RETRACTED ARTICLE: Normal families of solutions for modified equilibrium equations and their applications

Abstract

Using boundary behaviors of solutions for certain Laplace equation proved by Yan and Ychussie (Adv. Difference Equ. 2015:226, 2015) and applying a new method to dispose of the impulsive term with finite mass subject presented by Shi and Liao (J. Inequal. Appl. 2015:363, 2015) from another point of view, we prove that there exists a supra-open in $(X,\tau)$ for each $V \in\sigma$ in which the modified equilibrium equation has normal families of solutions. Moreover, we establish a new expression of a harmonic multifunction for the above equation. As applications, we not only prove the existence of normal families of solutions for modified equilibrium equations but also obtain several characterizations and fundamental properties of these new classes of superharmonic multifunctions.

1 Introduction

As in [2], the modified equilibrium equations for a self-gravitating fluid rotating about the $x_{3}$ axis with prescribed velocity $\Omega(r)$ can be defined as follows:

$$ \textstyle\begin{cases} \nabla P = \rho\nabla ( -\Phi+\int_{0}^{r} s\Omega^{2} (s)\,ds ), \\ \Delta\Phi=4\pi g \rho. \end{cases} $$

(1.1)

Here ρ, g, and Φ denote the density, gravitational constant, and gravitational potential, respectively, P is the pressure of the fluid at a point $x\in {\mathbb{R}}^{3}$, and $r=\sqrt{x_{1}^{2} +x_{2}^{2}}$. We want to find axisymmetric equilibria and therefore always assume that $\rho(x)=\rho(r,x_{3})$.

For a density ρ, from (1.1) we can obtain the induced potential [3]

$$ \Phi_{\rho}(x)= -g \int\frac{\rho(y)}{|x-y|}\,dy. $$

(1.2)

In the study of this model, Yan and Ychussie [1] proved the existence of the modified Laplace solution if the angular velocity satisfies certain decay conditions. For constant angular velocity, Huang et al. [4] have obtained that there exists an equilibrium solution if the angular velocity is less than certain constant and that there is no equilibrium for large velocity. The existence and uniqueness of the generalized solutions for the boundary value problems in elasticity of dipolar materials with voids were obtained in [5]. In particular, Marin and Lupu [6] solved the unknowns of the displacement and microrotation on harmonic vibrations in thermoelasticity of micropolar bodies. Similar procedures were used by Marin et al. [7, 8] in dealing with thermoelasticity of micropolar bodies. Many important physical phenomena on the engineering and science fields are frequently modeled by nonlinear differential equations. Such equations are often difficult or impossible to solve analytically. Nevertheless, analytical approximate methods to obtain approximate solutions have gained importance in recent years [9]. Recently, Ji et al. [3] talked about the exact numbers for solutions of modified equilibrium equations.

In 1977, Husain [10] initiated the concept of supra-open sets, which is considered as a wider class of some known types of near-open sets. In 1983, Mashhour et al. [11] defined the concept of S-continuity for a single-valued function $f:(X,\tau)\rightarrow (Y,\sigma)$. Many topological properties of the above-mentioned concepts and others have been established in [3, 12]. The purpose of this paper is to present the upper (lower) supra-continuous harmonic multifunction as a generalization of upper (lower) semi-continuous harmonic multifunctions in the sense of Berge [13], the upper (lower) quasi-continuous and the upper (lower) precontinuous harmonic multifunctions defined by Popa [14], and also upper (lower) α-continuous and upper (lower) β-continuous harmonic multifunctions defined by Wine [15]. Moreover, characterization of these new harmonic multifunctions by many their properties has also been established.

2 Preliminaries

The topological spaces, or simply spaces, used here will be given by $(X, \tau)$ and $(Y, \sigma)$. By $\tau\mbox{-}\operatorname{cl}(W)$ and $\tau\mbox{-}\operatorname{int}(W)$ we denote the closure and interior of a subset W of X with respect to topology τ. In $(X, \tau)$, a class $\tau^{*}\subseteq P(X)$ is called a supra-topology on X if $X \in\tau^{*}$ and $\tau^{*}$ is closed under arbitrary union [10]. Then $(X, \tau^{*})$ is called a supra-topological space or simply supra-space. Each member of τ is supra-open, and its complement is supra-closed [11]. In $(X, \tau^{*})$, the supra-closure, the supra-interior, and supra-frontier of any $A\subseteq X$ are denoted by $\mbox{supra-}\operatorname{cl}(A)$, $\mbox{supra-}\operatorname{int}(A)$, and $\mbox{supra-}\operatorname{fr}(A)$, respectively, which are defined in [11] likewise the corresponding ordinary ones. We define

$$ \tau^{*}(x)=\bigl\{ W\subseteq X: W\in\tau^{*}, x \in W\bigr\} $$

(2.1)

for any $x \in X$.

