 Research
 Open Access
Structure of the solution set for a partial differential inclusion
 Yi Cheng^{1}Email author,
 Ravi P Agarwal^{2, 3},
 Afif Ben Amar^{4} and
 Donal O’Regan^{5}
https://doi.org/10.1186/s1366201507230
© Cheng et al. 2015
 Received: 17 November 2015
 Accepted: 8 December 2015
 Published: 18 December 2015
Abstract
In this paper, we consider the biharmonic problem of a partial differential inclusion with Dirichlet boundary conditions. We prove existence theorems for related partial differential inclusions with convex and nonconvex multivalued perturbations, and obtain an existence theorem on extremal solutions, and a strong relaxation theorem. Also we prove that the solution set is compact \(R_{\delta}\) if the perturbation term of the related partial differential inclusion is convex, and its solution set is pathconnected if the perturbation term is nonconvex.
Keywords
 biharmonic problem
 differential inclusion
 setvalued mapping
 pathconnected
 compact \(R_{\delta}\)
MSC
 34B15
 34B16
 37J40
1 Introduction
We also refer the reader to the works of PapageorgiouShahzad [29] for the firstorder evolution inclusion and PapageorgiouYannakakis [30] for the secondorder evolution inclusion where the structure of solution sets was discussed. Following their lead, in this paper, we obtain the \(R_{\delta}\)structure of the solution set for a biharmonic differential inclusion based on the space variable \(x\in\Omega\). We prove that the solution set of the biharmonic inclusion problem in the convexvalued case is compact \(R_{\delta}\) in \(C(\overline{\Omega})\), and the solution set is pathconnected in the case of a nonconvexvalued orientor field.
The plan of our paper is as follows. In Section 2, we collect some preliminary results which will be used in this work. In Section 3, we present some basic assumptions and existence theorems for the both convex and nonconvex multivalued terms. Here, our results are based on the LeraySchauder alternative. In Section 4, a relaxation theorem is established. Finally the properties of the solution set is given in Section 5.
2 Preliminaries
In this section, we introduce some basic definitions and facts which are essential tools in the later sections; see HuPapageorgiou [31] for details.
Let \(R^{N}\) (\(N\geq1\)) be the Ndimensional real Euclidean space. Throughout this paper the symbol Ω denotes a nonempty, bounded, open set of \(R^{N}\), with a smooth boundary ∂Ω. Moreover, from now on, ‘measurable’ simply means Lebesgue measurable. Given two nonnegative constants \(k, p\geq1\), we denote by \(W^{k,p}(\Omega)\) the space of all realvalued functions defined on Ω whose weak partial derivatives up to the order k lie in \(L^{p}(\Omega)\), equipped with \(W^{k,p}(\Omega)\) the usual norm \(\\cdot\_{k,p}\). If \(u\in W^{2,p}(\Omega)\), we set \(\Delta u=\sum^{n}_{i=1}\frac{\partial^{2}u}{\partial{x_{i}}^{2}}\), \(\nabla{u}=\operatorname{grad}{u}=(\frac{\partial u}{\partial{x_{i}}})_{i=1}^{N}\). For any real number \(p> 1\), we denote by q the dual exponent of p (and throughout the paper we assume \(p>1\)).
Definition 2.1
Let X be a Banach space. A multifunction \(F: \Omega\rightarrow{\mathcal{P}}_{f}(X)\) is said to be ‘measurable’, if for all \(y\in X\), the \(R_{+}\)valued function \(x\rightarrow d(y,F(x))=\inf\{\yv\,v\in F(x)\}\) is measurable.
Definition 2.2
Definition 2.3
Let Y, Z be Hausdorff topological spaces and \(\beta:Y\rightarrow2^{Z}\setminus\{\emptyset\}\). \(\beta(\cdot)\) is called ‘upper semicontinuous (USC)’ (resp., ‘lower semicontinuous (LSC)’), if for any nonempty closed set \(C\subseteq Z\), \(\beta^{}(C)=\{y\in Y : \beta(y)\cap C\neq\emptyset\}\) (resp., \(\beta^{+}(C)=\{y\in Y:\beta (y)\subseteq C\}\)) is closed in Y.
A multifunction which is both USC and LSC is said to be continuous. From Theorem 2.1 and Remark 2.6 of [32], we note the following result.
