Mild solutions to fractional differential inclusions with nonlocal conditions
 Tingting Lian^{1},
 Changfeng Xue^{1} and
 Shaozhong Deng^{2}Email authorView ORCID ID profile
Received: 18 October 2016
Accepted: 28 November 2016
Published: 8 December 2016
Abstract
This article is concerned with the existence of mild solutions for fractional differential inclusions with nonlocal conditions in Banach spaces. The results are obtained by using fractional calculus, Hausdorff measure of noncompactness, and the multivalued fixed point theorem. The results obtained in the present paper extend some related results on this topic.
Keywords
1 Introduction
Fractional differential equations and inclusions have gained considerable interest due to their applications in various fields, such as physics, mechanics, and engineering, in part because they have been found to be more realistic and practical to describe many natural phenomena [1–8]. For more details about fractional calculus and fractional differential equations, we refer the reader to the books by Podlubny [9], Sabatier et al. [10], Kilbas et al. [11], and the papers by Eidelman and Kochubei [12], Lakshmikantham and Vatsala [13], and Agarwal et al. [14].
The study of abstract nonlocal differential problems was initiated by Byszewski and Lakshmikantham [15], who gave three theorems on the existence and uniqueness of the mild, strong, and classical solutions of a semilinear evolution nonlocal Cauchy problem by using the method of semigroups and the Banach fixed point theorem and argued there that a nonlocal condition can be applied in physics with better effects than classical initial conditions. This work was then followed by a lot of other research, and some basic results on nonlocal differential problems have been obtained [16–18]. Most results, however, were obtained with the assumption that the involved semigroup is compact, and one of the difficulties in these nonlocal problems is how to deal with the compactness of the solution operator at \(t=0\). Many methods and techniques have been developed to avoid this difficulty. We refer the reader to papers [19–21] and the references therein.
From the mathematical point of view, it is natural to combine fractional differential equations with nonlocal conditions. For example, Zhou and Jiao [22] discussed the nonlocal fractional evolution equations based on the Krasnoselskii fixed point theorem with the assumption that the involved semigroup is compact and the nonlocal term is Lipschitz continuous. Li et al. [23] studied the existence of mild solutions to fractional differential equations by using the Hausdorff measure of noncompactness when the semigroup is equicontinuous and the nonlocal term is compact. Ji and Li [24] also studied nonlocal fractional differential equations in general Banach spaces but without any compactness assumptions to the operator semigroup. Recently, Ji [25] studied the control system governed by a class of abstract nonlocal fractional differential equations. To the best of the authors’ knowledge, however, few work has been reported on the existence of solutions for fractional differential inclusions with nonlocal conditions governed by a linear closed operator that generates an equicontinuous semigroup.
In this paper, we assume that X is a real Banach space with norm \(\ \cdot\\). Let \(J=[0, b]\) with \(b>0\). We denote by \(C(J, X)\) the Banach space of continuous functions from J into X with the norm \(\x\ =\sup_{t\in J}\{\x(t)\\}\) for \(x\in C(J, X)\). We further denote by \(L^{1}(J, X)\) the space of Bochnerintegrable functions from J into X with the norm given by \(\f\_{L^{1}} =\int^{b}_{0} \f(t)\\,\mathrm{d}t \) for \(f\in L^{1}(J, X)\).
Using the technique of Hausdorff measure of noncompactness and the multivalued fixed point theorem, we prove some existence results on the fractional semilinear nonlocal differential inclusion (1.1)(1.2). We assume that the semigroup \(\{T(t)\} _{t\geq0}\) is equicontinuous. Note that the case of compact \(\{T(t)\}_{t\geq0}\) is just a particular case of our assumption. Therefore, the results in the present paper extend to some extent those in [23, 26].
This paper is organized as follows. In Section 2, we present some relevant definitions and facts about fractional derivative and integral, the Hausdorff measure of noncompactness, and the setvalued analysis. In Section 3, we give the existence results of mild solutions for problem (1.1)(1.2). In Section 4, an example is given to briefly show a potential application of our results.
2 Preliminaries

\(P(X)= \{B\subseteq X: B \mbox{ is nonempty and bounded}\}\),

\(P_{cl}(X)=\{ B\subseteq X: B \mbox{ is nonempty and closed}\}\),

\(P_{cp}(X)=\{B\subseteq X: B \mbox{ is nonempty and compact}\}\),

\(P_{cl,cv}(X)=\{B\subseteq X: B \mbox{ is nonempty, closed, and convex}\} \), and