In $(X,\tau)$, $A \subseteq X$ is called semiopen [11] if there exists $U\in\tau$ such that $U\subseteq A \subseteq\tau\mbox{-}\operatorname{cl}(U)$, whereas A is preopen [14] if $A\subseteq\tau\mbox{-}\operatorname{int}(\tau\mbox{-}\operatorname{cl}(A))$. The families of all semiopen and preopen sets in $(X,\tau)$ are denoted by $\operatorname{SO}(X,\tau )$ and $\operatorname{PO}(X,\tau)$, respectively. Moreover,

$$\tau^{\alpha}=\operatorname{SO}(X,\tau)\cap \operatorname{PO}(X,\tau) $$

and

$$\beta O(X,\tau)\supset \operatorname{SO}(X,\tau) \cup \operatorname{PO}(X,\tau). $$

Sets $A\in \tau^{\alpha}$ and $A \in\beta O(X,\tau)$ are called α-sets [16] and β-open sets [17], respectively. A single-valued function $f:(X,\tau)\rightarrow(Y,\sigma)$ is called S-continuous [3] if the inverse image of each open set in $(Y,\sigma)$ is $\tau^{*}$-supra open in $(X,\tau)$. For a harmonic multifunction $F:(X,\tau)\rightarrow(Y,\sigma)$, the upper and lower inverses of any $B\subseteq Y$ are given by

$$ F^{+} (B)=\bigl\{ x\in X:F(x)\subseteq B\bigr\} $$

(2.2)

and

$$ F^{-}(B)=\bigl\{ x \in X:F(X)\cap B \ne\phi\bigr\} , $$

(2.3)

respectively. Moreover, $F:(X,\tau)\rightarrow(Y,\sigma)$ is called upper (resp. lower) semicontinuous [13] if for each $V\in\sigma$, $F^{+} (V)\in \tau$ (resp. $F^{-} (V)\in\tau$). If in τ semicontinuity is replaced by $\operatorname{SO}(X,\tau)$, $\tau^{\alpha}$, $\operatorname{PO}(X,\tau)$, or $\beta O(X,\tau)$, then F is upper (lower) quasi-continuous [18], upper (lower) α-continuous [14], upper (lower) precontinuous [4], and upper (lower) β-continuous [2], respectively. A space $(X,\tau)$ is called supra-compact [12] if every supra-open cover of X admits a finite subcover.

3 Supra-continuous harmonic multifunctions

Definition 3.1

A harmonic multifunction $F:(X,\tau )\rightarrow(Y,\sigma)$ is said to be:

• (a) upper supra-continuous at a point $x\in X$ if for each open set V containing $F(x)$, there exists $W \in\tau^{*}(x)$ such that

  $$ F(W)\subseteq V; $$

  (3.1)

• (b) lower supra-continuous at a point $x \in X$ if for each open set V containing $F(x)$, there exists $W \in\tau^{*}(x)$ such that

  $$ F(W)\cap V \neq\phi; $$

  (3.2)

• (c) upper (lower) supra-continuous if F has this property at every point of X.

Any single-valued function $F:(X, \tau)\rightarrow(Y,\sigma)$ can be considered as multivalued if, to any $x \in X$, it assigns the singleton $\{f(x)\}$. Applying the above definitions of both upper and lower supra-continuous harmonic multifunctions to a single-valued function, it is clear that they coincide with the notion of S-continuous functions given by Mashhour et al. [11]. One characterization of the harmonic multifunctions is established in the following result, the proof of which is straightforward and so is omitted.

Remark 3.1

For a harmonic multifunction $F:(X,\tau )\rightarrow(Y,\sigma)$, many properties of upper (lower) semicontinuity [13] (resp. upper (lower) F-continuity [4], upper (lower) quasi-continuity [14], upper (lower) precontinuity [12], and upper (lower) G-continuity [19] can be deduced from the upper (lower) supra-continuity by considering $\tau^{*}= \tau$ (resp. $\tau^{*} = \tau^{\alpha}$, $\tau^{*}= \operatorname{SO}(X,\tau)$, $\tau^{*}= \operatorname{PO}(X,\tau )$, and $\tau^{*}= \beta O(X,\tau)$).

Proposition 3.1

A harmonic multifunction $F:(X,\tau )\rightarrow(Y,\sigma)$is upper (resp. lower) supra-continuous at a point $x\in X$if and only if for $V \in\sigma$with $F(x)\subseteq V$ (resp. $F(x)\cap V\neq\phi$), we have $x\in\mathrm{supra}\mbox{-}\operatorname{int}(F^{+} (V))$ (resp. $x \in\mathrm{supra}\mbox{-}\operatorname{int}(F^{-} (V))$).

