A new technique to study the boundary behaviors of superharmonic multifunctions and their application
- Yong Lu^{1} and
- Jianguo Sun^{2}Email author
Received: 15 July 2017
Accepted: 26 August 2017
Published: 2 October 2017
Abstract
Using some recent results of the Riesz decomposition method for sharp estimates of certain boundary value problems of harmonic functions in (St. Cer. Mat. 27:323-328, 1975), the boundary behaviors of upper and lower superharmonic multifunctions are studied. Several fundamental properties of these new classes of these functions are shown. A new technique is proposed to find the exact boundary behaviors by using Levin’s type boundary behaviors for harmonic functions admitting certain lower bounds in (Pacific J. Math. 15:961-970, 1965). Finally, some examples are given to illustrate the applications of our results.
Keywords
1 Introduction
In 1977, Husain [3] has initiated the concept of superharmonic-open sets, which is considered as a wider class of some known types of near-open sets. In 1983, Mashhour et al. [4, 5] defined the concept of S-continuity, but 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 [6, 7]. The purpose of this paper is to present the upper (resp. lower) superharmonic-continuous multifunction as a generalization of each of upper (resp. lower) super-continuous superharmonic multifunction in the sense of Berge [7] the upper (resp. lower) sub-continuous and the upper (resp. lower) precontinuous superharmonic multifunction due to Popa [1, 8] and also upper (resp. lower) α-continuous and upper (resp. lower) β-continuous superharmonic multifunctions as given in [9, 10] recently. Moreover, these new superharmonic multifunctions are characterized and many of their properties have also been established.
2 Preliminaries
3 Supra-continuous superharmonic multifunctions
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
4 Supra-continuous superharmonic multifunctions and superharmonic-closed graphs
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
Declarations
Acknowledgements
The authors want to thank the reviewers for much encouragement, support, productive feedback, cautious perusal and making helpful remarks, which improved the presentation and comprehensibility of the article.
Authors’ contributions
All authors contributed to each part of this work equally and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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