Abstract
In this paper, we study the existence of solutions for a new class of boundary value problems for nonlinear multi-term fractional differential inclusions. Our main result relies on the multi-valued form of Krasnoselskii’s fixed point theorem. An illustrative example is also presented.
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1 Introduction and preliminaries
In this paper we study the existence of solutions for the following multi-term fractional differential inclusions:
supplemented with boundary conditions
where \({}^{c}D^{\alpha}\), \({}^{c}D^{q_{i}}\) denote the Caputo fractional derivatives, \(2 < \alpha\leq3 \), \(1 < q_{i} \leq2\), \(i=1,2,\ldots, k\), \(t\in J:=[0,1]\), \(1< p\leq2\), \(k\geq1\), and \(F, G: J\times\mathbb {R}^{k+3} \to{\mathcal{P}}(\mathbb{R})\) are multifunctions.
Many of published papers about fractional differential equations and inclusions apply the fixed point theory for proving the existence results. For instance, one can find a lot of papers in this field (see [1–25] and the references therein).
Let \(\alpha>0\), \(n-1<\alpha<n\), \(n=[\alpha]+1\), and \(u\in C([a,b], \mathbb{R})\). The Caputo derivative of fractional of order α for the function u is defined by \({}^{c}D^{\alpha} u(t)= \frac{1}{\Gamma (n-\alpha)}\int_{0}^{t} (t-\tau)^{n-\alpha-1}u^{(n)}(\tau)\, d\tau\) (see for more details [11, 23, 25–27]). Also, the Riemann-Liouville fractional order integral of the function u is defined by \(I^{\alpha}u(t)= \frac{1}{\Gamma(\alpha)} \int_{0}^{t} \frac{u(\tau)}{(t-\tau )^{1-\alpha}}\, d\tau\) (\(t>0\)) whenever the integral exists [11, 23, 25–27]. In [28], it has been proved that the general solution of the fractional differential equation \({}^{c}D^{\alpha}u( t)=0\) is given by \(u( t)=c_{0}+c_{1}t+c_{2}t^{2}+\cdots+c_{n-1}t^{n-1}\), where \(c_{0},\ldots,c_{n-1}\) are real constants and \(n=[\alpha]+1\). Also, for each \(T>0\) and \(u\in C([0,T])\) we have
where \(c_{0},\ldots,c_{n-1}\) are real constants and \(n=[\alpha]+1\) [28].
Now, we review some definitions and notations as regards multifunctions [29, 30].
For a normed space \((X, \|\cdot\|)\), let \({\mathcal{P}}_{\mathrm{cl}}(X)=\{Y \in {\mathcal{P}}(X) : Y \mbox{ is closed}\}\), \({\mathcal{P}}_{\mathrm{b}}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is bounded}\}\), \({\mathcal{P}}_{\mathrm{cp}}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is compact}\}\), and \({\mathcal{P}}_{\mathrm{cp}, \mathrm{cv}}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is compact and convex}\}\), \({\mathcal{P}}_{\mathrm{b}, \mathrm{cl}, \mathrm{cv}}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is bounded, closed, and convex}\}\). A multi-valued map \(G : X \to{\mathcal{P}}(X)\) is convex (closed) valued if \(G(x)\) is convex (closed) for all \(x \in X\). The map G is bounded on bounded sets if \(G(\mathbb{B}) = \bigcup_{x \in\mathbb{B}}G(x)\) is bounded in X for all \(\mathbb{B} \in{\mathcal{P}}_{\mathrm{b}}(X)\) (i.e., \(\sup_{x \in\mathbb{B}}\{\sup\{|y| : y \in G(x)\}\} < \infty\)). G is called upper semi-continuous (u.s.c.) on X if for each \(x_{0} \in X\), the set \(G(x_{0})\) is a nonempty closed subset of X, and if for each open set N of X containing \(G(x_{0})\), there exists an open neighborhood \(\mathcal{N}_{0}\) of \(x_{0}\) such that \(G(\mathcal{N}_{0}) \subseteq N\). G is said to be completely continuous if \(G(\mathbb{B})\) is relatively compact for every \(\mathbb{B} \in{\mathcal{P}}_{\mathrm{b}}(X)\). If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., \(u_{n} \to u_{*}\), \(y_{n} \to y_{*}\), \(y_{n} \in G(u_{n})\) imply \(y_{*} \in G(u_{*})\). G has a fixed point if there is \(x \in X\) such that \(x \in G(x)\). The fixed point set of the multi-valued operator G will be denoted by FixG. A multi-valued map \(G : J \to {\mathcal{P}}_{\mathrm{cl}}(\mathbb{R})\) is said to be measurable if for every \(y \in \mathbb{R}\), the function \(t \mapsto d(y,G(t)) = \inf\{|y-z|: z \in G(t)\}\) is measurable.