Lemma 3.1

For any $A \in(X,\tau)$, we have

$$ \tau\mbox{-}\operatorname{int}(A)\subseteq\mathrm{supra}\mbox{-} \operatorname{int} (A)\subseteq A \subseteq\mathrm{supra}\mbox{-}\operatorname{cl}(A) \subseteq\tau\mbox{-}\operatorname{cl}(A). $$

(3.3)

Theorem 3.1

The following statements are equivalent for a harmonic multifunction $F:(X,\tau)\rightarrow(Y,\sigma)$:

• (i) Fis upper supra-continuous.

• (ii) For each $x\in X$and each $V \in \sigma(F(x))$, we have $F^{+} (V)\in\tau^{*}(x)$.

• (iii) For each $x \in X$and each $V \in \sigma(F(x))$, there exists $W\in\tau^{*}$such that $F(W)\subseteq V$.

• (iv) $F^{+} (V)\in\tau^{*}$for every $V\in\sigma$.

• (v) $F^{-} (K)$is supra-closed for every closed set $K\subseteq Y$.

• (vi) $\mathrm{supra}\mbox{-}\operatorname{cl}(F^{-} (B))\subseteq F^{-} (\tau\mbox{-}\operatorname{cl}(B))$for every $B\subseteq Y$.

• (vii) $F^{+}(\tau\mbox{-}\operatorname{int}(B))\subseteq\mathrm{supra}\mbox{-}\operatorname{int}(F^{+} (B))$for every $B\subseteq Y$.

• (viii) $\mathrm{supra}\mbox{-}\operatorname{fr}(F^{-}(B))\subseteq F^{-}(\operatorname{fr}(B))$for every $B\subseteq Y$.

• (ix) $F:(X, \tau^{*}) \rightarrow (Y,\sigma)$is upper semicontinuous.

Proof

(i) ⇔ (ii) and (i) ⇒ (iv) follow from Proposition 3.1.

(ii) ⇔ (iii) is obvious since an arbitrary union of supra-open sets is supra-open.

(iv) = (v). Let K be closed in Y. The result holds since

$$ F^{+}(Y\setminus K)=X\setminus F^{-}(K). $$

(3.4)

(v) ⇒ (vi) follows by putting $K = \sigma\mbox{-}\operatorname{cl}(B)$ and applying Lemma 3.1.

(vi) ⇒ (vii). Let $B\Rightarrow Y$. Then $\sigma\mbox{-}\operatorname{int}(B) \in \sigma$, and so $Y \setminus \sigma\mbox{-}\operatorname{int}(B)$ is closed in $(Y,\sigma)$. Therefore by (vi) we get

$$ X\setminus \mbox{supra-}\operatorname{int}\bigl(F^{+} (B)\bigr) \subseteq\mbox{supra-}\operatorname{cl}\bigl(X\setminus F^{+} \bigl(\sigma\mbox{-} \operatorname{int}(B)\bigr)\bigr) $$

(3.5)

and

$$ \mbox{supra-}\operatorname{cl}\bigl(F^{-} \bigl(Y \sigma\mbox{-} \operatorname{int}(B)\bigr)\subseteq F-\bigl(Y\setminus \sigma\mbox{-} \operatorname{int}(B)\bigr)\subseteq X\setminus F^{+} \bigl(\sigma\mbox{-} \operatorname{int}(B)\bigr)\bigr). $$

(3.6)

These imply that

$$ F^{+} \bigl(\sigma\mbox{-}\operatorname{int}(B)\bigr) \subseteq\mbox{supra-}\operatorname{int}\bigl(F^{+}(B)\bigr). $$

(3.7)

(vii) ⇒ (ii). Let $x\in X$ be arbitrary, and let $V\in \sigma(F(x))$. Then

$$ F^{+} (V) \subseteq\mbox{supra-}\operatorname{int}\bigl(F^{+} (V) \bigr). $$

(3.8)

Hence $F^{+} (V) \in\tau^{*}(x)$.

(viii) ⇔ (v). It is clear since supra-frontier and frontier of any set is supra-closed and closed, respectively.

(ix) ⇔ (iv) follows directly. □

Theorem 3.2

For a harmonic multifunction $F:(X,\tau )\rightarrow(Y,\sigma)$, the following statements are equivalent:

• (i) Fis lower supra-continuous.

• (ii) For each $X\in X$and each $V \in\sigma$such that

  $$ F(x)\cap V \neq\phi, $$

  (3.9)

  we have

  $$ F^{-} (V) \in \tau^{*}(x). $$

  (3.10)

• (iii) For each $x\in X$and each $V \in\sigma$with $F(x)\cap V \neq\phi$, there exists $W \in\tau^{*}$such that $F(W)\cap V \neq \phi$.

• (iv) $F^{-} (V)\in\tau^{*}$for every $V \in\sigma$.

