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 supraopen 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.
Introduction
As in [2], the modified equilibrium equations for a selfgravitating fluid rotating about the \(x_{3}\) axis with prescribed velocity \(\Omega(r)\) can be defined as follows:
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]
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 supraopen sets, which is considered as a wider class of some known types of nearopen sets. In 1983, Mashhour et al. [11] defined the concept of Scontinuity for a singlevalued function \(f:(X,\tau)\rightarrow (Y,\sigma)\). Many topological properties of the abovementioned concepts and others have been established in [3, 12]. The purpose of this paper is to present the upper (lower) supracontinuous harmonic multifunction as a generalization of upper (lower) semicontinuous harmonic multifunctions in the sense of Berge [13], the upper (lower) quasicontinuous and the upper (lower) precontinuous harmonic multifunctions defined by Popa [14], and also upper (lower) αcontinuous and upper (lower) βcontinuous harmonic multifunctions defined by Wine [15]. Moreover, characterization of these new harmonic multifunctions by many their properties has also been established.
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 supratopology on X if \(X \in\tau^{*}\) and \(\tau^{*}\) is closed under arbitrary union [10]. Then \((X, \tau^{*})\) is called a supratopological space or simply supraspace. Each member of τ is supraopen, and its complement is supraclosed [11]. In \((X, \tau^{*})\), the supraclosure, the suprainterior, and suprafrontier 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
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,
and
Sets \(A\in \tau^{\alpha}\) and \(A \in\beta O(X,\tau)\) are called αsets [16] and βopen sets [17], respectively. A singlevalued function \(f:(X,\tau)\rightarrow(Y,\sigma)\) is called Scontinuous [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
and
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) quasicontinuous [18], upper (lower) αcontinuous [14], upper (lower) precontinuous [4], and upper (lower) βcontinuous [2], respectively. A space \((X,\tau)\) is called supracompact [12] if every supraopen cover of X admits a finite subcover.
Supracontinuous harmonic multifunctions
Definition 3.1
A harmonic multifunction \(F:(X,\tau )\rightarrow(Y,\sigma)\) is said to be:

(a)
upper supracontinuous 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 supracontinuous 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) supracontinuous if F has this property at every point of X.
Any singlevalued 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 supracontinuous harmonic multifunctions to a singlevalued function, it is clear that they coincide with the notion of Scontinuous 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) Fcontinuity [4], upper (lower) quasicontinuity [14], upper (lower) precontinuity [12], and upper (lower) Gcontinuity [19] can be deduced from the upper (lower) supracontinuity 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) supracontinuous 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
Theorem 3.1
The following statements are equivalent for a harmonic multifunction \(F:(X,\tau)\rightarrow(Y,\sigma)\):

(i)
Fis upper supracontinuous.

(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 supraclosed 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 supraopen sets is supraopen.
(iv) = (v). Let K be closed in Y. The result holds since
(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
and
These imply that
(vii) ⇒ (ii). Let \(x\in X\) be arbitrary, and let \(V\in \sigma(F(x))\). Then
Hence \(F^{+} (V) \in\tau^{*}(x)\).
(viii) ⇔ (v). It is clear since suprafrontier and frontier of any set is supraclosed 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 supracontinuous.