Consider the Pompeiu-Hausdorff metric \(H_{d} : {\mathcal{P}}(X) \times {\mathcal{P}}(X) \to\mathbb{R} \cup\{\infty\}\) given by
where \(d(A,b) = \inf_{a\in A}d(a;b)\) and \(d(a,B) = \inf_{b\in B}d(a;b)\). A multi-valued operator \(N : X \to{\mathcal{P}}_{\mathrm{cl}}(X)\) is called contraction if there exists \(\gamma \in(0,1)\) such that \(H_{d}(N(x),N(y)) \le\gamma d(x,y)\) for each \(x, y \in X\).
We say that \(F: J\times\mathbb{R}^{k+3} \rightarrow{\mathcal{P}}(\mathbb{R})\) is a Carathéodory multifunction if \(t\mapsto F(t,u_{1}, \ldots, u_{k+3})\) is measurable for all \(u_{i} \in\mathbb{R}\) and \((u_{1}, \ldots, u_{k+3})\mapsto F(t,u_{1}, \ldots, u_{k+3}) \) is upper semi-continuous for almost all \(t\in J\) [29, 31]. Also, a Carathéodory multifunction \(F: J\times \mathbb{R}^{k+3} \rightarrow{\mathcal{P}}(\mathbb{R})\) is called \(L^{1}\)-Carathéodory if for each \(\rho>0\) there exists \(\phi_{\rho}\in L^{1}(J,\mathbb{R}^{+})\) such that
for all \(|u_{1}|, \ldots, |u_{k+3}|\leq\rho\) and for almost all \(t\in J\) [29, 31].
Define the set of selections of F and G at \(u \in C(J,\mathbb{R})\) by
and
for almost all \(t\in J \). If F is an arbitrary multifunction, then it has been proved that \(S_{F}(u)\neq\emptyset\) for all \(u\in C(J,X)\) if \(\dim X<\infty\) [32].
The graph of a function F is the set \(\operatorname{Gr}(F)=\{ (x,y)\in X\times Y: y\in F(x)\} \) [29]. The graph \(\operatorname{Gr}(F)\) of \(F:X\to\mathcal {P}_{\mathrm{cl}}(Y)\) is said to be a closed subset of \(X\times Y\), if for every sequence \(\{u_{n}\}_{n \in\mathbb{N}} \subset X\) and \(\{y_{n}\}_{n \in \mathbb{N}} \subset Y\), when \(n \to\infty\), \(u_{n} \to u_{0}\), \(y_{n} \to y_{0}\), and \(y_{n} \in F(u_{n})\), then \(y_{0} \in F(u_{0})\) [29].
We will use the following lemmas and theorem in our main result.
Lemma 1.1
([29], Proposition 1.2)
If \(F : X \to\mathcal{P}_{\mathrm{cl}}(Y)\) is u.s.c., then \(\operatorname{Gr}(F)\) is a closed subset of \(X \times Y\). Conversely, if F is completely continuous and has a closed graph, then it is upper semi-continuous.
Lemma 1.2
([32])
Let X be a separable Banach space. Let \(F : [0, 1] \times X^{k+3} \to{\mathcal{P}}_{\mathrm{cp},\mathrm{cv}}(X)\) be an \(L^{1}\)-Carathéodory function. Then the operator
is a closed graph operator.