• (v) $F^{+} (K)$is supra-closed for every closed set $K\subseteq Y$.

• (vi) $\mathrm{supra}\mbox{-}\operatorname{cl}(F^{+} (B)) \subseteq F^{+} (\sigma\mbox{-}\operatorname{cl}(B))$for any $B\subseteq Y$.

• (vii) $F^{-} (\sigma\mbox{-}\operatorname{int}(B))\subseteq\mathrm{supra}\mbox{-}\operatorname{int}(F^{-} (B))$for any $B\subseteq Y$.

• (viii) $\mathrm{supra}\mbox{-}\operatorname{fr}(F^{+} (B)) \subseteq F^{+} (\operatorname{fr}(B))$for every $B \subseteq Y$.

• (ix) $F:(X, \tau^{*})\rightarrow(Y, \sigma)$is lower semicontinuous.

Proof

The proof is a quite similar to that of Theorem 3.1. Recalling that the net $(\chi_{i})_{(i\in l)}$ supra-converges to $x_{0}$ if, for each $W \in\tau^{*} (x_{O})$, there exists $i_{o} \in I$ such that $x_{i} \in W$ for all $i\ge i_{o}$. □

Theorem 3.3

A harmonic multifunction $F : (X, \tau )\rightarrow(Y,\sigma)$is upper supra-continuous if and only if, for each net $(\chi_{i})_{(i\in l)}$supra-convergent to $x_{o}$and for each $V\in\sigma$with $F(x_{o})\subseteq V$, there is $i_{o} \in I$such that $F(X_{i}) \subseteq V$for all $i \ge i_{o}$.

Proof

Necessity. Let $V\in\sigma$ with $F(x_{o})\subseteq V$. By the upper supra-continuity of F there is $W\in\tau^{*}(X_{O})$ such that $F(W)\subseteq V$. Since by hypothesis a net $(\chi _{i})_{(i\in l)}$ is supra-convergent to $x_{o}$ and $W \in\tau^{*}(x_{o})$, there is $i_{o} \in I$ such that $x_{i} \in W$ for all $i > i_{o}$, and then $F(X_{i}) \subseteq V$ for all $i > i_{o}$. Sufficiency. Assume the converse, that is, there is an open set V in Y with $F(x_{o} )\subseteq V$ such that for each $W\in\tau^{*}$, $F(W)\nsubseteq V$, that is, there is $x_{w} \in W$ such that $F(x_{w}) \nsubseteq V$. Then all $x_{w}$ form a net in X with directed set W of $\tau^{*}(x_{o})$. Clearly, this net is supra-convergent to $x_{o}$. However, $F(x_{w})\nsubseteq V$ for all $W \in\tau^{*}(x_{o})$. This leads to a contradiction, which completes the proof. □

Theorem 3.4

A harmonic multifunction $F : (X,\tau )\rightarrow(Y, \sigma)$is lower supra-continuous if and only if, for each $y_{o} \in F(x_{o})$and for every net $(\chi_{i})_{(i\in l)}$supra-convergent to $x_{o}$, there exist a subset $(Z_{j})_{(j\in J)}$of the net $(\chi_{i})_{(i\in l)}$and a net $(y_{i})_{(j,v)\in J}$inYsuch that $(y_{i})_{(j,v)\in J}$is supra-convergent toyand $y_{j} \in F(z_{j})$.

Proof

For the necessity, suppose that F is lower supra-continuous, $(\chi_{i})_{(i\in l)}$ is a net supra-convergent to $x_{o}$, $y \in F(x_{o})$, and $V \sigma(y)$. So we have $F(x_{o}) \cap V \ne\phi$, and by lower supra-continuity of F at $x_{o}$ there is a supra-open set $W \subseteq X$ containing $x_{o}$ such that $W \subseteq F^{-}(V)$. Since the net $(\chi_{i})_{(i\in l)}$ is supra-convergent to $x_{0}$, for this W, there is $i_{o} \in I$ such that, for $i > i_{o}$, we have $x_{i} \in W$, and therefore $x_{i} \in F^{-}(V)$. For each $V\in\sigma(y)$, define the sets

$$ I_{v} =\bigl\{ i_{o} \in I:i >i_{o} \Rightarrow x_{i} \in F^{-}(V)\bigr\} $$

(3.11)

and

$$ J =\bigl\{ (i,V):V\in D(y),i \in I_{v}\bigr\} . $$

(3.12)

