 Research
 Open Access
Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions
 M. Kamenskii^{1},
 V. Obukhovskii^{2},
 G. Petrosyan^{2} and
 JenChih Yao^{3}Email author
https://doi.org/10.1186/s1366301806521
© The Author(s) 2019
 Received: 22 November 2018
 Accepted: 21 December 2018
 Published: 21 January 2019
Abstract
Keywords
 Fractional differential equation
 Semilinear differential equation
 Cauchy problem
 Approximation
 Semidiscretization
 Index of the solution set
 Fixed point
 Condensing map
 Measure of noncompactness
MSC
 34A45
 34A08
 34G20
 47H08
 47H10
 47H11
1 Introduction
In the last years, the theory of differential equations and inclusions of fractional order attracted the attention of a large number of researchers. To a large extent, this is caused by its interesting applications in physics, enginery, biology, economics, and other sciences (see, e.g., monographs [1, 4, 9, 13, 22, 26, 28, 29, 31, 35], and the references therein). It is worth noting, in this connection, that one of the most important advantages of fractional order models in comparison with those of integer order is that a fractional order derivative of a function depends on its past values and hence becomes a powerful tool for the description of memory and hereditary properties of some media. A particular advantage of such an approach appears in the investigation of nonlocal boundary value problems, that is, in the association with the differential equation or inclusion of that type an initial condition depending on the behavior of the whole solution.
 (i)
\(\Delta (x) = \frac{1}{T}\int_{0}^{T} x(t)\,dt\) (mean value condition);
 (ii)
\(\Delta (x) = \sum_{i=1}^{n} \alpha _{i} x(t_{i}) + \xi \), with \(\xi \in E\), \(\alpha _{i} \neq 0\), \(t_{i} \in [0,T]\), \(i = 1, \dots ,n\) (multipoint discrete mean condition);
 (iii)
\(\Delta (x) \equiv \mathcal{M}\), with \(\mathcal{M} \subset E\) being a prescribed set (the generalized Cauchy problem).
Among a large amount of papers dedicated to fractionalorder equations and inclusions in Banach spaces, let us mention [3, 5, 15–17, 20, 21, 23, 24, 27, 33, 34, 36], where various existence results were obtained. In particular, in [5] a technique based on the weak topology methods was used to study a semilinear fractional differential inclusion subjected to a nonlocal initial condition. Notice that the results on the existence of solutions to the Cauchy and the periodic problems for semilinear differential inclusions in a Banach space were obtained in the authors’ papers [15, 17] by applying the methods of the theory of condensing multivalued maps. In [16] and [25] the authors justified the scheme of semidiscretization of the Cauchy problem for differential equations of the same type and presented results on the approximation of solutions to this problem. Notice also that the semidiscretization method for initial and periodic problems of ODEs in a Banach space was studied in [6, 11, 12, 30, 32], among other works. In the present work, we develop and extend the investigations in the same direction.
The structure of the paper is as follows. In the next section, we recall necessary notions and facts from the theory of differential equations of fractional order, measures of noncompactness and condensing maps. In the third section, we introduce the translation multivalued operator along the trajectories of the problem under consideration and prove that it is condensing with respect of the Hausdorff measure of noncompactness (Theorem 3). Based on this result, we show that this multivalued operator has a fixed point and therefore our problem has a solution (Theorem 4). In the last section, we develop the semidiscretization scheme and apply it to justify the approximation of solutions to the considered nonlocal boundary value problem (Theorem 6).
2 Preliminaries
2.1 Differential equations of fractional order
Recall some notions and definitions, which we will need in the sequel (details can be found, e.g., in [22, 28, 29, 35]).
Let E be a real Banach space.
Definition 1
Definition 2
Definition 3
2.2 Measures of noncompactness and condensing maps

\(\operatorname{Pb}(\mathcal{E}) = \{A\subseteq \mathcal{E}: A\neq \emptyset \mbox{ is bounded} \}\);

\(\operatorname{Pv}(\mathcal{E}) = \{A\in \operatorname{Pb}(\mathcal{E}): A \mbox{ is convex} \}\);

\(K(\mathcal{E}) = \{A\in \operatorname{Pb}(\mathcal{E}): A \mbox{ is compact} \}\);