Theorem 2.1
Let X be a Banach Space with the weak topology, and \(D\subseteq X\) a weakly compact, convex subset of X, Then any weakly sequentially upper semicontinuous map \(F:D\rightarrow{\mathcal{P}}_{wkc}(D)\) has a fixed point, i.e., there exists \(x\in D\), such that \(x\in F(x)\).
Remark 2.1
Recall \(F:D\rightarrow{\mathcal{P}}_{wkc}(D)\) is weakly sequentially upper semicontinuous if for any weakly closed set A of D, \(F^{1}(A)\) is sequentially closed for the weak topology on D.
We now use Theorem 2.1 to obtain the following result.
Lemma 2.1
Let Ω be a nonempty, closed, convex subset of a Banach space X. Suppose \(F:\Omega\rightarrow{\mathcal{P}}_{c}(\Omega)\) has a weakly sequentially closed graph and \(F(\Omega)\) is weakly relatively compact. Then F has a fixed point.
Proof
Let \(D=\overline{\operatorname{co}}(F(\Omega))\). It follows from the KreinŠmulian theorem that D is a weakly compact convex set. Note \(F(\Omega)\subseteq D \subseteq\Omega\). Notice also that \(T=F_{D}:D\rightarrow{\mathcal{P}}_{c}(D)\). First we prove that GrT is weakly compact. Since \((X\times X)_{w} = X_{w}\times X_{w}\) (\(X_{w}\) is the space X endowed its weak topology), it follows that \(D\times D\) is a weakly compact subset of \(X\times X\). Also, \(\operatorname{Gr}T =\{(x,y)\in D\times X:y\in T(x)\}\subset D\times D\), so, GrT is weakly relatively compact. Let \((x,y) \in D\times D\) be weakly adherent to GrT. Then from the EberleinŠmulian theorem we can find \(\{ (\{x_{n}\}, \{y_{n}\})\}_{n}\subseteq \operatorname{Gr}T\) such that \(y_{n}\in T(x_{n})\), \(x_{n}\rightarrow x\) weakly and \(y_{n}\rightarrow y\) weakly in X. Because F has weakly sequentially closed graph, \(y \in T(x)\) and so \((x,y)\in \operatorname{Gr}T\). Therefore, GrT is a weakly closed subset of \(D \times D\) and so weakly compact. Consequently \(T(x)\) is weakly closed and so a weakly compact subset of D for every \(x \in D\). In view of Theorem 2.1 it suffices to show that T is weakly sequentially upper semicontinuous. First we note that GrT is weakly closed and therefore is sequentially weakly closed. Let \(A\subset D\) be a weakly closed set and let \(x_{n}\in T^{1}(A)\) with \(x_{n} \rightarrow x\) weakly. Now since \(T(x_{n})\cap A \neq \emptyset\) and \(T(x_{n})\subset D\), then for \(y_{n}\in T(x_{n})\cap A\) we may assume \(y_{n}\rightarrow y\) weakly for some \(y\in A\). Since \((x_{n}, y_{n})\in \operatorname{Gr}T\) and GrT is sequentially weakly closed, we have \(y\in T(x)\cap A\) and so \(x\in T^{1}(A)\). Thus \(T^{1}(A)\) is sequentially weakly closed. Applying Theorem 2.1, we see that T has a fixed point \(x \in D \subset\Omega\). Therefore F has a fixed point. □
Let X be a Banach space and \(L^{p}(\Omega,X)\) be the Banach space of all functions \(u:\Omega\rightarrow X\) which are Bochner integrable. Let \(D(L^{p}(\Omega,X))\) be the collection of nonempty decomposable subsets of \(L^{p}(\Omega,X)\). Next is the BressanColombo continuous selection theorem.
Lemma 2.2
(see, e.g., [33])
Let X be a separable metric space and let \(F:X\rightarrow D(L^{p}(\Omega,X))\) be a LSC multifunction with closed decomposable values. Then F has a continuous selection.
Let X be a separable Banach Space and \(C(\Omega,X)\) be the Banach space of all continuous functions. A multifunction \(F:\Omega\times X\rightarrow{\mathcal{P}}_{wkc}(X)\) is said to be of Carathéodory type, if for every \(u\in X\), \(F(\cdot,u)\) is measurable, and for almost all \(x\in\Omega\), \(F(x,\cdot)\) is hcontinuous. A nonempty subset \(\eta_{0}\subset C(\Omega,X)\) is called σcompact if there is a sequence \(\{\eta_{k}\}_{k\geq1}\) of compact subsets \(\eta_{k}\) such that \(\eta_{0}=\bigcup_{k\geq1}\eta_{k}\). Let \(\eta_{0}\subset \eta\), such that \(\eta_{0}\) is dense in η and σcompact. The following continuous selection theorem in the extreme point case is due to Tolstonogov [34].