We write $(i',V') \ge(i,V)$ if and only if $i' > i$ and $V' \subseteq V$. Also, define $\zeta: J \rightarrow I$ by $\zeta((j,V))= j$. Then ζ, increasing and cofinal in I, defines a subset of $(\chi_{i})_{(i\in l)}$ denoted by $(z_{i})_{(j,v)\in J}$. On the other hand, for any $(j,V) \in J$, since $j > j_{o}$ implies $x_{j} \in F^{-}(V)$, we have $F(Z_{j})\cap V = F(X_{j}) \cap V\ne\phi$. Pick $y_{j} \in F(Z_{j}) \cap V \ne\phi$. Then the net $(y_{i})_{(j,v)\in J}$ is supra-convergent to y. To see this, let $V_{0} \in\sigma(y)$. Then there is $j_{0} \in I$ with $j_{o} = \zeta( j_{o}, V_{o} )$; $(j_{o}, V_{o}) \in J$ and $y_{jo} \in V$. If $(j,V) > (j_{o},V_{o})$, then $j > j_{o}$ and $V \subseteq V_{o}$. Therefore, $y_{j} \in F(z_{j}) \cap V \subseteq F(x_{j})\cap V \subseteq F(x_{j}) \cap V_{o}$, and so $y_{j}\in V_{o}$. Thus $(y_{i})_{(j,v)\in J}$ is supra-convergent to y, which shows the result. To show the sufficiency, assume the converse, that is, F is not lower supra-continuous at $x_{o}$. Then there exists $V \in\sigma$ such that $F(x_{o}) \cap V\ne\phi$, and for any supra-neighborhood $W \subseteq X$ of $x_{o}$, there is $x_{w} \in W$ for which $F(x_{w}) \cap V = \phi$. Let us consider the net $(\chi_{w})_{W\in \tau^{*}(\chi_{0})}$ which obviously is supra-convergent to $x_{o}$. Suppose $y_{o} \in F(x_{o}) \cap V$. By hypothesis there are a subnet $(z_{k})_{k\in K}$ of $(\chi_{w})_{W\in\tau^{*}(\chi_{0})}$ and $y_{k} \in F(z_{k})$ such that $(y_{k})_{k\in K}$ is supra-convergent to $y_{o}$. As $y_{o} \in V \in\sigma$, there is $k_{0}' \in K$ such that $k>k_{0}'$ implies $y_{k} \in V$. On the other hand, $(z_{k})_{kEK}$ is a subnet of the net $(\chi^{w})_{W\in\tau ^{*}(\chi_{0})}$, and so there is a function $\Omega: K \rightarrow\tau ^{*}(x_{o})$ such that $z_{k}=\chi_{\Omega(k)}$, and for each $W \in \tau^{*}(x_{o})$, there exists $k_{0}'' \in K$ such that $\Omega(k_{0}'') \ge W$. If $k\ge k_{0}''$, then $\Omega(k) \ge\Omega(k_{0}'') \ge W$. Considering $k_{0} \in K$ such that $k_{o} \ge k_{0}'$ and $k_{o} \ge k_{0}''$. Therefore $y_{k} \in V$, and by the meaning of the net $(\chi_{W})_{W\in\tau^{*}(\chi_{0})}$ we have

$$ F(z_{k}) \cap V = F(\chi_{\Omega(K)}) \cap V = \phi. $$

(3.13)

This gives $y_{k} \notin V$, which contradicts the hypothesis, and so the requirement holds. □

Definition 3.2

A subset W of a space $(X, \tau)$ is called supra-regular if, for any $x \in W$ and any $H \in\tau^{*}(x)$, there exists $U \in\tau$ such that

$$ x \in U \subseteq \tau\mbox{-}\operatorname{cl}(U) \subseteq H . $$

(3.14)

Recall that $F: (X, \tau) \rightarrow(Y,\sigma)$ is punctually supra-regular if, for each $X\in X$, $F(x)$ is supra-regular.

Lemma 3.2

In a space $(X,\tau)$, if $W \subseteq X$is supra-regular and contained in a supra-open setH, then there exists $U \in\tau$such that

$$ W \subseteq U \subseteq\tau\mbox{-}\operatorname{cl}(U) \subseteq H . $$

(3.15)

For a harmonic multifunction $F:(X, \tau)\rightarrow(Y,\sigma)$, a harmonic multifunction $\mathrm{supra}\mbox{-}\operatorname{cl}(F):(X, \tau)\rightarrow(Y,\sigma)$is defied as $(\mathrm{supra}\mbox{-}\operatorname{cl} F)(x) = \mathrm{supra}\mbox{-}\operatorname{cl}(F(x))$for each $x \in X$.

Proposition 3.2

For a punctuallyα-paracompact and punctually supra-regular harmonic multifunction $F: (X, \tau) \rightarrow(Y, \sigma)$, we have $(\mathrm{supra}\mbox{-}\operatorname{cl}(F)^{+} (W)) = F^{+} (W)$for each $W\in \sigma^{*}$.