\(P_{cp, cv}(X)=\{B\subseteq X: B \mbox{ is nonempty, compact, and convex}\}\).
Now let us recall the definitions of fractional derivative and integral.
Definition 1
Definition 2
Now using the probability density function and its Laplace transform developed in [27], we give the following definition of mild solutions to problem (1.1)(1.2).
Definition 3
Lemma 1
 (1)For any fixed \(t\geq0\), both \(\mathcal{T}_{q}(t)\) and \(\mathcal{S}_{q}(t)\) are bounded operators, that is, for any \(x\in X\), we have [24]$$\begin{aligned} & \bigl\ \mathcal{T}_{q}(t) x\bigr\ \leq M\ x\, \end{aligned}$$(2.10)where \(M=\sup_{t\in J}\T(t)\\).$$\begin{aligned} & \bigl\ \mathcal{S}_{q}(t)x\bigr\ \leq\frac{Mq}{\Gamma(1+q)}\x\, \end{aligned}$$(2.11)
 (2)
Both \(\mathcal{T}_{q}(t)\) and \(\mathcal{S}_{q}(t)\) are equicontinuous for \(t\in J\) if \(\{T(t)\}_{t\geq0}\) is equicontinuous.
Proof
In the same way we can prove that \(\\mathcal{S}_{q}(t+h)x\mathcal {S}_{q}(t)x\\rightarrow0\) as \(h\rightarrow0\), uniformly for all \(x\in B\), and thus \(\mathcal{S}_{q}(t)\) is also equicontinuous. □
Lemma 2
[32]
 (1)
B is relatively compact if and only if \(\beta(B)=0\);
 (2)
\(\beta(B)=\beta(\overline{B})=\beta(\operatorname{conv}B)\), where B̅ and convB represent the closure and the convex hull of B, respectively;
 (3)
\(\beta(B)\leq\beta(C)\) when \(B\subseteq C\);
 (4)
\(\beta(B+C)\leq\beta(B)+\beta(C)\), where \(B+C=\{ x+y:x\in B,y\in C\}\);
 (5)
\(\beta (B\cup C )\leq\max\{\beta(B),\beta (C)\}\);
 (6)
\(\beta(\lambda(B))\leq\lambda\beta(B)\) for any \(\lambda\in \mathbb{R}\);
 (7)
If the map \(Q:D(Q)\subseteq X\rightarrow Z\) is Lipschitz continuous with constant k, then \(\beta_{Z}(QB) \leq k \beta(B)\) for any bounded subset \(B\subseteq D(Q)\), where Z is a Banach space. and \(\beta_{Z}(\cdot)\) is the Hausdorff measure of noncompactness associated with Z;
 (8)
If \(\{W_{n}\} _{n=1} ^{\infty}\) is a decreasing sequence of bounded and closed nonempty subsets of X and \(\lim_{n\rightarrow\infty} \beta(W_{n})=0\), then \(\bigcap_{n=1}^{\infty}W_{n}\) is nonempty and compact in X.
Lemma 3
[32]
Lemma 4
[33]
Lemma 5
[34]
In addition, for completeness, we further include some basic definitions and results on multivalued maps. For more details about the multivalued maps, see the books by Deimling [35] and Hu and Papageorgiou [36].
Definition 4
 (1)
A multivalued map \(F: X \rightarrow P(Y)\) is said to be convex (closed) valued if \(F(x)\) is convex (closed) in Y for all \(x\in X\). Recall that \(P(Y)\) represents the set of all nonempty and bounded subsets of Y.
 (2)
F is said to be completely continuous if \(F(B)\) is relatively compact for every bounded subset B of X.
 (3)
F is said to have a fixed point if there is \(x\in X\) such that \(x\in F(x)\).
 (4)
F is said to be upper semicontinuous (u.s.c.) on X if \(F^{1}(V)=\{x\in X: F(x)\subseteq V\}\) is an open subset of X for every open subset V of Y.
 (5)
F is said to be closed if its graph \(G_{F}=\{(x,y)\in X\times Y: y\in F(x)\}\) is a closed subset of the topological space \(X\times Y\), that is, \(x_{n}\rightarrow x\), \(y_{n}\rightarrow y\), and \(y_{n}\in F(x_{n})\) imply \(y\in F(x)\).
Remark 1
Note that if \(D\subset X\) is closed, \(F(x)\) is closed for all \(x\in D\), and \(\overline{F(D)}\) is compact, then F is u.s.c. if and only if F is closed.
Definition 5
 (1)It is integrably bounded, that is, there is \(\omega\in L^{1}(J, \mathbb{R}^{+})\) such that$$\bigl\ f_{n}(t)\bigr\ \leq\omega(t) \quad\mbox{for a.e. } t\in J. $$
 (2)
The set \(\{f_{n}(t):n\in\mathbb{N}\}\) is relatively compact in X for a.e. \(t\in J\).
Lemma 6
[37]
Every semicompact sequence in \(L^{1}(J, X)\) is weakly compact in the space \(L^{1}(J, X)\).
Lemma 7
[26]
3 Main results
 (HA):