\(Kv(\mathcal{E}) = \operatorname{Pv}(\mathcal{E})\cap K(\mathcal{E})\).
Definition 4
 (1)
monotone if for each \(\varOmega _{0},\varOmega _{1}\in \operatorname{Pb}(\mathcal{E})\), \(\varOmega _{0}\subseteq \varOmega _{1}\) implies \(\beta (\varOmega _{0})\leq \beta (\varOmega _{1})\);
 (2)
nonsingular if for each \(a\in \mathcal{E}\) and each \(\varOmega \in \operatorname{Pb}(\mathcal{E})\), we have \(\beta (\{a\}\cup \varOmega )=\beta (\varOmega )\);
 (3)
regular if \(\beta (\varOmega )=0\) is equivalent to the relative compactness of \(\varOmega \in \operatorname{Pb}(\mathcal{E})\);
 (4)
real if \(\mathcal{A}\) is the set of all real numbers \(\mathbb{R}\) with the natural ordering;
 (5)
algebraically semiadditive if \(\beta (\varOmega _{0} + \varOmega _{1}) \leq \beta (\varOmega _{0}) + \beta (\varOmega _{1})\) for every \(\varOmega _{0}, \varOmega _{1} \in \operatorname{Pb}(\mathcal{E})\).
Definition 5
Definition 6
 (i)
t is proper, i.e., \(t^{1}(K)\) is compact for every compact \(K \subset X\);
 (ii)
for each \(x \in X\) the set \(t^{1}(x)\) is acyclic, i.e., it has the same homologies as a onepoint space;
 (iii)
\(\mathcal{F}(x) = r(t^{1}(x))\), \(\forall x \in X\).
The class of Vietoris multivalued maps is sufficiently broad. To demonstrate this, recall the following notions.
Definition 7
A metric space X is called contractible if there exist a point \(x_{0} \in X\) and a continuous map (homotopy) \(h: [0,1] \times X \rightarrow X\) such that \(h(0,x) = x\) and \(h(1,x) = x_{0}\) for all \(x \in X\).
It is obvious that convex and, more generally, starshaped sets are contractible.
Definition 8
(see [14])
Notice that an \(R_{\delta }\)set is acyclic, but need not be contractible (see an example in [10]).
Definition 9
Let X be a metric space, \(\mathcal{E}\) a Banach space. A u.s.c. multivalued map \(\mathcal{F}: X \rightarrow K(\mathcal{E})\) is called an \(R_{\delta }\)multivalued map if for every \(x \in X\) the set \(\mathcal{F}(x)\) is \(R_{\delta }\).
Definition 10
From Proposition 3.4.1(a) of [18] it follows that every \(R^{c}_{\delta }\)multivalued map is a Vietoris map.
Then, by applying Corollary 3.4.3 in [18], we get the following fixed point theorem, which we will need in the sequel.
Theorem 1
Recall some notions (see, e.g., [7, 18]). Let E be a Banach space.
Definition 11

\(L^{p}\)integrable if it admits an \(L^{p}\)Bochner integrable selection, i.e., there exists a function \(g\in L^{p} ((0,\tau ); E )\) such that \(g(t)\in G(t)\) for a.e. \(t\in [0,\tau ]\);

\(L^{p}\)integrably bounded if there exists a function \(\xi \in L^{p}((0,\tau ))\) such thatfor a.e. \(t\in [0,\tau ]\).$$ \bigl\Vert G(t) \bigr\Vert \leq \xi (t) $$
The set of all \(L^{p}\)integrable selections of a multivalued function \(G: [0,\tau ] \to K(E)\) is denoted by \(\mathcal{S}^{p}_{G}[0,\tau ]\).
Definition 12
In the sequel we will need the following important property on the χestimation of the integral of a multivalued function.
Lemma 1
(see Theorem 4.2.3 in [18])
We will need the following auxiliary assertion which is an analogue of the known Gronwall lemma on integral inequalities.
Lemma 2
([17], Lemma 13)
3 Existence result
 \((A)\) :

