A new technique to study the boundary behaviors of superharmonic multifunctions and their application

Definition 3.1

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

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

Any single-valued superharmonic function \(f:(X, \tau)\rightarrow (Y,\sigma)\) can be considered as a multi-valued one which assigns to any \(x \in X\) the singleton \(\{f(x)\}\). We apply the above definitions of both upper and lower superharmonic-continuous multifunctions to the single-valued case. It is clear that they coincide with the notion of S-continuous due to Mashhour et al. [5]. One characterization of the above superharmonic multifunction is established throughout the following result, of which the proof is straightforward, so it is omitted.

Remark 3.1

For a superharmonic multifunction \(F:(X,\tau )\rightarrow(Y,\sigma)\), many properties of upper (resp. lower) semicontinuity [7] (resp. upper (lower)) F-continuity [9], upper (resp. lower) sub-continuity [1], upper (resp. lower) precontinuity [10] and upper (resp. lower) (G-continuity [10]) can be deduced from the upper (resp. lower) superharmonic-continuity by considering \(\tau^{*}= \tau\) (resp. \(\tau^{*} = \tau^{\alpha}\), \(\tau^{*}= SO(X,\tau)\), \(\tau^{*}= PO(X,\tau)\) and \(\tau^{*}= \beta O(X,\tau)\)).

Proposition 3.1

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

Lemma 3.1

Theorem 3.1

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

(i) F is upper superharmonic-continuous;

(ii) for each \(x\in X\) and each \(V \in \sigma(F(x))\), we have \(F^{+} (V)\in\tau^{*}(x)\);

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

(v) \(F^{-} (K)\) is superharmonic-closed for every closed set \(K\subseteq Y\);

(vi) \(\operatorname{superharmonic-cl}(F^{-} (B))\subseteq F^{-} (\operatorname{\tau-cl}(B))\) for every \(B\subseteq Y\);

(vii) \(F^{+}(\operatorname{\tau-int}(B))\subseteq \operatorname{superharmonic-int}(F^{+} (B))\) for every \(B\subseteq Y\);

(viii) \(\operatorname{superharmonic-fr}(F^{-}(B))\subseteq F^{-}(\operatorname{fr}(B))\) for every \(B\subseteq Y\);

(ix) \(F:(X, \tau^{*}) \rightarrow (Y,\sigma)\) is upper superharmonic-continuous.

Proof

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

(ii) ⇔ (iii): This is obvious, since the arbitrary union of superharmonic-open set is superharmonic-open.

(v) ⇒ (vi): By putting \(K = \operatorname{\sigma-cl}(B)\) and applying Lemma 3.1.

(viii) ⇔ (v): Clearly, a suprafrontier and frontier of any set is superharmonic-closed and closed, respectively.

(ix) ⇔ (iv): Follows immediately. □

Theorem 3.2

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

(i) F is lower superharmonic-continuous;

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

(v) \(F^{+} (K)\) is superharmonic-closed for every closed set \(K\subseteq Y\);

(vi) \(\operatorname{superharmonic-cl}(F^{+} (B)) \subseteq F^{+} (\sigma \operatorname{cl-}(B))\) for any \(B\subseteq Y\);

(vii) \(F^{-} (\operatorname{\sigma-{int}}(B))\subseteq \operatorname{superharmonic-int}(F^{-} (B))\) for any \(B\subseteq Y\);

(viii) \(\operatorname{superharmonic-fr}(F^{+} (B)) \subseteq F^{+} (\operatorname{fr}(B))\) for every \(B \subseteq Y\);

(ix) \(F:(X, \tau^{*})\rightarrow(Y, \sigma)\) is lower superharmonic-continuous.