Lemma 2.3
(see, e.g., [34])
Let the multifunction \(F:\Omega\times X\rightarrow{\mathcal{P}}_{wkc}(X)\) be of Carathéodory type and be integrably bounded. Then there exists a continuous function \(g:\eta\rightarrow L^{p}(\Omega,X)\) such that for almost \(x\in\Omega\), if \(u(\cdot)\in\eta_{0}\), then \(g(u)(x)\in \operatorname{ext}F(x,u(x))\), and if \(u(\cdot)\in\eta\setminus\eta_{0}\), then \(g(u)(x)\in \overline{\operatorname{ext}}\, F(x,u(x))\).
Definition 2.4
A nonempty subset D of Y is said to be contractible if there exist a point \(y_{0}\in D\) and a continuous function \(h:[0,1]\times D\rightarrow D\) such that \(h(0,y)=y_{0}\) and \(h(1,y)=y\) for every \(y\in D\).
Definition 2.5
Note that a compact \(R_{\delta}\) set D is nonempty, compact, and connected. However, in contrast to contractible sets, a compact \(R_{\delta}\) set D need not be pathconnected. We also need the following approximation result that can be proved from Proposition 4.1 of [29] with minor modifications to accommodate the presence of \(x\in\Omega\).
Lemma 2.4
 (i)
\(\forall(u,s)\in R\times R^{N} \), \(x\rightarrow G(x,u,s)\) is measurable;
 (ii)
\(\forall x\in\Omega\), \((u,s)\rightarrow G(x,u,s)\) is USC;
 (iii)
\(\forall(x,u,s)\in\Omega\times R\times R^{N}\), \(G(x,u,s)\leq \varphi(x)\) a.e. with \(\varphi(x)\in L_{+}^{q}(\Omega)\).
 (a)
For every \(x\in\Omega\), and \((u,s)\in R\times R^{N}\) there exist \(\mu_{n}(u,s)> 0\) and \(\varepsilon _{n}> 0\) such that if \(u_{1}, u_{2}\in B_{\varepsilon _{n}}(u)=\{y\in R:uy\leq \varepsilon _{n}\}\), \(s_{1}, s_{2}\in B_{\varepsilon _{n}}(s)\), then \(h(G_{n}(x,u_{1},s_{1}), G_{n}(x,u_{2},s_{2}))\leq\mu _{n}(x,u,s)\varphi(x)(u_{1}u_{2}+\s_{1}s_{2}\)\) a.e. (i.e., \(G_{n}(x,u,s)\) is locally hLipschitz with respect to \((u,s)\)).
 (b)
\(G(x,u,s)\subseteq\cdots \subseteq G_{n}(x,u,s)\subseteq G_{n1}(x,u,s)\subseteq\cdots\), \(G_{1}(x,u,s)\leq\varphi(x)\) a.e. \(n\geq1\), \(G_{n}(x,u,s)\rightarrow G(x,u,s)\) as \(n\geq1\) for every \((x,u,s)\in\Omega\times R\times R^{N}\), and finally there exists \(g_{n}: \Omega\times R\times R^{N}\rightarrow R\), measurable in x, locally Lipschitz in \((u,s)\) and \(g_{n}(x,u,s)\in G_{n}(x,u,s)\) for every \((x,u,s)\in\Omega\times R\times R^{N}\). Moreover, if \(G(x,\cdot,\cdot)\) is hcontinuous, then \(x\rightarrow G_{n}(x,u,s)\) is measurable (hence \((x,u,s)\rightarrow G_{n}(x,u,s)\) is measurable too; see [35]).
Theorem 2.2
(see, e.g., [36])
Let X and Y be two normed spaces. If \(T:X\rightarrow Y\) is a compact linear operator and \(\{x_{n}\}_{n}\) is a sequence in X such that \(x_{n}\rightarrow x\) weakly then \(T(x_{n})\rightarrow T(x)\) strongly.
3 Existence theorems of solutions
Definition 3.1
 \(H(F)_{1}\)::

\(H:\Omega\times R\times R^{N}\times R\rightarrow{\mathcal{P}}_{k}(R)\) is a multifunction satisfying the following properties:
 (a)
\((x,u,s,t)\rightarrow H(x,u,s,t)\) is graph measurable.