\(A:D(A) \subseteq E\rightarrow E\) is a linear closed (not necessarily bounded) operator generating a bounded \(C_{0}\)semigroup \(\{U(t) \}_{t\geq 0}\) of linear operators in E.
 \((F1)\) :

for each \(x \in E\) the multivalued function \(F (\cdot ,x ): [0,T ]\rightarrow Kv (E ) \) admits a measurable selection;
 \((F2)\) :

for a.e. \(t\in [0,T]\) the multivalued map \(F(t,\cdot ):E \rightarrow Kv ( E ) \) is u.s.c.;
 \((F3)\) :

there exists a function \(\alpha \in L^{\infty }_{+} ([0,T])\) such that$$ \bigl\Vert F(t,x) \bigr\Vert _{E}\leq \alpha (t) \bigl(1+ \bigl\Vert x(t) \bigr\Vert _{E}\bigr) \quad \mbox{for a.e. } t\in [0,T], $$
 \((F4)\) :

there exists a function \(\mu \in L^{\infty }([0,T])\) such that for each bounded set \(\varOmega \subset E\) we havefor a.e. \(t \in [0,T]\), where χ is the Hausdorff MNC in E.$$ \chi \bigl(F(t,\varOmega )\bigr) \leq \mu (t) \chi (\varOmega ), $$
 \((\Delta 1)\) :

there exists a functional \(\mathfrak{f} \colon C([0,T]; \mathbb{R})\rightarrow \mathbb{R}_{+}\) and a constant \(C\geq 0\) which is:
 (i)
sublinear in the sense that \(\mathfrak{f}(\lambda _{0} \psi _{0} + \lambda _{1} \psi _{1}) \leq \lambda _{0} \mathfrak{f}(\psi _{0}) + \lambda _{1} \mathfrak{f} (\psi _{1})\), \(\forall \lambda _{0} \geq 0\), \(\lambda _{1} \geq 0\), \(\psi _{0}, \psi _{1} \in C([0,T];\mathbb{R})\);
 (ii)
monotone nondecreasing in the sense that \(\psi _{0}, \psi _{1} \in C([0,T];\mathbb{R})\), \(\psi _{0}(t) \leq \psi _{1}(t)\), \(\forall t \in [0,T]\) implies \(\mathfrak{f}(\psi_{0}) \leq \mathfrak{f}(\psi_{1})\)
$$ \bigl\Vert \Delta (x) \bigr\Vert _{E}\leq \mathfrak{f}\bigl( \bigl\Vert x(\cdot ) \bigr\Vert _{E}\bigr)+C. $$  (i)
 \((\Delta 2)\) :

Let \(\varOmega \subset C([0,T];E)\) be a nonempty bounded set and x a solution of scalar problem (2.4)–(2.5) with \(\lambda =  \eta \), \(\eta >0\) and \(x_{0} = \chi (\varOmega (0))\) such thatThen$$ \chi \bigl(\varOmega (t)\bigr)\leq x(t), \quad \forall t \in [0,T]. $$$$ \chi \bigl(\Delta (\varOmega )\bigr)\leq \mathfrak{f}(x). $$
Remark 1
 (i)
\(\mathfrak{f}(\varphi )=\frac{1}{T}\int ^{T}_{0}\varphi (s)\,ds\);
 (ii)
\(\mathfrak{f}(\varphi )=\sum_{i=1}^{n} \alpha _{i}  \varphi (t_{i})\), \(\alpha _{i} \neq 0\), \(t_{i} \in [0,T]\), \(i = 1, \dots ,n\);
 (iii)
\(\mathfrak{f}(\varphi )=0\).
Definition 13
(see, e.g., [15])
Remark 2
\(\xi _{q} (\theta )\geq 0\), \(\int ^{\infty }_{0} \xi _{q} (\theta ) \,d\theta =1\), \(\int ^{\infty }_{0} \theta \xi _{q} (\theta ) \,d\theta =\frac{1}{\varGamma (q+1)}\).
Lemma 3
 (1)for each \(t \in [0,T]\), \(\mathcal{G}(t)\) and \(\mathcal{T}(t)\) are linear bounded operators, more precisely, for each \(x \in E\) we have$$\begin{aligned}& \bigl\Vert \mathcal{G}(t)x \bigr\Vert _{E}\leq M \Vert x \Vert _{E}, \end{aligned}$$(3.5)where$$\begin{aligned}& \bigl\Vert \mathcal{T}(t)x \bigr\Vert _{E}\leq \frac{qM}{\varGamma (1+q)} \Vert x \Vert _{E}, \end{aligned}$$(3.6)$$ M = \sup_{t \geq 0} \bigl\Vert U(t) \bigr\Vert . $$
 (2)
the operator functions \(\mathcal{G}(\cdot )\) and \(\mathcal{T}(\cdot )\) are strongly continuous, i.e., functions \(t\in [0,T] \to \mathcal{G}(t)x\) and \(t\in [0,T] \to \mathcal{T}(t)x\) are continuous for each \(x \in E\).
Remark 3
By the symbol \(\varSigma ^{F}_{x_{0}}\) we will denote the set of all mild solutions to the Cauchy problem (3.1), (3.3) on the interval \([0,T]\).
From the results of [15, 17] about the existence and topological structure of solutions to the Cauchy problem (3.1) and (3.3), the next assertion follows.
Theorem 2
Now we will consider the translation multivalued operator \(\varTheta \colon D \subseteq E \multimap E\) along the trajectories of problem (3.1)–(3.2) defined as \(\varTheta = \Delta \circ \varSigma \), where \(D \subset E\) is an open subset. It is clear that Θ is an \(R^{c}_{\delta }\)multivalued map.
Theorem 3
 \((A1)\) :

