here. To this end, let $K$$K$KK$K$ be a convex cone in $Y$$Y$YY$Y$, i.e a nonempty convex set closed under multiplication by positive scalars. If $f:M\to Y$$f:M\to Y$f:M rarr Yf: M \rightarrow Y$f:M\to Y$ is a function satisfying

whenever $a\in A$$a\in A$a in Aa \in A$a\in A$ and $x,y\in M$$x,y\in M$x,y in Mx, y \in M$x,y\in M$, then $F:M\to {\mathcal{P}}_0(Y)$$F:M\to {\mathcal{P}}_0(Y)$F:M rarrP_(0)(Y)F: M \rightarrow \mathscr{P}_{0}(Y)$F:M\to {\mathcal{P}}_0(Y)$, defined by $F(x)=f(x)++K$$F(x)=f(x)++K$F(x)=f(x)++KF(x)=f(x)+ +K$F(x)=f(x)++K$, is an $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-convex set-valued function. In particular, when $Y=\mathbf{R}$$Y=\mathbf{R}$Y=RY=\mathbf{R}$Y=\mathbf{R}$ and $K={\mathbf{R}}_{+}$$K={\mathbf{R}}_{+}$K=R_(+)K=\mathbf{R}_{+}$K={\mathbf{R}}_{+}$, then (1.1) reduces to an inequality which obviously yields (for suitable $A$$A$AA$A$ and $s$$s$ss$s$ ) diverse convexity concepts for real-valued functions used in optimization and functional analysis.

The goal of the present paper is to characterize the continuity of $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-convex and $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-concave set-valued functions. The results we shall prove reveal, for each of these classes of functions, the connection between continuity, on the one hand, and lower semi-continuity;.upper semicontinuity, local boundedness, uniform boundedness, on the other hand. They highlight the usefulness of the two types of set-valued functions introduced here. In a very general setting both necessary and sufficient conditions are obtained for continuity which have been missing even for those particular classes of functions which suggested the general concept of convexity and concavity, respectively. Another feature of these results is that they are a set-valued counterpart to well-known theorems concerning the continuity of convex and concave real-valued functions. By specializing our results, some sufficient conditions for continuity, recently given by K. Nikodem [3], [4], can be obtained.

To make our paper self-contained we mention in this section all the notions concerning set-valued functions we shall need.

Let $X$$X$XX$X$ and $Y$$Y$YY$Y$ be real topological linear spaces, let $M$$M$MM$M$ be a nonempty subset of $X$$X$XX$X$, and let $F$$F$FF$F$ be a function from $M$$M$MM$M$ to $P_0(Y)$$P_0(Y)$P_(0)(Y)P_{0}(Y)$P_0(Y)$. For simplicity we shall write