 (b)
For almost all \(x\in\Omega\), \((u,s,t)\rightarrow H(x,u,s,t)\) is LSC.
 (c)For every \((u,s,t)\in R\times R^{N}\times R\), there exist \(\omega _{0}(x)\in L^{p}(\Omega)\), \(\omega_{1}(x)\in L^{\frac{p}{1\alpha }}(\Omega)\), \(\omega_{2}(x)\in L^{\frac{p}{1\beta}}(\Omega)\), \(\omega_{3}(x)\in L^{\frac{p}{1\gamma}}(\Omega)\) such thatwhere \(0\leq\alpha,\beta,\gamma<1\).$$\begin{aligned} \bigl\vert H(x,u,s,t)\bigr\vert =&\bigl\{ v:v\in H(x,u,s,t)\bigr\} \\ \leq& \omega_{0}(x)+\omega_{1}(x)u^{\alpha}+ \omega_{2}(x)\s\ ^{\beta}+\omega_{3}(x)t^{\gamma}\quad \mbox{a.e. } x\in\Omega, \end{aligned}$$
 (a)
Theorem 3.1
If assumption \(H(F)_{1}\) holds, then the partial differential inclusion (1.1) has a solution \(u\in W_{p}(\Omega)\).
Proof
Hence, by Lemma 2.2, there exists a continuous map \(g:W_{p}(\Omega)\rightarrow L^{p}(\Omega)\subset W_{0}^{2,p}(\Omega)\) such that \(g(u)\in S^{p}_{H}(u)\). To complete the proof, we need to consider the fixed point problem: \(u=L^{1}\circ g(u)\).
 \(H(F)_{2}\)::

\(H:\Omega\times R\times R^{N}\times R\rightarrow {\mathcal{P}}_{kc}(R)\) is a multifunction satisfying the following properties:
 (a)
\((x,u,s,t)\rightarrow H(x,u,s,t)\) is graph measurable.
 (b)
For almost all \(x\in\Omega\), \((u,s,t)\rightarrow H(x,u,s,t)\) is USC; and \(H(F)_{1}\)(iii) holds.
 (a)
Theorem 3.2
If assumption \(H(F)_{2}\) holds, then the solution set of the partial differential inclusion (1.1) is nonempty in \(W_{p}(\Omega)\). Moreover, the solution set is weakly compact in \(W_{p}(\Omega)\).
Proof
In view of the proof of Theorem 3.1, we only need to emphasis those steps where the proofs differ.
4 Relation theorem of solutions
 \(H(F)_{3}\)::

\(H:\Omega\times R\times R^{N}\times R\rightarrow {\mathcal{P}}_{kc}(R)\) is multifunction such that for almost all \(x\in\Omega\), \((u,s,t)\rightarrow H(x,u,s,t)\) is hcontinuous, and \(H(F)_{1}\)(i), (iii) holds, where \(p>\frac{N}{2}\geq 2\).
In the following let S denote the solution set of (1.1), and \(S_{e}\) denote the solution set of (4.1).
Theorem 4.1
If assumption \(H(F)_{3}\) holds, then the partial differential inclusion (4.1) has a solution \(u\in W_{p}(\Omega)\cap C(\overline{\Omega})\).
Proof
To prove our next result, we need the following definition.
Definition 4.1
(see [38])
The multifunction \(H:\Omega \times R\times R^{N}\times R\rightarrow{\mathcal{P}}_{k}(R)\) is called ‘onesided Lipschitz (OSL)’ continuous if there is an integrable function \({\mathcal{L}}:\Omega \rightarrow R\) such that for every \(u_{1}, u_{2}\in R\), \(x\in\Omega\), \(s\in R^{N}\), \(t\in R\), and \(v_{1}\in H(x, u_{1}, s, t)\) there exists \(v_{2}\in H(x, u_{2}, s, t)\) such that \((v_{2}v_{1})\cdot (u_{2}u_{1})\leq{\mathcal{L}}(x)u_{2}u_{1}^{2}\).
Theorem 4.2
 (i)
H is onesided Lipschitz (OSL) continuous;
 (ii)
\({\mathcal{L}}(x)\leq\alpha<{\frac{1}{\lambda}}\), where \({\mathcal{L}}\) is from Definition 4.1 and λ from (4.2);
Proof
5 Properties of the solutions set
Remark 5.1
The results obtained above for the problem (1.1) hold for the partial differential inclusion (5.1).
