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Topological structure of solution sets for fractional evolution inclusions of Sobolev type
 Pengxian Zhu^{1} and
 Qiaomin Xiang^{2}Email author
 Received: 5 August 2018
 Accepted: 8 November 2018
 Published: 15 November 2018
Abstract
The paper is devoted to establishing the solvability and topological property of solution sets for the fractional evolution inclusions of Sobolev type. We obtain the existence of mild solutions under the weaker conditions that the semigroup generated by \(AE^{1}\) is noncompact as well as F is weakly upper semicontinuous with respect to the second variable. On the same conditions, the topological structure of the set of all mild solutions is characterized. More specifically, we prove that the set of all mild solutions is compact and the solution operator is u.s.c. Finally, an example is given to illustrate our abstract results.
Keywords
 Fractional evolution inclusion
 Sobolev type
 Weakly upper semicontinuous
 Measures of noncompactness
1 Introduction
Fractional calculus has been used successfully to study many complex systems in various fields of science and engineering, which mainly rely on the nonlocal character of the fractional differentiation, we refer the readers to [10, 19]. Fractional differential equations and inclusions have recently been proved to be valuable tools in the mathematical modeling of systems and processes in the fields of physics [8], chemistry, aerodynamics, electrodynamics of a complex medium, polymer rheology, etc., involves derivatives of fractional order. In recent years, there has been a significant development on differential equations involving fractional derivatives. We refer the reader to [22–24] and the references therein.
It is worth mentioning that evolution equations of Sobolev type have been extensively studied due to their various applications such as in the flow of fluid through fissured rocks, thermodynamics and shear in second order fluids (cf. [3, 6, 12]). The fractional evolution equations of Sobolev type, which arise in the theory of control of dynamical systems when the controlled system or the controller is described by a fractional evolution equation of Sobolev type, provide the mathematical modeling and simulations of controlled systems and processes. For the research of fractional evolution equations of Sobolev type, we refer the readers to [9, 13].
Since a differential inclusion usually has many solutions starting at a given point, new issues, such as the investigation of topological properties of solution sets, selection of solutions with given properties, and evaluation of the reachability sets, appear. An important aspect of the topological structure is the compactness of solution sets. Regarding the topological structure of solution sets for differential equations and inclusions, we may cite, among others [2, 11, 21, 25]. However, to the best of our knowledge, the topological structure of the solution sets for fractional evolution inclusions of Sobolev type has not been explored.
In this work, motivated by the above consideration, we are interested in investigating the topological structure of the solution set for (1.1) under rather mild conditions. Our purpose is to prove that the solution set for (1.1) is nonempty and compact with the semigroup generated by \(AE^{1}\) being noncompact.
By assuming that \(E^{1}\) is compact or the resolvent set \(R(\lambda ,AE^{1})\) of \(AE^{1}\) is compact for every \(\lambda \in \rho (AE ^{1})\), which guarantees that the semigroup generated by \(AE^{1}\) is compact, there has been many works devoted to solvability and controllability of fractional (or integer order) evolution equations of Sobolev type such as [1, 7, 9]. After reviewing the previous research on the fractional evolution equations and inclusions of Sobolev type, we find that most of the works assume that the semigroup generated by \(AE^{1}\) is compact. However, much less is known about the fractional evolution equations and inclusions of Sobolev type with the noncompact semigroup. We will prove that the solution set for (1.1) is nonempty and compact when the semigroup generated by \(AE^{1}\) is noncompact and the multivalued function F is weakly upper semicontinuous with respect to the second variable. We also prove that the solution operator is u.s.c.
Let us give a description of our approach. When dealing with the solvability of (1.1), the key point is to find a compact convex subset which is invariant under the operator \(\mathfrak{L} \) (defined in the proof of Theorem 3.1). It is noted that when the semigroup is compact, the compactness of the convex subset becomes a direct consequence by a standard argument. In this paper, however, we assume that the semigroup is noncompact. To overcome this difficulty, we suppose that the multivalued function F satisfies some regular properties expressed by the measure of noncompactness. Hence, by utilizing the measure of noncompactness, multivalued analysis, and fixed point theory, we obtain the solvability of (1.1). The method of proving the result of the topological structure of the solution set of (1.1) mainly comes from [21].
The paper is organized as follows. In Sect. 2, we recall some concepts and facts which are broadly used for deriving the main results of the paper. In Sect. 3, the nonemptiness of solution set of (1.1) is proved, and then we show that the solution set of (1.1) is a compact set; moreover, we prove that the solution operator is u.s.c. An example is given to illustrate our abstract results in Sect. 4.
2 Preliminaries
As usual, for a Banach space V, \(2^{V}\) stands for the collection of all nonempty subsets of V. \(C(J;V)\) denotes the Banach space of all continuous functions from J to V equipped with its usual norm. For \(1\leq p<\infty \), let \(L^{p}(J;V)\) stand for the Banach space consisting of all Bochner integrable functions \(u:J\rightarrow V\). \(W^{1,p}(J;V)\) is the subspace of \(L^{p}(J;V)\) consisting of functions such that the weak derivative \(u_{t}\) belongs to \(L^{p}(J;V)\). Both spaces \(L^{p}(J;V)\) and \(W^{1,p}(J;V)\) are endowed with their standard norms.
We present the criterion of weak compactness in \(L^{p}(J;V)\) for \(1< p<+\infty \), which is more useful further.
Lemma 2.1
([20, Corollary 1.3.1])
Let V be reflexive and \(1< p< +\infty \). A subset \({K}\subset L^{p}(J;V)\) is weakly relatively sequentially compact in \(L^{p}(J;V)\) if and only if K is bounded in \(L^{p}(J;V)\).
Definition 2.1
([21])
Definition 2.2
([21])
 (\(H_{1}\)):