for any subset $T$$T$TT$T$ of $M$$M$MM$M$.
If $x_0$$x_0$x_(0)x_{0}$x_0$ is a point in $M$$M$MM$M$, we say that $F$$F$FF$F$ is:
(i) lower semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$ if for every neighbourhood $V$$V$VV$V$ of the origin of $Y$$Y$YY$Y$ there exists a neighbourhood $U$$U$UU$U$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F\left(x_0\right)\subset F(x)++V$$F\left(x_0\right)\subset F(x)++V$F(x_(0))sub F(x)++VF\left(x_{0}\right) \subset F(x)+ +V$F\left(x_0\right)\subset F(x)++V$ for all $x\in U\cap M$$x\in U\cap M$x in U nn Mx \in U \cap M$x\in U\cap M$;
(ii) upper semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$ if for every neighbourhood $V$$V$VV$V$ of the origin of $Y$$Y$YY$Y$ there exists a neighbourhood $U$$U$UU$U$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F(x)\subset F\left(x_0\right)++V$$F(x)\subset F\left(x_0\right)++V$F(x)sub F(x_(0))++VF(x) \subset F\left(x_{0}\right)+ +V$F(x)\subset F\left(x_0\right)++V$ for all $x\in U\cap M$$x\in U\cap M$x in U nn Mx \in U \cap M$x\in U\cap M$;
(iii) continuous at $x_0$$x_0$x_(0)x_{0}$x_0$ if $F$$F$FF$F$ is both lower semicontinuous and upper semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$;
(iv) locally prebounded at $x_0$$x_0$x_(0)x_{0}$x_0$ if for every neighbourhood $V$$V$VV$V$ of the origin of $Y$$Y$YY$Y$ there exist a positive integer $n$$n$nn$n$ and a neighbourhood $U$$U$UU$U$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F(x)\cap nV\ne \mathrm{\varnothing }$$F(x)\cap nV\ne \mathrm{\varnothing }$F(x)nn nV!=O/F(x) \cap n V \neq \emptyset$F(x)\cap nV\ne \mathrm{\varnothing }$ for all $x\in U\cap M$$x\in U\cap M$x in U nn Mx \in U \cap M$x\in U\cap M$;
(v) locally bounded at $x_0$$x_0$x_(0)x_{0}$x_0$ if for every neighbourhood $V$$V$VV$V$ of the origin of $Y$$Y$YY$Y$ there exist a positive integer $n$$n$nn$n$ and a neigbourhood $U$$U$UU$U$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F(U\cap M)\subset nV$$F(U\cap M)\subset nV$F(U nn M)sub nVF(U \cap M) \subset n V$F(U\cap M)\subset nV$;
(vi) bounded at $x_0$$x_0$x_(0)x_{0}$x_0$ if the set $F\left(x_0\right)$$F\left(x_0\right)$F(x_(0))F\left(x_{0}\right)$F\left(x_0\right)$ is bounded, i.e. if for every neighbourhood $V$$V$VV$V$ of the origin of $Y$$Y$YY$Y$ there exists a positive integer $n$$n$nn$n$ such that $F\left(x_0\right)\subset n\mathrm{\nabla }$$F\left(x_0\right)\subset n\mathrm{\nabla }$F(x_(0))sub n gradF\left(x_{0}\right) \subset n \nabla$F\left(x_0\right)\subset n\mathrm{\nabla }$.

The function $F$$F$FF$F$ is called :
(j) lower semicontinuous (resp. upper semicontinuous, continuous, locally prebounded, locally bounded) on $M$$M$MM$M$ if it is lower semicontinuous (resp. upper semicontinuous, continuous, locally prebounded, locally bounded) at each point of $M$$M$MM$M$;
(jj) pointwise bounded on $M$$M$MM$M$ if it is bounded at each point of $M$$M$MM$M$;
(jjj) uniformly bounded on $M$$M$MM$M$ if for every neighbourhood $V$$V$VV$V$ of the origin of $Y$$Y$YY$Y$ there exist a positive integer $n$$n$nn$n$ and a nonempty open subset $T$$T$TT$T$ of $M$$M$MM$M$ such that $F(T)\subset nV$$F(T)\subset nV$F(T)sub nVF(T) \subset n V$F(T)\subset nV$.

We note that the above-mentioned concepts of lower semicontinuity, upper semicontinuity and continuity of a set-valued function are well known, local preboundedness seems to be used here for the first time, while the other above-introduced concepts have been inspired by concepts used in functional analysis for families of functions.

Lemma 2.1. Let $X$$X$XX$X$ and $Y$$Y$YY$Y$ be real topological linear spaces, let $M$$M$MM$M$ be a nozempty subset of $X$$X$XX$X$, let $x_0$$x_0$x_(0)x_{0}$x_0$ be a point in $M$$M$MM$M$, and let $F:M\to {\mathcal{P}}_0(Y)$$F:M\to {\mathcal{P}}_0(Y)$F:M rarrP_(0)(Y)F: M \rightarrow \mathscr{P}_{0}(Y)$F:M\to {\mathcal{P}}_0(Y)$ be a function which is bounded at $x_0$$x_0$x_(0)x_{0}$x_0$. Then the following assertions are true:
$1^{\circ }$$1^{\circ }$1^(@)1^{\circ}$1^{\circ }$ If $F$$F$FF$F$ is lower semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$, then it is locally prebounded at $x_0$$x_0$x_(0)x_{0}$x_0$.
$2^{\circ }$$2^{\circ }$2^(@)2^{\circ}$2^{\circ }$ If: $F$$F$FF$F$ is upper semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$, then it is locally bounded at $x_0$$x_0$x_(0)x_{0}$x_0$.