Theorem 1.3
([33], Krasnoselskii’s fixed point theorem)
Let X be a Banach space, \(Y\in{\mathcal{P}}_{\mathrm{b}, \mathrm{cl}, \mathrm{cv}}(X)\) and \(A, B: Y\to{\mathcal{P}}_{\mathrm{cp},\mathrm{cv}}(X)\) two multi-valued operators. If the following conditions are satisfied:
-
(i)
\(Ay+By\subset Y\) for all \(y\in Y\);
-
(ii)
A is a contraction;
-
(iii)
B is u.s.c. and compact,
then there exists \(y\in Y\) such that \(y\in Ay+By\).
2 Main results
Now, we are ready to prove our main result. Let \(X=\{u: u,u', u'', {}^{c}D^{q_{i}} u\in C(J,\mathbb{R}), i=1,2,\ldots, k\}\). Then \((X, \Vert \cdot\Vert)\) endowed with the norm
is a Banach space [34].
We need the following auxiliary lemma. See also [35, 36].
Lemma 2.1
Let \(y\in C(J,{\mathbb{R}})\) and \(u\in C^{2}([0,1],\mathbb{R})\) is a solution to the fractional boundary value problem
then
and vice versa, where \(\Delta= \frac{\Gamma(3-p)}{4\Gamma(3-p) + 2} \neq0 \).
Proof
It is well known that the solution of equation \({}^{c}D^{\alpha}u(t)=y(t)\) can be written as
where \(c_{0}, c_{1}, c_{2}\in\mathbb{R}\). Then we get
and
By using the boundary value conditions, we obtain \(c_{0}=0\) and
and
Substituting the values of \(c_{0}\), \(c_{1}\), and \(c_{2}\) in (2.3) we get (2.2).
Conversely, applying the operator \({}^{c}D^{\alpha}\) on (2.2) and taking into account (2.1), it follows that \({}^{c}D^{\alpha }u(t)=y(t)\). From (2.2) it is easily to verify that the boundary conditions \(u(0)=0\), \(u'(0)=-u(1)- u'(1)\), \(u''( 0) = - u''(1)- {}^{c}D^{p}u(1)\) are satisfied. This establishes the equivalence between (2.1) and (2.2). The proof is completed. □
Definition 2.2
A function \(u\in C^{2}([0,1],\mathbb{R})\) is called a solution for the problem (1.1)-(1.2) if it satisfies the boundary value conditions \(u(0)=0\), \(u'(0)=-u(1)- u'(1)\), and \(u''( 0) = - u''(1) - {}^{c}D^{p}u(1)\), there exist functions \(v, v_{1}\in L^{1}(J, {\mathbb{R}})\) such that \(v(t)\in F(t, u(t), u'(t), u''(t), {}^{c}D^{q_{1}}u(t), \ldots, {}^{c}D^{q_{k}}u(t))\), \(v_{1}(t) \in G(t, u(t), u'(t), u''(t), {}^{c}D^{q_{1}}u(t), \ldots, {}^{c}D^{q_{k}}u(t))\) for almost all \(t\in J\) and
Remark 2.3
For the sake of brevity, we set
and, for each \(i=1,\ldots, k\),
Also in the following we use the notation \(\|x\|_{\infty}=\sup\{|x(t)|: t\in J\}\).