Proof

Let $x \in(\mbox{supra-}\operatorname{cl}(F))^{+}(W)$ for any $W\in \sigma^{*}$. This means $F(x) \subseteq\mbox{supra-}\operatorname{cl}(F(x)) \subseteq W$, which leads to $x \in F^{+} (W)$. Hence one inclusion holds. To show the other, let $X\in F^{+} (W)$, where $W \in \sigma^{*} (x)$. Then $F(x) \subseteq W$, and by hypothesis on F and the fact that $\sigma\subseteq\sigma^{*}$, applying Lemma 3.2, we get that there exists $G \in\sigma$ such that

$$ F(x)\subseteq G\in\sigma\mbox{-}\operatorname{cl}(G)\subseteq W. $$

(3.16)

Therefore, $\mbox{supra-}\operatorname{cl}(F(x)) \subseteq W$, which means that $x \in(\mbox{supra-}\operatorname{cl} F)^{+} (W)$. Hence the equality holds. □

Theorem 3.5

Let $F (X, \tau)\rightarrow(Y, \sigma)$be a punctuallya-paracompact and punctually supra-regular harmonic multifunction. ThenFis upper supra-continuous if and only if $(\mathrm{supra}\mbox{-}\operatorname{cl}F): (X, \tau)\rightarrow(Y, \sigma)$is upper supra-continuous.

Proof

Necessity. Suppose that $V \in\sigma$ and $x \in (\mbox{supra-}\operatorname{cl} F)^{+} (V) = F^{+} (V)$ (see Proposition 3.2). By upper supra-continuity of F there exists $H \in\tau^{*}(x)$ such that $F(H) \subseteq V$. Since $\sigma\in\sigma^{*}$, by Lemma 3.2 and the assumption on F there exists $G \in\sigma$ such that

$$ F(h) \subseteq G \subseteq\sigma-\operatorname{cl}(G) \subseteq W $$

(3.17)

for each $h \in H$.

Hence

$$ \mbox{supra-}\operatorname{cl}\bigl(F(h)\bigr) = (\mbox{supra-} \operatorname{cl} F) (h) \subseteq\mbox{supra-}\operatorname{cl} (G) \subseteq \sigma\mbox{-}\operatorname{cl}(G) \subseteq V $$

(3.18)

for each $h \in H$, which gives that

$$ (\mbox{supra-}\operatorname{cl} F) (H) \subseteq V. $$

(3.19)

Thus $(\mbox{supra-}\operatorname{cl} F)$ is upper supra-continuous.

For sufficiency, assume that $V\in\sigma$ and $X \in F^{+} (V) = (\mbox{supra-}\operatorname{cl} F)^{+} (V)$. By hypothesis on F in this case, there is $H\in\tau^{*}(x)$ such that $(\mbox{supra-}\operatorname{cl} F)(H) \subseteq V$, which obviously gives that $F(H) \subseteq V$. This completes the proof. □

Lemma 3.3

In a space $(X,\tau)$, for any $x \in X$and $A\subseteq X$, $X \in\mathrm{supra}\mbox{-}\operatorname{cl}(A)$if and only if $A\cap W\ne \phi$for each $W\in \tau^{*}(x)$.

Proposition 3.3

For a harmonic multifunction $F: (X, \tau ) \rightarrow(Y, \sigma)$, $(\mathrm{supra}\mbox{-}\operatorname{cl} F)^{-} (W) = F^{-} (W)$for each $W \in \sigma^{*}$.

Proof

Let $x \in(\mbox{supra-}\operatorname{cl} F)^{-} (W)$. Then $W \cap \mbox{supra-}\operatorname{cl}(F(x)) \neq\phi$, where $W\in\sigma^{*}$. So Lemma 3.3 gives $W\cap F(x) \neq\phi$, and hence $x \in F^{-}(W)$. Conversely, let $x \in F^{-}(W)$. Then

$$ \phi\neq F(x)\cap W\subseteq(\mbox{supra-}\operatorname{cl} F)^{-}(x) \cap W, $$

(3.20)

and so

$$ x \in(\mbox{supra-}\operatorname{cl} F)^{-}(W). $$

(3.21)

Hence

$$ x \in(\mbox{supra-}\operatorname{cl} F)^{+} (W), $$

(3.22)

and this completes the proof. □

Theorem 3.6

A harmonic multifunction $F: (X, \tau )\rightarrow(Y, \sigma)$is lower supra-continuous if and only if $(\mathrm{supra}\mbox{-}\operatorname{cl} F): (X, \tau) \rightarrow(Y, \sigma)$is lower supra-continuous.

Proof

This is an immediate consequence of Proposition 3.2 taking into consideration that $\tau\subseteq\tau^{*}$ and (iv) of Theorem 3.2. □

Theorem 3.7

If $F:(X,\tau)\rightarrow(Y, \sigma)$is an upper supra-continuous surjection, then $F(x)$is compact relative toYfor each $x\in X$. If $(X,\tau)$is supra-compact, then $(Y,\sigma )$is compact.