Proof. $1^{\circ }$$1^{\circ }$1^(@)1^{\circ}$1^{\circ }$ Let $V$$V$VV$V$ be any neighbourhood of the origin of $Y$$Y$YY$Y$. Choose a neighbourhood $V_0$$V_0$V_(0)V_{0}$V_0$ of the origin of $Y$$Y$YY$Y$ such that $V_0-V_0\subset V$$V_0-V_0\subset V$V_(0)-V_(0)sub VV_{0}-V_{0} \subset V$V_0-V_0\subset V$. Since $F$$F$FF$F$ is bounded at $x_0$$x_0$x_(0)x_{0}$x_0$, there exists a positive integer $n$$n$nn$n$ such that $F\left(x_0\right)\subset n{V}_0$$F\left(x_0\right)\subset n{V}_0$F(x_(0))sub nV_(0)F\left(x_{0}\right) \subset n V_{0}$F\left(x_0\right)\subset n{V}_0$. Taking now into consideration that $F$$F$FF$F$ is lower semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$, we can conclude that there exists a neighbourhood $U$$U$UU$U$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F\left(x_0\right)\subset \subset F(x)+n{V}_0$$F\left(x_0\right)\subset \subset F(x)+n{V}_0$F(x_(0))⊂⊂F(x)+nV_(0)F\left(x_{0}\right) \subset \subset F(x)+n V_{0}$F\left(x_0\right)\subset \subset F(x)+n{V}_0$ for all $x\in U\cap M$$x\in U\cap M$x in U nn Mx \in U \cap M$x\in U\cap M$. Thus we have

for all $x\in U\cap M$$x\in U\cap M$x in U nn Mx \in U \cap M$x\in U\cap M$, and hence $F(x)\cap nV\ne \varnothing$$F(x)\cap nV\ne \varnothing$F(x)nn nV!=O/F(x) \cap n V \neq \varnothing$F(x)\cap nV\ne \varnothing$ for all $x\in U\cap M$$x\in U\cap M$x in U nn Mx \in U \cap M$x\in U\cap M$. Consequently, $F$$F$FF$F$ is locally prebounded at $x_0$$x_0$x_(0)x_{0}$x_0$.
$2^{\circ }$$2^{\circ }$2^(@)2^{\circ}$2^{\circ }$. Let $V$$V$VV$V$ be any neighbourhood of the origin of $Y$$Y$YY$Y$. Choose a neighbourhood $V_0$$V_0$V_(0)V_{0}$V_0$ of the origin of $Y$$Y$YY$Y$ such that $V_0+V_0\subset V$$V_0+V_0\subset V$V_(0)+V_(0)sub VV_{0}+V_{0} \subset V$V_0+V_0\subset V$. Since $F\left(x_0\right)$$F\left(x_0\right)$F(x_(0))F\left(x_{0}\right)$F\left(x_0\right)$ is bounded, there exists a positive integer $n$$n$nn$n$ such that $F\left(x_0\right)\subset n{V}_0$$F\left(x_0\right)\subset n{V}_0$F(x_(0))sub nV_(0)F\left(x_{0}\right) \subset n V_{0}$F\left(x_0\right)\subset n{V}_0$. In view of the upper semicontinuity of $F$$F$FF$F$ at $x_0$$x_0$x_(0)x_{0}$x_0$ there exists a neighbourhood $U$$U$UU$U$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F(U\cap M)\subset F\left(x_0\right)+n{V}_0$$F(U\cap M)\subset F\left(x_0\right)+n{V}_0$F(U nn M)sub F(x_(0))+nV_(0)F(U \cap M) \subset F\left(x_{0}\right)+n V_{0}$F(U\cap M)\subset F\left(x_0\right)+n{V}_0$. This inclusion implies $F(U\cap M)\subset n{V}_0+n{V}_0\subset nV$$F(U\cap M)\subset n{V}_0+n{V}_0\subset nV$F(U nn M)sub nV_(0)+nV_(0)sub nVF(U \cap M) \subset n V_{0}+n V_{0} \subset n V$F(U\cap M)\subset n{V}_0+n{V}_0\subset nV$. Hence $F$$F$FF$F$ is locally bounded at $x_0$$x_0$x_(0)x_{0}$x_0$.

First we investigate the continuity of an $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-convex set-valued function at an interior point of its domain.