Theorem 2.4
Suppose that:
- (H1):
-
\(F: J\times\mathbb{R}^{k+3} \to{\mathcal{P}}_{\mathrm{cp},c}(\mathbb{R})\) is a multifunction and \(G:J\times\mathbb {R}^{k+3} \to{\mathcal{P}}_{\mathrm{cp},c}(\mathbb{R})\) is a Carathéodory multifunction;
- (H2):
-
there exist continuous functions \(p, m: J \to(0,\infty )\) such that \(t\mapsto F(t,w_{1},w_{2},w_{3},z_{1}, \ldots, z_{k} ) \) is measurable and
$$\bigl\Vert F(t,w_{1},w_{2},w_{3},z_{1}, \ldots, z_{k} ) \bigr\Vert \leq m(t),\qquad \bigl\Vert G(t,w_{1},w_{2},w_{3},z_{1},\ldots, z_{k} ) \bigr\Vert \leq p(t); $$ - (H3):
-
there exists a continuous function \(h:J\to(0,\infty)\) such that
$$\begin{aligned}& H_{d} \bigl(F(t,w_{1},w_{2},w_{3},z_{1}, \ldots, z_{k}) , F\bigl(t,w'_{1},w'_{2},w'_{3},z'_{1}, \ldots, z'_{k}\bigr)\bigr) \\& \quad \leq h(t) \Biggl[ \sum_{i=1}^{3} \bigl\vert w_{i} -w'_{i} \bigr\vert + \sum _{i=1}^{k} \bigl\vert z_{i} -z'_{i} \bigr\vert \Biggr] \end{aligned}$$for all \(t\in J\) and for each \(w_{1},w_{2},w_{3},z_{1},\ldots, z_{k}, w'_{1},w'_{2},w'_{3},z'_{1},\ldots, z'_{k} \in\mathbb{R}\).
If
for \(i=1,2,\ldots,k\), where the \(\Lambda_{j}\) (\(j=1,\ldots,4\)) are defined in (2.5)-(2.8), then the inclusion problem (1.1)-(1.2) has at least one solution.
Proof
We define the subset Y of X by \(Y=\{ u\in X: \Vert u\Vert\leq M \} \), where
It is clear that Y is closed, bounded, and convex subset of Banach space X. We define the multi-valued operators \(A,B:Y\to{\mathcal{P}}(X)\) such that for some \(v\in S_{F,u}\),
and for some \(v_{1}\in S_{G,u}\),
In this way, the fractional differential inclusion (1.1)-(1.2) is equivalent to the inclusion problem \(u\in Au+Bu\). We show that the multi-valued operators A and B satisfy the conditions of Theorem 1.3 on Y.
First, we show that the operators A and B define the multi-valued operators \(A,B: Y\to{\mathcal{P}}_{\mathrm{cp},\mathrm{cv}}(X)\). First we prove that A is compact-valued on Y. Note that the operator A is equivalent to the composition \({\mathcal{L}} \circ S_{F}\), where \({\mathcal{L}} \) is the continuous linear operator on \(L^{1}(J, \mathbb{R})\) into X, defined by
Suppose that \(u\in Y\) is arbitrary and let \(\{v_{n}\}\) be a sequence in \(S_{F,u}\). Then, by definition of \(S_{F,u}\), we have \(v_{n}(t)\in F(t, u(t), u'(t), u''(t), {}^{c}D^{q_{1}}u(t), \ldots, {}^{c}D^{q_{k}}u(t) )\) for almost all \(t\in J\). Since \(F(t, u(t), u'(t), u''(t), {}^{c}D^{q_{1}}u(t), \ldots, {}^{c}D^{q_{k}}u(t) )\) is compact for all \(t\in J\), there is a convergent subsequence of \(\{v_{n}(t)\}\) (we denote it by \(\{v_{n}(t)\}\) again) that converges in measure to some \(v(t)\in S_{F,u}\) for almost all \(t\in J\). On the other hand, \({\mathcal{L}} \) is continuous, so \({\mathcal{L}} (v_{n})(t)\to{\mathcal{L}} (v)(t)\) pointwise on J.
In order to show that the convergence is uniform, we have to show that \(\{{\mathcal{L}} (v_{n})\}\) is an equi-continuous sequence. Let \(t_{1}, t_{2} \in J\) with \(t_{1}< t_{2} \). Then we have
Continuing this process, we have
and
and, finally, for every \(i=1,\ldots, k\),
We see that the right-hand sides of the above inequalities tend to zero as \(t_{2}\to t_{1}\). Thus, the sequence \(\{{\mathcal{L}} (v_{n})\}\) is equi-continuous and by using the Arzelá-Ascoli theorem, we see that there is a uniformly convergent subsequence. So, there is a subsequence of \(\{v_{n}\}\) (we denote it again by \(\{v_{n}\}\)) such that \({\mathcal{L}} (v_{n})\to{\mathcal{L}} (v)\). Note that \({\mathcal{L}} (v) \in {\mathcal{L}} (S_{F,u})\). Hence, \(A(u) ={\mathcal{L}} (S_{F,u})\) is compact for all \(u\in Y\). So \(A(u)\) is compact.