From Theorem 3.1, it is easy to show that for every \(f\in H(x,u,\nabla u)\subseteq L^{p}(\Omega)\), problem (5.1) has at least one weak solution \(u=L^{1}(f)\in W_{p}(\Omega )\) and \(\u\_{W}\leq C\f\_{p}\), where C is a constant independent of u and f. Let \(L^{p}(\Omega)_{w}\) denote the LebesgueBochner space furnished with the weak topology. From the proof of Theorem 3.1, it follows that the map \(P=L^{1}:L^{p}(\Omega )_{w}\rightarrow W_{p}(\Omega)\) is sequentially continuous.
Remark 5.2
Since the embedding of \(W_{p}(\Omega)\hookrightarrow C^{1}(\overline{\Omega})\) is compact when \(p\geq N\), \(P:L^{p}(\Omega)_{w}\rightarrow C^{1}(\overline{\Omega})\) is completely continuous.
 \(H(F)_{4}\)::

\(H:\Omega\times R\times R^{N} \rightarrow{\mathcal{P}}_{kc}(R)\) is a multifunction satisfying the following properties:
 (i)
\((x,s,t)\rightarrow H(x,s,t)\) is graph measurable.
 (ii)
For almost all \(x\in\Omega\), \((s,t)\rightarrow H(x,s,t)\) is USC.
 (iii)For every \((s,t)\in R\times R^{N}\), there exist \(\xi_{0}(x)\in L^{p}(\Omega)\), \(\xi_{1}(x)\in L^{\frac{p}{1\gamma}}(\Omega)\), \(\xi_{2}(x)\in L^{\frac{p}{1\eta}}(\Omega)\), such thatfor a.e. \(x\in\Omega\), where \(0\leq\gamma, \eta<1\).$$\bigl\vert H(x,s,t)\bigr\vert =\sup\bigl\{ v:v\in H(x,s,t)\bigr\} \leq \xi_{0}(x)+\xi_{1}(x)u^{\gamma}+\xi_{2}(x)s^{\eta}$$
 (i)
Theorem 5.1
If hypothesis \(H(F)_{4}\) holds and \(p\geq N\), then the solution set S of problem (5.1) is an \(R_{\delta}\) set in \(C(\overline{\Omega})\).
Proof
In general we can always get a subsequence of \(\{\delta_{m}\}_{m\geq1}\) satisfying either Case 1 or Case 2. Thus in conclusion, \(\mu(\delta,u)\) is continuous, and hence for every \(n\geq1\), \(S_{n}\subseteq C(\overline{\Omega}) \) is compact and contractible.
Next we claim that \(S=\bigcap_{n\geq1}S_{n}\). Obviously, \(S\subseteq\bigcap_{n\geq1}S_{n}\). Let \(u\in\bigcap_{n\geq1}S_{n}\). Then from definition \(u=P(v_{n})\), \(v_{n}\in S^{p}_{H_{n}(\cdot, u_{n},\nabla u_{n})}\) for some \(n\geq1\). On passing to a subsequence if necessary we may assume that \(v_{n}\rightarrow v\) weakly in \(L^{p}(\Omega)\). Then \(v\in S^{p}_{H(\cdot, u,\nabla u)}\) (see Theorem 3.2). Thus \(u=P(v)\) with \(v\in S^{p}_{H_{n}(\cdot, u,\nabla u)}\), from which we can conclude that \(u\in S\) i.e., \(S=\bigcap_{n\geq1}S_{n}\). Finally from Hyman’s result [41] we see that S is an \(R_{\delta}\) set in \(C(\overline{\Omega})\). □
The following remark is given as an immediate consequence of Theorem 5.1 for the multivalued problem (5.1).
Remark 5.3
If hypothesis \(H(F)_{4}\) holds, then for every \(x\in\Omega\), \(S(x)=\{u(x)u\in S\}\) (the reachable set at \(x\in\Omega\)) is compact and connected in R.
 \(H(F)_{5}\)::

\(H:\Omega\times R\times R^{N}\rightarrow {\mathcal{P}}_{k}(R)\) is a multifunction such that for almost all \(x\in \Omega\), \((u,s)\rightarrow H(x,u,s)\) is LSC, and \(H(F)_{4}\)(i), (iii) holds.
Theorem 5.2
Proof
Declarations
Acknowledgements
This work is partially supported by National Natural Science Foundation of China (No. 11401042).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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