Theorem 3.1. If $X$$X$XX$X$ and $Y$$Y$YY$Y$ are real topological linear spaces, $M$$M$MM$M$ a convex subset of $X,x_0$$X,x_0$X,x_(0)X, x_{0}$X,x_0$ an interior point of $M$$M$MM$M$, and $F:M\to {\mathcal{P}}_0(Y)$$F:M\to {\mathcal{P}}_0(Y)$F:M rarrP_(0)(Y)F: M \rightarrow \mathscr{P}_{0}(Y)$F:M\to {\mathcal{P}}_0(Y)$ an $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$ -
-convex function, then the following statements are equivalent:
$1^{\circ }F$$1^{\circ }F$1^(@)F1^{\circ} F$1^{\circ }F$ is both continuous and bounded at $x_0$$x_0$x_(0)x_{0}$x_0$.
$2^{\circ }{F}^1$$2^{\circ }{F}^1$2^(@)F^(1)2^{\circ} F^{1}$2^{\circ }{F}^1$ is both upper semicontinuous and bounded at $c_0$$c_0$c_(0)c_{0}$c_0$.
$3^{\circ }F$$3^{\circ }F$3^(@)F3^{\circ} F$3^{\circ }F$ is locally bounded at $x_0$$x_0$x_(0)x_{0}$x_0$.
$4^{\circ }F$$4^{\circ }F$4^(@)F4^{\circ} F$4^{\circ }F$ is both locally prebounded and bounded at $x_0$$x_0$x_(0)x_{0}$x_0$.
$5^{\circ }F$$5^{\circ }F$5^(@)F5^{\circ} F$5^{\circ }F$ is both lower semicontinuous and bounded at $x_0$$x_0$x_(0)x_{0}$x_0$.
Proof. The implications $1^{\circ }\Rightarrow 2^{\circ }$$1^{\circ }\Rightarrow 2^{\circ }$1^(@)=>2^(@)1^{\circ} \Rightarrow 2^{\circ}$1^{\circ }\Rightarrow 2^{\circ }$ and $3^{\circ }\Rightarrow 4^{\circ }$$3^{\circ }\Rightarrow 4^{\circ }$3^(@)=>4^(@)3^{\circ} \Rightarrow 4^{\circ}$3^{\circ }\Rightarrow 4^{\circ }$ are obvious, while the implication $2^{\circ }\Rightarrow 3^{\circ }$$2^{\circ }\Rightarrow 3^{\circ }$2^(@)=>3^(@)2^{\circ} \Rightarrow 3^{\circ}$2^{\circ }\Rightarrow 3^{\circ }$ has been stated in Lemma 2.1.

Suppose now that $4^{\circ }$$4^{\circ }$4^(@)4^{\circ}$4^{\circ }$ is true. Under this assumption $F^{\prime }$$F^{\prime }$F^(')F^{\prime}$F^{\prime }$ is lower semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$. Indeed, let $V$$V$VV$V$ be a neighbourhood of the origin of $Y$$Y$YY$Y$. Choose a balanced neighbourhood $V_0$$V_0$V_(0)V_{0}$V_0$ of the origin of $Y$$Y$YY$Y$ such that $V_0++V_0\subset V$$V_0++V_0\subset V$V_(0)++V_(0)sub VV_{0}+ +V_{0} \subset V$V_0++V_0\subset V$. Since $F$$F$FF$F$ is bounded at $x_0$$x_0$x_(0)x_{0}$x_0$, there exists a positive integer $m$$m$mm$m$ such that $F\left(x_0\right)\subset m{V}_0$$F\left(x_0\right)\subset m{V}_0$F(x_(0))sub mV_(0)F\left(x_{0}\right) \subset m V_{0}$F\left(x_0\right)\subset m{V}_0$, and since $F$$F$FF$F$ is locally prebounded at $x_0$$x_0$x_(0)x_{0}$x_0$, there exist a positive integer $n$$n$nn$n$ and a neighbourhood $U_0$$U_0$U_(0)U_{0}$U_0$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F(x)\cap nn{V}_0\ne \mathrm{\varnothing }$$F(x)\cap nn{V}_0\ne \mathrm{\varnothing }$F(x)nn nnV_(0)!=O/F(x) \cap n n V_{0} \neq \emptyset$F(x)\cap nn{V}_0\ne \mathrm{\varnothing }$ for all $x\in U_0\cap M$$x\in U_0\cap M$x inU_(0)nn Mx \in U_{0} \cap M$x\in U_0\cap M$. Take now, on the one hand, a balanced neighbourhood $W$$W$WW$W$ of the origin of $X$$X$XX$X$ such that $x_0+W\subset U_0\cap M$$x_0+W\subset U_0\cap M$x_(0)+W subU_(0)nn Mx_{0}+W \subset U_{0} \cap M$x_0+W\subset U_0\cap M$, and, on the other hand, a number $a\in A$$a\in A$a in Aa \in A$a\in A$ which satisfies the following inequalities:

Of course, $U=x_0+aW$$U=x_0+aW$U=x_(0)+aWU=x_{0}+a W$U=x_0+aW$ is a neighbourhood of $x_0$$x_0$x_(0)x_{0}$x_0$ contained in $U_0\cap M$$U_0\cap M$U_(0)nn MU_{0} \cap M$U_0\cap M$. We claim that

Let $x$$x$xx$x$ be a point in $U$$U$UU$U$. Then there exists $y\in W$$y\in W$y in Wy \in W$y\in W$ such that $x=x_0+$$x=x_0+$x=x_(0)+x=x_{0}+$x=x_0+$ + ay. From $T(x_0+y)\cap n{V}_0\ne \varnothing$$T\left(x_0+y\right)\cap n{V}_0\ne \varnothing$T(x_(0)+y)nn nV_(0)!=O/T\left(x_{0}+y\right) \cap n V_{0} \neq \varnothing$T(x_0+y)\cap n{V}_0\ne \varnothing$ it follows that $0\in F(x_0+y)-n{V}_0$$0\in F\left(x_0+y\right)-n{V}_0$0in F(x_(0)+y)-nV_(0)0 \in F\left(x_{0}+y\right)-n V_{0}$0\in F(x_0+y)-n{V}_0$. Consequently, we have

Taking into account this result as well as the $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-convexity of $F$$F$FF$F$, we obtain

Hence (3.1) is proved. Therefore $F$$F$FF$F$ is lower semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$. So the implication $4^{\circ }\Rightarrow 5^{\circ }$$4^{\circ }\Rightarrow 5^{\circ }$4^(@)=>5^(@)4^{\circ} \Rightarrow 5^{\circ}$4^{\circ }\Rightarrow 5^{\circ }$ is stated.

To complete the proof we show now that $5^{\circ }$$5^{\circ }$5^(@)5^{\circ}$5^{\circ }$ implies $1^{\circ }$$1^{\circ }$1^(@)1^{\circ}$1^{\circ }$. Let $V$$V$VV$V$ be any neighbourhood of the origin of $Y$$Y$YY$Y$. Choose a balanced neighbourhood $V_0$$V_0$V_(0)V_{0}$V_0$ of the origin of $Y$$Y$YY$Y$ such that $V_0+V_0+V_0\subset V$$V_0+V_0+V_0\subset V$V_(0)+V_(0)+V_(0)sub VV_{0}+V_{0}+V_{0} \subset V$V_0+V_0+V_0\subset V$. Since $F^{\prime }$$F^{\prime }$F^(')F^{\prime}$F^{\prime }$ is bounded at $x_0$$x_0$x_(0)x_{0}$x_0$, there exists a positive integer $n$$n$nn$n$ such that $F^{\prime }\left(x_0\right)\subset n{V}_0$$F^{\prime }\left(x_0\right)\subset n{V}_0$F^(')(x_(0))sub nV_(0)F^{\prime}\left(x_{0}\right) \subset n V_{0}$F^{\prime }\left(x_0\right)\subset n{V}_0$. On the other hand, since $F$$F$FF$F$ is lower semicontinuous at $x_0$$x_0$x_(0)x_{0}$x_0$, there exists for $V_0$$V_0$V_(0)V_{0}$V_0$ a neighbourhood $U_0$$U_0$U_(0)U_{0}$U_0$ of $x_0$$x_0$x_(0)x_{0}$x_0$ such that $F\left(x_0\right)\subset F(x)+V_0$$F\left(x_0\right)\subset F(x)+V_0$F(x_(0))sub F(x)+V_(0)F\left(x_{0}\right) \subset F(x)+V_{0}$F\left(x_0\right)\subset F(x)+V_0$ for all $x\in U_0\cap M$$x\in U_0\cap M$x inU_(0)nn Mx \in U_{0} \cap M$x\in U_0\cap M$.