Proof

Let

$$ \{V_{i} : i \in I, V_{i} \in\sigma\} $$

(3.23)

be a cover of Y, and since $F(x)$ is compact relative to Y for each $x \in X$, there exists a finite subset $I_{o}(x)$ of I such that

$$ F(x)\subseteq U\bigl(V_{i}: i \in I_{o} (x) \bigr). $$

(3.24)

The upper supra-continuity of F gives that there exists $W(x) \in\tau^{*}(X,x)$ such that

$$ F\bigl(W(x)\bigr) \subseteq\bigcup\bigl(V_{i}:i \in I_{o} (x)\bigr). $$

(3.25)

Since $(X, \tau)$ is supra-compact, there exist ${x_{1},x_{2}, \ldots,x_{n}}$ such that

$$ X=\bigcup\bigl(W(x_{j}):1 \le j \le n\bigr). $$

(3.26)

So

$$ Y = F(X) = \bigcup\bigl(F\bigl(W(x_{j})\bigr): 1 \le j \le n\bigr)\subseteq\bigcup\bigl(V_{i} : i \in I_{0} (x_{j}), 1 \le j \le n\bigr). $$

(3.27)

Hence $(Y,\sigma)$ is compact. □

4 Supra-continuous harmonic multifunctions and supra-closed graphs

Definition 4.1

A harmonic multifunction $F:(X, \tau )\rightarrow(Y, \sigma)$ is said to have a supra-closed graph if, for each pair $(x,y) \notin G(F)$, there exist $W\in \tau^{*}(X)$ and $H\notin\sigma^{*}(y)$ such that

$$ (W H)\cap G(F) =\phi. $$

(4.1)

A harmonic multifunction $F:(X, \tau) \rightarrow(Y, \sigma)$ is point-closed (supra-closed) if, for each $x \in X$, $F(x)$ is closed (supra-closed) in Y.

Proposition 4.1

A harmonic multifunction $F : (X,\tau )\rightarrow(Y, \sigma)$has a supra-closed graph if and only if, for all $x \in X$and $y \in Y$such that $y \notin F(x)$, there exist two supra-open setsH, Wcontainingxandy, respectively, such that

$$ F(H)\cap W = \phi. $$

(4.2)

Proof

Necessity. Let $x \in X$ and $y \in Y$ with $y \notin F(x)$. Then since F has a supra-closed graph, there are $H\in\tau^{*} (x)$ and $W\in\sigma^{*}$ containing $F(x)$ such that $(H\times W)\cap G(F) = \phi$. This implies that, for every $x \in H$ and $y \in W$, we have $y \notin F(x)$, and so $F(H) \cap W =\phi$.

Sufficiency. Let $(x,y) \notin G(F)$, which means $y \notin F(x)$. Then there are two disjoint supra-open sets H, W containing x and y, respectively, such that $F(H)\cap W = \phi$. This implies that $(H W) \cap G(F) = \phi$, which completes the proof. □

Theorem 4.1

If $F:(X, \tau)\rightarrow(Y,\sigma)$is upper supra-continuous and point-closed harmonic multifunction and $(Y, \sigma)$is regular, then $G(F)$is supra-closed.

Proof

Suppose that

$$ (x,y)\notin G(F). $$

(4.3)

Then $y \notin F(x)$. Since Y is regular, there exists disjoint $V_{i} \in \sigma$ ($i =1,2$) such that $y \in V_{1}$ and

$$ F(x) \subseteq V_{2}. $$

(4.4)

Since F is upper supra-continuous at x, there exists

$$ W\in\tau^{*}(x) $$

(4.5)

such that $F(W)\subseteq V_{2}$. As $V_{1} \cap V_{2} = \phi$, we have

$$ \bigcap_{i=1}^{2} \mbox{supra-}\operatorname{int}(V_{i})\ne\phi, $$

(4.6)

and therefore

$$\begin{aligned}& x \in \mbox{supra-}\operatorname{int}(W) = W, \end{aligned}$$

(4.7)

$$\begin{aligned}& y \in\mbox{supra-}\operatorname{int}(V_{1}), \end{aligned}$$

(4.8)

and

$$ (x, y) \in W \times\mbox{supra-}\operatorname{int} (V_{1} ) \subseteq (X\times Y)\setminus G(F). $$

(4.9)

Thus

$$ (X \times Y) \setminus G(F) \in \tau^{*} (X \times Y), $$

(4.10)

which gives the desired result. □

Definition 4.2

A subset W of a space $(X,\tau)$ is called α-paracompact [15] if, for every open cover v of W in $(X,\tau)$, there exists a locally finite open cover ξ of W that refines v.