Proof

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

Theorem 3.3

A superharmonic multifunction \(F : (X, \tau )\rightarrow(Y,\sigma)\) is upper superharmonic-continuous if and only if for each net \((\chi_{i})_{(i\in l)}\) superharmonic-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 upper superharmonic-continuity of F, there is \(W\in\tau^{*}(X_{O})\) such that \(F(W)\subseteq V\). Since from the hypothesis a net \((\chi _{i})_{(i\in l)}\) is superharmonic-convergent to \(x_{o}\) and \(W \in\tau ^{*}(x_{o})\) there is one \(i_{o} \in I\) such that \(x_{i} \in W\) for all \(i > i_{o}\) and then \(F(X_{i}) \subseteq V\) for all \(i > i_{o}\). As regards sufficiency, assume the converse, i.e. there is an open set V in Y with \(F(x_{o} )\subseteq V\) such that for each \(W\in\tau^{*}\) under inclusion we have the relation \(F(W)\nsubseteq V\), i.e. there is \(x_{w} \in W \) such that \(F(x_{w}) \nsubseteq V\). Then all of \(x_{w}\) will form a net in X with directed set W of \(\tau^{*}(x_{o})\), clearly this net is superharmonic-convergent to \(x_{o}\). But \(F(x_{w})\nsubseteq V\) for all \(W \in\tau^{*}(x_{o})\). This leads to a contradiction which completes the proof. □

Theorem 3.4

A superharmonic multifunction \(F : (X,\tau )\rightarrow(Y, \sigma)\) is lower superharmonic-continuous if and only if for each \(y_{o} \in F(x_{o})\) and for every net \((\chi_{i})_{(i\in l)}\) superharmonic-convergent to \(x_{o}\), there exists a subnet \((Z_{j})_{(j\in J)}\) of the net \((\chi_{i})_{(i\in l)}\) and a net \((y_{i})_{(j,v)\in J}\) in Y so that \((y_{i})_{(j,v)\in J}\) superharmonic-convergent to y and \(y_{j} \in F(z_{j})\).

Proof

Definition 3.2

Lemma 3.2

Proposition 3.2

Proof

This means that \(x \in(\operatorname{superharmonic-cl} F)^{+} (W)\). Hence the equality holds. □

Theorem 3.5

Proof

Lemma 3.3

Proposition 3.3

Proof

Theorem 3.6

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

Proof

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

Theorem 3.7

If \(F:(X,\tau)\rightarrow(Y, \sigma)\) is an upper superharmonic-continuous surjection and for each \(x\in X,F(x)\) is compact relative to Y. If \((X,\tau)\) is superharmonic-compact, then \((Y,\sigma)\) is compact.

Proof

Definition 4.1

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

Proposition 4.1

Proof

As regards necessity, let \(x \in X\) and \(y \in Y\) with \(y \notin F(x)\). Then by the superharmonic-closed graph of F, there are \(H\in\tau^{*} (x)\) and \(W\in\sigma^{*}\) containing \(F(x)\) such that \((HxW)\cap G(F) = \phi\). This implies that for every \(x \in H\) and \(y \in W\) where \(y \notin F(x)\) we have \(F(H) \cap W =\phi\).

As regards sufficiency, let \((x,y) \notin G(F)\), this means \(y \notin F(x)\); then there are two disjoint superharmonic-open sets H, W containing x and y, respectively, such that \(F(H)\cap W = \phi\). This implies that \((H \times W) \cap G(F) = \phi\), which completes the proof. □

Theorem 4.1

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

Proof

Definition 4.2

A subset W of a space \((X,\tau)\) is called α-paracompact [12] if for every open cover v of W in \((X,\tau)\) there exists a locally finite open cover ξ of W which refines v.

Theorem 4.2

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

Proof

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

Lemma 4.1

[14]

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

Theorem 4.3

Theorem 4.4

If the graph \(G_{F}\) of a superharmonic multifunction \(F:(X,\tau)\rightarrow(Y,\sigma)\) is lower superharmonic-continuous, then F is also.

Proof

Authors’ contributions

All authors contributed to each part of this work equally and read and approved the final manuscript.