the semigroup U is exponentially decreasing in the sense thatfor some \(\eta > \\mu \_{\infty }\), where \(\mu (\cdot )\) is the function from condition \((F4)\).$$ \bigl\Vert U(t) \bigr\Vert \leq e^{\eta t}, \quad t \geq 0 $$
Proof
Now we are in position to prove the main result of this section.
Theorem 4
Proof
Let \(b(t)=\x_{0}\_{E} E_{q}((\eta + \lambda )t^{q}) + \mathcal{C}\).
4 Approximation of solutions
In this section we will apply a semidiscretization scheme for the approximation of solutions to problem (3.1)–(3.2).
Notice that a function \(\overline{x}_{h} \in C([0,T];E_{h})\) is a solution of Eq. (4.7) if and only if it is a mild solution of problem (4.1), (4.6) with \(y_{0}^{h} = \overline{x}_{h}(0)\).
 \((H1)\) :

for each \(x\in E\),as \(h\rightarrow 0\) uniformly in \(t\in [0,T]\);$$ Q_{h}U_{h}(t)P_{h}x\rightarrow U(t)x $$
 \((H2)\) :

there exists \(k>0\) such thatfor each \(t \in [0,T]\) and bounded \(\varOmega \subset E\).$$ \chi _{E} \biggl(\bigcup _{h\in H}Q_{h}F_{h} \bigl(t,P_{h}(\varOmega )\bigr) \biggr) \leq k\chi _{E} (\varOmega ) $$
 \((H3)\) :

the multivalued map \((h,x)\multimap Q_{h}F_{h}(t,P_{h}x)\) is u.s.c. for a.e. \(t\in [0,T]\).
Remark 4
In the sequel we will need the following assertion.
Lemma 4
([16], Lemma 2)
Theorem 5
 \((A1_{h})\) :

the semigroups \(U_{h}\) are exponentially decreasing in the sense thatfor some \(\eta > k\), where k is the constant from condition \((H2)\).$$ \bigl\Vert U_{h}(t) \bigr\Vert \leq e^{\eta t}, \quad t \geq 0 $$
Proof
Let \(\varOmega \subset E\) be a nonempty bounded set.
Theorem 6
Under condition (3.12), the sequence \(\{x^{h_{n}}\}\) is relatively compact and its limit points are initial values of solutions to inclusion (3.1) satisfying boundary value condition (3.2).
Proof
So, the sequence \(\{\nu ^{h_{n}}\}\) is relatively compact in \(C([0,T];E)\). We will assume, w.l.o.g., that \(\nu ^{h_{n}} \rightarrow \nu ^{0}\).
Declarations
Acknowledgements
The authors are grateful to the anonymous referee for his/her helpful remarks. The work on the paper was carried out during the visit of V. Obukhovskii and M. Kamenskii to the Center for Fundamental Science, Kaohsiung Medical University and the Department of Applied Mathematics, National Sun YatSen University, Kaohsiung, Taiwan in 2018. They would like to express their gratitude to the members of the Center and the Department for their kind hospitality.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
The work is supported by the joint Taiwan MOST—Russia RFBR grant 175152022. The work of the first, second and third authors is supported by the Ministry of Education and Science of the Russian Federation in the frameworks of the project part of the state work quota (Project No 1.3464.2017/4.6). J.C. Yao was supported by the Grant MOST 1062923E039001MY3.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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