Theorem 4.2

Let $F :(X, \tau)\rightarrow(Y, \sigma)$be an upper supra-continuous harmonic multifunction from $(X,\tau)$into a Hausdorff space $(Y,\sigma)$. If $F(x)$isα-paracompact for each $x \in X$, then $G(F)$is supra-closed.

Proof

Let $(x_{o}, y_{o}) \notin G(F)$. Then $y_{o} \notin F(x_{o})$. Since $(Y,\sigma)$ is Hausdorff, then, for each $y \in F(x_{o})$, there exist $V_{y} \in\sigma(y)$ and $V_{y}^{*} \in \sigma(y_{o})$ such that

$$ V_{y} \cap V_{y}^{*} =\phi. $$

(4.11)

So the family $\{V_{y}: y\in F(x_{0})\}$ is an open cover of $F(x_{o})$. Thus, by the α-paracompactness of $F(x_{o})$,there is a locally finite open cover $\{U_{i}:i \in I\}$ that refines $\{V_{y}:y\in F(x_{o})\}$. Therefore there exists $H_{o} \in\sigma (y_{o})$ such that $H_{o}$ intersects only finitely many members $U_{i_{1}},U_{i_{2}},\ldots,U_{i_{n}}$ of h. Choose $y_{1}, y_{2},\ldots,y_{n}$ in $F(x_{o})$ such that $U_{i_{j}}\subseteq U_{y_{j}}$ for each $1 < j < n$ and the set

$$ H= H_{o}\cap \biggl(\bigcup _{i\in I}V_{y_{i}} \biggr). $$

(4.12)

Then $H \in \sigma(y_{o})$ is such that

$$ H \cap\biggl(\bigcup_{i\in I}V_{i} \biggr) = \phi. $$

(4.13)

The upper supra-continuity of F means that there exists $W \in\tau^{*}(xo)$ such that

$$ x_{o} \in W\subseteq F^{+}\biggl(\bigcup _{i\in I}V_{i}\biggr). $$

(4.14)

It follows that $(W \times H) \cap G(F)=\phi$, and hence $G(F)$ is supra-closed. □

Lemma 4.1

([18])

The following hold for $F:(X,\tau) \rightarrow(Y,\sigma)$, $A \subseteq X$and $B \subseteq Y$:

• (i)
  $$ G_{F}^{+}(A\times B) = A \cap F^{+} (B); $$

  (4.15)
• (ii)
  $$ G_{F}^{-}(A\times B) = A \cap F^{-} (B). $$

  (4.16)

Theorem 4.3

For a harmonic multifunction $F:(X,\tau )\rightarrow(Y,\sigma)$, ifGFis upper supra-continuous, thenFis upper supra-continuous.

Proof

Let $x\in X$ and $V\in \sigma(F(x))$. Since $X \times V\in \tau\times\sigma$ and

$$G_{F}(x) \subseteq X \times V, $$

by Theorem 3.1 there exists $W\in \tau^{*}(x)$ such that $G_{F}(W)\subseteq X \times V$. Therefore, by Lemma 4.1 we get

$$ W\subseteq G_{F}^{-}(X\times V) =X\cap G_{f}^{+}(V)=F^{+}(V), $$

(4.17)

and so $F(W)\subseteq V$. Hence Theorem 3.1 gives that also F upper supra-continuous. □

Theorem 4.4

If the graph $G_{F}$of a harmonic multifunction $F:(X,\tau)\rightarrow(Y,\sigma)$is lower supra-continuous, then so isF.

Proof

Let $x \in X$ and $V \in\sigma(F(x))$ with $F(x) \cap V \neq\phi$. Since

$$ X\times V \in\tau\times\sigma, $$

(4.18)

we have

$$ G_{F} (x) \cap(X \times V) = {{x} \times F(x)}\cap(X \times V) = {x} \times\bigl(F(x) \cap V\bigr)\neq\phi. $$

(4.19)

Theorem 3.2 shows that there exists $W \in\tau^{*}(x)$ such that

$$ G_{F}(w) \subseteq(X \times V) \neq\phi $$

(4.20)

for each $w \in W$.

Hence Lemma 4.1 gives that

$$ W \subseteq G^{-} (X\times V) = X\cap G^{-} (V)=F^{-} (V). $$

(4.21)

So

$$ F(w)\cap V \neq\phi $$

(4.22)

for each $w \in W$, which, together with Theorem 3.2, completes the proof. □

5 Conclusions

In this paper, we proved that there exists a supra-open set in $(X,\tau )$ for each $V \in\sigma$ in which the modified equilibrium equation has normal families of solutions. Moreover, we also established a new expression of harmonic multifunctions for the above equation. Meanwhile, we discussed the relationships between superharmonic multifunctions and superharmonic-closed graphs. As applications, we not only proved the existence of normal families of solutions for modified equilibrium equations but also obtained several characterizations and fundamental properties of these new classes of superharmonic multifunctions.