(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 supraclosed 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 \((\chi_{i})_{(i\in l)}\) supraconverges 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 supracontinuous if and only if, for each net \((\chi_{i})_{(i\in l)}\)supraconvergent 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 supracontinuity 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 supraconvergent 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 supraconvergent 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 supracontinuous if and only if, for each \(y_{o} \in F(x_{o})\)and for every net \((\chi_{i})_{(i\in l)}\)supraconvergent 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 supraconvergent toyand \(y_{j} \in F(z_{j})\).
Proof
For the necessity, suppose that F is lower supracontinuous, \((\chi_{i})_{(i\in l)}\) is a net supraconvergent 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 supracontinuity of F at \(x_{o}\) there is a supraopen set \(W \subseteq X\) containing \(x_{o}\) such that \(W \subseteq F^{}(V)\). Since the net \((\chi_{i})_{(i\in l)}\) is supraconvergent 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
and
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 supraconvergent 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 supraconvergent to y, which shows the result. To show the sufficiency, assume the converse, that is, F is not lower supracontinuous at \(x_{o}\). Then there exists \(V \in\sigma\) such that \(F(x_{o}) \cap V\ne\phi\), and for any supraneighborhood \(W \subseteq X\) 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 supraconvergent 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 supraconvergent 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
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 supraregular if, for any \(x \in W\) and any \(H \in\tau^{*}(x)\), there exists \(U \in\tau\) such that
Recall that \(F: (X, \tau) \rightarrow(Y,\sigma)\) is punctually supraregular if, for each \(X\in X\), \(F(x)\) is supraregular.
Lemma 3.2
In a space \((X,\tau)\), if \(W \subseteq X\)is supraregular and contained in a supraopen setH, then there exists \(U \in\tau\)such that
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 punctuallyαparacompact and punctually supraregular 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
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 punctuallyaparacompact and punctually supraregular harmonic multifunction. ThenFis upper supracontinuous if and only if \((\mathrm{supra}\mbox{}\operatorname{cl}F): (X, \tau)\rightarrow(Y, \sigma)\)is upper supracontinuous.
Proof
Necessity. Suppose that \(V \in\sigma\) and \(x \in (\mbox{supra}\operatorname{cl} F)^{+} (V) = F^{+} (V)\) (see Proposition 3.2). By upper supracontinuity 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
for each \(h \in H\).
Hence
for each \(h \in H\), which gives that
Thus \((\mbox{supra}\operatorname{cl} F)\) is upper supracontinuous.
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
and so
Hence
and this completes the proof. □
Theorem 3.6
A harmonic multifunction \(F: (X, \tau )\rightarrow(Y, \sigma)\)is lower supracontinuous if and only if \((\mathrm{supra}\mbox{}\operatorname{cl} F): (X, \tau) \rightarrow(Y, \sigma)\)is lower supracontinuous.
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 supracontinuous surjection, then \(F(x)\)is compact relative toYfor each \(x\in X\). If \((X,\tau)\)is supracompact, then \((Y,\sigma )\)is compact.
Proof
Let
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
The upper supracontinuity of F gives that there exists \(W(x) \in\tau^{*}(X,x)\) such that
Since \((X, \tau)\) is supracompact, there exist \({x_{1},x_{2}, \ldots,x_{n}}\) such that
So
Hence \((Y,\sigma)\) is compact. □
Supracontinuous harmonic multifunctions and supraclosed graphs
Definition 4.1
A harmonic multifunction \(F:(X, \tau )\rightarrow(Y, \sigma)\) is said to have a supraclosed graph if, for each pair \((x,y) \notin G(F)\), there exist \(W\in \tau^{*}(X)\) and \(H\notin\sigma^{*}(y)\) such that
A harmonic multifunction \(F:(X, \tau) \rightarrow(Y, \sigma)\) is pointclosed (supraclosed) if, for each \(x \in X\), \(F(x)\) is closed (supraclosed) in Y.
Proposition 4.1
A harmonic multifunction \(F : (X,\tau )\rightarrow(Y, \sigma)\)has a supraclosed graph if and only if, for all \(x \in X\)and \(y \in Y\)such that \(y \notin F(x)\), there exist two supraopen setsH, Wcontainingxandy, respectively, such that
Proof
Necessity. Let \(x \in X\) and \(y \in Y\) with \(y \notin F(x)\). Then since F has a supraclosed 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 supraopen 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 supracontinuous and pointclosed harmonic multifunction and \((Y, \sigma)\)is regular, then \(G(F)\)is supraclosed.
Proof
Suppose that
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
Since F is upper supracontinuous at x, there exists
such that \(F(W)\subseteq V_{2}\). As \(V_{1} \cap V_{2} = \phi\), we have
and therefore
and
Thus
which gives the desired result. □
Definition 4.2
A subset W of a space \((X,\tau)\) is called αparacompact [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 supracontinuous harmonic multifunction from \((X,\tau)\)into a Hausdorff space \((Y,\sigma)\). If \(F(x)\)isαparacompact for each \(x \in X\), then \(G(F)\)is supraclosed.
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
So the family \(\{V_{y}: y\in F(x_{0})\}\) is an open cover of \(F(x_{o})\). Thus, by the αparacompactness 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
Then \(H \in \sigma(y_{o})\) is such that
The upper supracontinuity of F means that there exists \(W \in\tau^{*}(xo)\) such that
It follows that \((W \times H) \cap G(F)=\phi\), and hence \(G(F)\) is supraclosed. □
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 supracontinuous, thenFis upper supracontinuous.
Proof
Let \(x\in X\) and \(V\in \sigma(F(x))\). Since \(X \times V\in \tau\times\sigma\) and
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
and so \(F(W)\subseteq V\). Hence Theorem 3.1 gives that also F upper supracontinuous. □
Theorem 4.4
If the graph \(G_{F}\)of a harmonic multifunction \(F:(X,\tau)\rightarrow(Y,\sigma)\)is lower supracontinuous, then so isF.
Proof
Let \(x \in X\) and \(V \in\sigma(F(x))\) with \(F(x) \cap V \neq\phi\). Since
we have
Theorem 3.2 shows that there exists \(W \in\tau^{*}(x)\) such that
for each \(w \in W\).
Hence Lemma 4.1 gives that
So
for each \(w \in W\), which, together with Theorem 3.2, completes the proof. □
Conclusions
In this paper, we proved that there exists a supraopen 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 superharmonicclosed 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.
Change history
19 July 2020
A Correction to this paper has been published: https://doi.org/10.1186/s1366102001428y
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Acknowledgements
The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper. The corresponding author expresses her appreciation for kind hospitality of Universidad de Granada, where portions of this paper were written.
Funding
This work was supported by the Natural Science Foundation of China (Grant No. 11401160) and the Natural Science Foundation of Hebei Province (No. A2015209040).
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YT designed the solution methodology. WD prepared the revised manuscript. YW participated in the design of the study. ZJ drafted the manuscript. All authors read and approved the final manuscript.
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Tan, Y., Ding, W., Wang, Y. et al. RETRACTED ARTICLE: Normal families of solutions for modified equilibrium equations and their applications. Bound Value Probl 2017, 176 (2017). https://doi.org/10.1186/s1366101709066
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Keywords
 normal family
 modified equilibrium equation
 modified Laplace equation