Now, we show that \(A(u)\) is convex for all \(u\in X\). Let \(z_{1},z_{2}\in A(u)\). We select \(f_{1},f_{2}\in S_{F,u}\) such that
for almost all \(t\in J\). Let \(0\leq\lambda\leq1\). Then we have
Since F has convex values, \(S_{F,u}\) is convex and \(\lambda f_{1}(s) + (1-\lambda)f_{2}(s)\in S_{F,u}\). Thus
Consequently, A is convex-valued. Similarly, B is compact and convex-valued.
Here, we show that \(A(u)+ B(u)\subset Y\) for all \(u\in Y\). Suppose that \(u\in Y\) and \(z_{1}\in A(u)\), \(z_{2}\in B(u)\) are arbitrary elements. Choose \(v_{1}\in S_{F,u}\) and \(v_{2}\in S_{G,u}\) such that
and
for almost all \(t\in J\). Hence, we get
Hence, \(\sup_{t\in J}\vert z_{1}(t)+z_{2}(t) \vert\leq(\Vert p\Vert_{\infty}+\Vert m\Vert_{\infty})\Lambda_{1}\). Also we have
which implies that \(\sup_{t\in J}\vert z'_{1}(t)+z'_{2}(t) \vert\leq (\Vert p\Vert_{\infty}+\Vert m\Vert_{\infty})\Lambda_{2} \) and
from which \(\sup_{t\in J}\vert z''_{1}(t)+z''_{2}(t) \vert\leq(\Vert p\Vert_{\infty}+\Vert m\Vert_{\infty})\Lambda_{3}\). Finally, for all \(i=1, \ldots, k\), we have
and so \(\sup_{t\in J}\vert{}^{c}D^{q_{i}}z_{1}(t)+{}^{c}D^{q_{i}} z_{2}(t) \vert\leq(\Vert p\Vert_{\infty}+\Vert m\Vert_{\infty})\Lambda_{4}^{i}\), \(i=1,2,\ldots, k\). Hence, it follows that
Now, we show that the operator B is compact on Y. To do this, it is enough to prove that \(B(Y)\) is uniformly bounded and equi-continuous in X. Let \(z\in B(Y)\) be arbitrary. For some \(u\in Y\), choose \(v_{1}\in S_{G,u} \) such that
Hence,
for \(i=1,\ldots, k\). Hence, \(\Vert z\Vert \leq \Vert p\Vert_{\infty}(\Lambda_{1}+\Lambda_{2}+\Lambda_{3} +\Lambda_{4}^{i} )\), \(i=1,\ldots, k\).
Now, we show that B maps Y to equi-continuous subsets of X. Let \(t_{1}, t_{2}\in J\) with \(t_{1} < t_{2}\), \(u\in Y\), and \(z \in B(u)\). Choose \(v_{1}\in S_{G,u} \) such that \(z(t)\) is given by (2.9). Then we have
and
for each \(i=1,\ldots, k\). It is seen that the right-hand sides of the above inequalities tend to zero as \(t_{2}\to t_{1}\). Hence, by using the Arzelá-Ascoli theorem, B is compact.
Next, we prove that B has a closed graph. Let \(u_{n}\in Y\) and \(z_{n}\in B(u_{n})\) for all n such that \(u_{n}\to u_{0}\) and \(z_{n}\to z_{0}\). We show that \(z_{0}\in B(u_{0})\). Associated with \(z_{n}\in B(u_{n})\) for each \(n\in \mathbb{N}\), there exists \(v_{n}\in S_{G,u_{n}}\) such that
for all \(t\in J\). It suffices to show that there exists \(v_{0}\in S_{G,u_{0}}\) such that, for each \(t\in J\),
Consider the continuous linear operator \(\Theta:L^{1}(J,\mathbb{R})\to X\) by
Notice that
By using Lemma 1.2, \(\Theta\circ S_{G}\) is a closed graph operator. Since \(z_{n}(t)\in\Theta(S_{G,u_{n}})\) for all n, and \(u_{n}\to u_{0}\), there is \(v_{0}\in S_{G,u_{0}}\) such that
Hence, \(z_{0}\in B(u_{0})\). So, it follows that B has a closed graph and this implies that the operator B is upper semi-continuous.