E and A are closed linear operators.
 (\(H_{2}\)):

\(D(E)\subset D(A)\) and E is bijective.
 (\(H_{3}\)):

\(E^{1}:Y\rightarrow D(E)\) is continuous.
Remark 2.1
We can deduce from assumptions \((H_{1})\) and \((H_{3})\) that \(E^{1}\) is bounded.
Assumptions \((H_{1})\), \((H_{2})\) and a closed graph theorem imply the boundedness of the linear operator \(AE^{1}:Y\rightarrow Y\), and then \(AE^{1}\) generates a semigroup \(\{T(t),t\geq 0\}\) which is continuous for \(t > 0\) in the uniform operator topology (see [18, Theorem 1.2]). Throughout this paper, we assume that there exists a constant \(M>0\) such that \(\sup \{ \Vert T(t) \Vert ,t\geq 0\}\leq M\).
We use the following definition of mild solution of (2.1) which comes from [9].
Definition 2.3
By using the same argument as in the proof of [9, Lemma 3.2], we have some additional properties of the two families \(\{\mathcal{Q}(t),t\geq 0\}\) and \(\{\mathcal{P}(t),t\geq 0\}\) of operators.
Lemma 2.2
 (i)for every \(t\geq 0\), \(\mathcal{Q}(t)\) and \(\mathcal{P}(t)\) are linear and bounded operators on Y, more precisely,$$\begin{aligned} \bigl\Vert \mathcal{Q}(t)\omega \bigr\Vert \leq M \bigl\Vert E^{1} \bigr\Vert \Vert \omega \Vert , \quad \quad \bigl\Vert \mathcal{P}(t)\omega \bigr\Vert \leq \frac{q M \Vert E^{1} \Vert }{\varGamma (1+q)} \Vert \omega \Vert , \quad t\geq 0, \omega \in Y; \end{aligned}$$
 (ii)
\(\mathcal{Q}(t)\) and \(\mathcal{P}(t)\), \(t> 0\), are continuous in the uniform operator topology.
Remark 2.2
From the proof of [21, Lemma 3.1], we can derive the following characterization.
Lemma 2.3
Lemma 2.4
Proof
 (i)
φ is called closed if \(\operatorname{Gra}(\varphi )\) is closed in \(U\times Z\);
 (ii)
φ is called quasicompact if \(\varphi (K)\) is relatively compact for each compact set \(K\subset U\);
 (iii)
φ is called upper semicontinuous (shortly, u.s.c.) if \(\varphi^{1}(D)\) is closed for each closed set \(D\subset Z\), and lower semicontinuous (shortly, l.s.c.) if \(\varphi^{1}(D)\) is open for each open set \(D\subset Z\).
The following lemma gives a sufficient condition for u.s.c. multivalued maps.
Lemma 2.5
([14, Theorem 1.1.12])
Let \(\varphi :U \rightarrow K(Z)\) be a closed and quasicompact multivalued map. Then φ is u.s.c.
Furthermore, in the case when U and Z are Banach spaces, a multivalued map \(\varphi : D\subset U \rightarrow 2^{Z}\) is called weakly upper semicontinuous (shortly, weakly u.s.c.) if \(\varphi^{1}( \mathcal{B})\) is closed in D for every closed set \(\mathcal{B} \subset Z\).
Lemma 2.6
([5, Lemma 2.2(ii)])
Let \(\varphi :D\subset U\rightarrow 2^{Z}\) be a multivalued map with convex weakly compact values. Then φ is weakly u.s.c. if and only if, for each sequence \(\{(u_{n},z_{n})\}\subset D\times Z\) such that \(u_{n}\rightarrow u\) in U and \(z_{n}\in \varphi (u_{n})\), \(n\geq 1\), it follows that there exists a subsequence \(\{z_{n_{k}}\}\) of \(\{z_{n}\}\) and \(z\in \varphi (u)\) such that \(z_{n_{k}}\rightarrow z\) weakly in Z.
We state the following fixed point result which will be used in the proof of the existence result.
Lemma 2.7
([4, Lemma 1])
Let D be a nonempty, compact, and convex subset of a Banach space and \(\varphi :D\rightarrow 2^{D}\) u.s.c. with contractible values. Then φ has at least one fixed point.
Now, we recall some facts about the measure of noncompactness (MNC). The definition of MNC can be found in lots of literature works, for example, [14]. Here, we only introduce some specific properties of the Hausdorff MNC.
The Hausdorff MNC enjoys the following properties (see [14]).
 (i)for any bounded linear operators T from \(\mathcal{E}\) to \(\mathcal{E}\) and \(\varOmega \subset \mathcal{E}\), it follows that$$ \chi (T\varOmega )\leq \Vert T \Vert \chi (\varOmega ); $$(2.2)
 (ii)for every bounded subset \(D\subset \mathcal{E}\) and \(\epsilon >0\), there is a sequence \(\{w_{n}\}\subset D\) such that$$ \chi (D)\leq 2\chi \bigl(\{w_{n}\} \bigr) +\epsilon . $$(2.3)
We need the following statement which provides us with a basic MNC estimate.
Lemma 2.8
([17])
3 Main results
 \((H_{4})\) :