The \(C_{0}\) semigroup \(\{T(t)\}_{t\geq0}\) generated by the linear operator A is equicontinuous. We denote$$M=\sup_{t\in{J}} \bigl\{ \bigl\Vert T(t)\bigr\Vert \bigr\} . $$
 (Hg):

The nonlocal term \(g: C(J, X)\rightarrow X\) is continuous and compact, and there exists a constant \(N>0\) such that \(\g(u)\\leq N\) for all \(u\in{C(J,X)}\).
 (HF):

The multivalued operator \(F: J\times X\rightarrow P_{cp,cv}(X)\) satisfies the following hypotheses:
 (1)F is measurable to t for every \(x\in X\) and u.s.c. to x for a.e. \(t\in J\). For every \(u \in C(J, X)\), the setis nonempty.$$S_{F}(u)= \bigl\{ f\in L^{1}(J, X): f(t)\in F\bigl(t,u(t) \bigr), \mbox{ a.e. }t\in J \bigr\} $$
 (2)There exists a function \(m\in L^{{1}/{q_{1}}}(J, \mathbb {R}^{+})\) with \(q_{1}\in(0, q)\) such that, for any \(x \in X\),for a.e. \(t\in J\).$$ \bigl\Vert F(t,x)\bigr\Vert =\sup \bigl\{ \Vert y\Vert : y \in F(t,x) \bigr\} \leq m(t) $$(3.1)
 (3)There exists a constant \(L>0\) such that, for any bounded subset D of X, we havefor a.e. \(t\in J\).$$ \beta\bigl(F(t, D)\bigr) \leq L\beta(D) $$(3.2)
 (1)
 (Hη):

There exists a function \(\eta\in C(J, \mathbb {R}^{+})\) such that, for each constant \(\lambda\in(1,0)\), we havewhere \(M, N, q_{1}\), and m are from (HA), (Hg), and (HF)(2).$$ \eta(t)\geq MN+\frac{qM}{\Gamma(1+q)}\frac{t^{(1+\lambda )(1q_{1})}}{(1+\lambda)^{(1q_{1})}}\m \_{L^{{1}/{q_{1}}}[0,t]},\quad t\in J, $$(3.3)
In the proof of the existence results, we also need the following lemmas.
Lemma 8
[38]
Lemma 9
[39] (Fixed point theorem)
If W is a bounded, closed, convex, and compact nonempty subset of X and the map \(F: W\rightarrow2^{W}\) is upper semicontinuous with \(F(x)\) being a closed and convex nonempty subset of W for each \(x\in W\), then F has at least one fixed point in W.
Lemma 10
[40]
Now we are ready to prove the following existence result for the nonlocal fractional differential inclusion (1.1)(1.2).
Theorem 1
If hypotheses (HA), (Hg), (HF)(1)(2)(3), and (Hη) are satisfied, then the fractional differential inclusion (1.1)(1.2) has at least one mild solution on J.
Proof
Step 1. We show that the values of R are convex and closed subsets in \(C(J, X)\).
Step 2. We construct a bounded, convex, closed, and compact nonempty subset \(W\subseteq C(J, X)\) such that R maps W into itself.
Step 3. We show that the graph of R is closed.
Step 4. We show that R is u.s.c. on W.
As a consequence of the previous proof, we have that W is closed and \(R(u)\) is closed for all \(u\in W\). The set \(\overline{R(W)}\subseteq W\) is compact. Moreover, R is closed. According to Remark 1, we can come to the conclusion that R is u.s.c.
Finally, due to the fixed point of Lemma 9, R has at least one point \(u \in R (u)\), and u is a mild solution to the fractional semilinear differential inclusion (1.1) with the nonlocal condition (1.2). □
Remark 2
If A generates an analytic semigroup or a differential semigroup \(\{ T(t)\}_{t\geq0}\), then \(\{T(t)\}_{t\geq0}\) is equicontinuous [41]. In applications of partial differential equations, such as parabolic equations and strongly damped wave equations, the corresponding solution semigroups are analytic. Therefore, the results in this paper have wide applicability.
 (Hg′):

The nonlocal term \(g: C(J, X)\rightarrow X\) is continuous and compact.
 (HF)(2′):

There exist a function \(\alpha\in L^{{1}/{q_{1}}}(J, \mathbb{R}^{+})\) for some given \(q_{1}\in(0,q)\) and an increasing function \(\Omega :\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) such thatfor a.e. \(t\in J\) and all \(x\in X\).$$\bigl\Vert F(t,x)\bigr\Vert \leq\alpha(t)\Omega\bigl(\Vert x\Vert \bigr) $$
Theorem 2
Proof
4 An example
Declarations
Acknowledgements
The authors thank the support of the National Natural Science Foundation of China (Grant No. 11471281) for the work reported in this paper.
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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