Finally, we show that A is a contraction multifunction. Let \(u,w\in X\) and \(z_{1}\in A(w)\) is given. Then we can select \(v_{1}\in S_{F,w}\) such that
for all \(t\in J\). Since
for almost all \(t\in J\), there exists \(y\in F(t, u(t), u'(t), u''(t), {}^{c}D^{q_{1}}u(t), \ldots, {}^{c}D^{q_{k}}u(t))\) such that
for almost all \(t\in J\). Consider the multifunction \(U:J\to{\mathcal{P}}(\mathbb{R})\) by
where
Since \(v_{1}\) and \(\varphi= mg\) are measurable, \(U(\cdot)\cap F(\cdot, u(\cdot), u'(\cdot), u''(\cdot), {}^{c}D^{q_{1}}u(\cdot), \ldots, {}^{c}D^{q_{k}} u(\cdot) ) \) is a measurable multifunction. Thus, we can choose
such that
and
for all \(t\in J\). Now, we have
Similarly,
Hence,
for each \(1\leq i\leq k\). So
This implies that \(H_{d}(A(u), A(w) )\leq L \Vert u-w\Vert\). Thus A and B satisfy all the conditions of Theorem 1.3 and so the inclusion \(u\in A(u) +B(u) \) has a solution in Y. Therefore the inclusion problem (1.1)-(1.2) has a solution in Y and the proof is completed. □
Finally, we give an example to illustrate the validity of our main result.
Example 2.5
Consider the following fractional differential inclusion:
with the following boundary conditions:
where \(t\in[0,1]\). In the above inclusion problem, we have \(\alpha= 5/2\), \(p=3/2\), \(k=2\), and \(q_{1}=q_{2}=3/2\). Also, we have \(\Delta=0.1597\).
Now, we define \(F: [0,1] \times\mathbb{R} \times\mathbb{R} \times \mathbb{R} \times\mathbb{R} \times\mathbb{R} \to\mathcal{P}(\mathbb {R}) \) by
and \(G: [0,1] \times\mathbb{R} \times\mathbb{R} \times\mathbb{R} \times\mathbb{R} \times\mathbb{R} \to\mathcal{P}(\mathbb{R}) \) by
Then there exist continuous functions \(m, p:[0,1]\to(0, \infty)\) given by
On the other hand, we can easily check that, for every \(x_{i}, y_{i} , z_{i} , v_{i}, w_{i} \in\mathbb{R}\) (\(i=1,2\)),
where \(h: [0,1]\to(0, \infty)\) is defined by \(h(t)= \frac{t}{100}\). It can easily be found that \(\Lambda_{1} = 0.7369\), \(\Lambda_{2} = 1.4434\), \(\Lambda_{3} = 1.8102\), \(\Lambda_{4}^{1} = 1.7687\), and \(\Lambda_{4}^{2}=1.7687\). Since \(\Vert h\Vert_{\infty}= \frac{1}{100}\), we have \(L:= \Vert h\Vert _{\infty}(\Lambda_{1} +\Lambda_{2} + \Lambda_{3} +\Lambda_{4}^{1} + \Lambda_{4}^{2} ) = 0.01 \times7.5279=0.075279 < 1\). Consequently all assumptions and conditions of Theorem 2.4 are satisfied. Hence, Theorem 2.4 implies that the fractional differential inclusion problem (2.10)-(2.11) has a solution.
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Ntouyas, S.K., Etemad, S. & Tariboon, J. Existence results for multi-term fractional differential inclusions. Adv Differ Equ 2015, 140 (2015). https://doi.org/10.1186/s13662-015-0481-z
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DOI: https://doi.org/10.1186/s13662-015-0481-z