Take now a balanced neighbourhood $W$$W$WW$W$ of the origin of $X$$X$XX$X$ such that $x_0+W\subset U_0\cap M$$x_0+W\subset U_0\cap M$x_(0)+W subU_(0)nn Mx_{0}+W \subset U_{0} \cap M$x_0+W\subset U_0\cap M$. Also select a number $a\in A$$a\in A$a in Aa \in A$a\in A$ which satisfies the following inequalities:

The set

is a nieghbourhood of $x_0$$x_0$x_(0)x_{0}$x_0$ contained in $U_0\cap M$$U_0\cap M$U_(0)nn MU_{0} \cap M$U_0\cap M$. Furthermore, we have

Indeed, if $x$$x$xx$x$ is in $U$$U$UU$U$, it can expressed in the form

where $y\in W$$y\in W$y in Wy \in W$y\in W$. Since $F\left(x_0\right)\subset F(x_0+y)+V_0$$F\left(x_0\right)\subset F\left(x_0+y\right)+V_0$F(x_(0))sub F(x_(0)+y)+V_(0)F\left(x_{0}\right) \subset F\left(x_{0}+y\right)+V_{0}$F\left(x_0\right)\subset F(x_0+y)+V_0$ and $x_0=(1-a)x+a(x_0+y)$$x_0=(1-a)x+a\left(x_0+y\right)$x_(0)=(1-a)x+a(x_(0)+y)x_{0}=(1-a) x+a\left(x_{0}+y\right)$x_0=(1-a)x+a(x_0+y)$, it follows by the $(A,s)$$(A,s)$(A,s)(A, s)$(A,s)$-convexity of $F^{\prime }$$F^{\prime }$F^(')F^{\prime}$F^{\prime }$ that
$(1-a{)}^{s}F(x)+a^{s}F\left(x_0\right)\subset (1-a{)}^{s}F(x)+a^{s}F(x_0+y)+a^s{V}_0\subset F\left(x_0\right)+a^s{V}_0$$(1-a{)}^{s}F(x)+a^{s}F\left(x_0\right)\subset (1-a{)}^{s}F(x)+a^{s}F\left(x_0+y\right)+a^s{V}_0\subset F\left(x_0\right)+a^s{V}_0$(1-a)^(s)F(x)+a^(s)F(x_(0))sub(1-a)^(s)F(x)+a^(s)F(x_(0)+y)+a^(s)V_(0)sub F(x_(0))+a^(s)V_(0)(1-a)^{s} F(x)+a^{s} F\left(x_{0}\right) \subset(1-a)^{s} F(x)+a^{s} F\left(x_{0}+y\right)+a^{s} V_{0} \subset F\left(x_{0}\right)+a^{s} V_{0}$(1-a{)}^{s}F(x)+a^{s}F\left(x_0\right)\subset (1-a{)}^{s}F(x)+a^{s}F(x_0+y)+a^s{V}_0\subset F\left(x_0\right)+a^s{V}_0$, and henoe
$F(x)+{\left(\frac{a}{1-a}\right)}^{s}F\left(x_0\right)\subset {\left(\frac{1}{1-a}\right)}^{s}F\left(x_0\right)+{\left(\frac{a}{1-a}\right)}^s{V}_0\subset {\left(\frac{1}{1-a}\right)}^{s}F\left(x_0\right)+V_0$$F(x)+{\left(\frac{a}{1-a}\right)}^{s}F\left(x_0\right)\subset {\left(\frac{1}{1-a}\right)}^{s}F\left(x_0\right)+{\left(\frac{a}{1-a}\right)}^s{V}_0\subset {\left(\frac{1}{1-a}\right)}^{s}F\left(x_0\right)+V_0$F(x)+((a)/(1-a))^(s)F(x_(0))sub((1)/(1-a))^(s)F(x_(0))+((a)/(1-a))^(s)V_(0)sub((1)/(1-a))^(s)F(x_(0))+V_(0)F(x)+\left(\frac{a}{1-a}\right)^{s} F\left(x_{0}\right) \subset\left(\frac{1}{1-a}\right)^{s} F\left(x_{0}\right)+\left(\frac{a}{1-a}\right)^{s} V_{0} \subset\left(\frac{1}{1-a}\right)^{s} F\left(x_{0}\right)+V_{0}$F(x)+{\left(\frac{a}{1-a}\right)}^{s}F\left(x_0\right)\subset {\left(\frac{1}{1-a}\right)}^{s}F\left(x_0\right)+{\left(\frac{a}{1-a}\right)}^s{V}_0\subset {\left(\frac{1}{1-a}\right)}^{s}F\left(x_0\right)+V_0$.
This result implies