\(F(t,\cdot )\) is weakly u.s.c. for a.e. \(t\in J\) and \(F(\cdot ,v)\) has an \(L^{p}\)integral selection for each \(v\in X\);
 \((H_{5})\) :

there exists a function \(\eta \in L^{p}(J; \mathbb{R}^{+})\) such thatfor a.e \(t\in J\) and each \(v\in X\);$$ \bigl\Vert F(t,v) \bigr\Vert :=\sup \bigl\{ \Vert y \Vert :y\in F(t,v) \bigr\} \leq \eta (t) \bigl(1+ \Vert v \Vert \bigr) $$
 \((H_{6})\) :

there exists \(\mu \in L^{p}(J;\mathbb{R}^{+})\) such thatfor a.e. \(t\in J\) and all bounded subsets \(\varOmega \subset X\).$$ \chi \bigl(F(t,\varOmega ) \bigr)\leq \mu (t)\chi (\varOmega ) $$
Lemma 3.1
([21, Lemma 3.3])
Assume that \((H_{4})\)–\((H_{5})\) hold and suppose that Y is reflexive. Then \(\operatorname {Sel}_{F}\) is weakly u.s.c with nonempty, convex, and weakly compact values.
Now, we are ready to give the existence result. Here, \(x\in C(J;X)\) is a mild solution of (1.1) if x is a mild solution of (2.1) with \(f\in \operatorname {Sel}_{F}(x)\).
Theorem 3.1
Let \(p>1\), \(pq>1\), and assumptions \((H_{1})\)–\((H_{6})\) hold. Assume further that Y is reflexive, then (1.1) has at least one solution.
Proof
In what follows, we focus on the compactness of \(\mathcal{M}\). We only need to prove that \(\mathcal{M}\) is relatively compact since \(\mathcal{M}\) is closed.
For each \(k=0,1,\ldots \) , thanks to \((H_{5})\), we know that \(\operatorname {Sel}_{F}(\mathcal{M}_{k})\) is integrably bounded. Then, applying Lemma 2.3, we get \(\mathfrak{L}(\mathcal{M}_{k})\) is equicontinuous. From this, we prove that \(\mathcal{M}_{k+1}\) is equicontinuous. Therefore, \(\mathcal{M}\) is also equicontinuous.
A direct calculation yields that, for every \(\nu \in \mathfrak{L}(x)\), \(h(0,\nu )= \mathcal{S}f^{*}\) and \(h(1,\nu )=\nu \). Also, it is not difficult to verify that h is continuous. Hence, \(\mathfrak{L}\) has contractible values.
Therefore, an application of Lemma 2.7 enables us to conclude that \(\mathfrak{L}\) has at least one fixed point in \(\mathcal{M}\). The proof is complete. □
Now, we are in a position to give the topological structure of \(\varPhi (x_{0})\).
Theorem 3.2
Let the hypotheses in Theorem 3.1 be satisfied. For each \(x_{0}\in D(E)\), \(\varPhi (x_{0})\) is a compact set.
Proof
Theorem 3.3
If the hypotheses in Theorem 3.1 are satisfied, then Φ is u.s.c.
Proof
Assume that \(K\subset D(E)\) is a compact subset and take \(\{x_{n}\} \subset \varPhi (K)\). As in Theorem 3.1, we know \(\{x_{n}\}\) is equicontinuous.
4 An example

\(f_{1}\) is l.s.c. and \(f_{2}\) is u.s.c.,

\(f_{1}(t,\xi ,x)\leq f_{2}(t,\xi ,x)\) for each \((t,\xi ,x)\in [0,T]\times [0,\pi ] \times \mathbb{R}\),

there exist \(l_{1},l_{2}\in L^{\infty }(0,T; \mathbb{R}^{+})\) such thatfor each \((t,\xi ,x)\in [0,T]\times [0,\pi ] \times \mathbb{R}\).$$ \bigl\vert f_{i}(t,\xi ,x) \bigr\vert \leq l_{1}(t) \vert x \vert +l_{2}(t),\quad i=1,2 $$
Therefore, hypotheses \((H_{1})\)–\((H_{5})\) are satisfied. If F satisfies \((H_{6})\), then Theorems 3.1, 3.2 enable us to obtain that the set of all mild solutions of system (4.1) is nonempty and it is a compact set.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their insightful and valuable comments.
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Funding
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Authors’ contributions
Both authors have contributed equally in this paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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