1 Introduction and preliminaries

Fractional differential equations and inclusions are generalizations of ordinary differential equations and inclusions to arbitrary non-integer orders. Fractional differential equations and inclusions appear naturally in a number of fields such as physics, engineering, biophysics, chemistry, biology, economics, control theory, etc. Recently, many papers have been published about fractional differential equations and inclusions by researchers which apply the fixed point theory in their existence theorems. For instance, one can find a lot of papers in this field (see [1–24] 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 order α for the function u is defined by \({}^{\mathrm{c}}D^{\alpha} u(t)= \frac {1}{\Gamma(n-\alpha)}\int_{0}^{t} (t-\tau)^{n-\alpha-1}u^{(n)}(\tau)\, d\tau \) (see for more details [22, 24–28]). 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 [22, 24–28]. In [27], it has been proved that the general solution of the fractional differential equation \({}^{\mathrm{c}}D^{\alpha}u( t)=0\) 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

$$I^{\alpha} {}^{\mathrm{c}}D^{\alpha}u( t)=u(t)+c_{0}+c_{1}t+c_{2}t^{2}+ \cdots+c_{n-1}t^{n-1}, $$

where \(c_{0},\ldots,c_{n-1}\) are real constants and \(n=[\alpha]+1\) [27].

Now, we review some definitions and notations about 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 \({\mathcal{P}}_{\mathrm{cp}, \mathrm{c}}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is compact and convex}\}\). A multivalued 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 semicontinuous (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 multivalued 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 multivalued operator G will be denoted by FixG. A multivalued 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

$$H_{d}(A, B) = \max\Bigl\{ \sup_{a \in A}d(a,B), \sup _{b \in B}d(A,b)\Bigr\} , $$

where \(d(A,b) = \inf_{a\in A}d(a;b)\) and \(d(a,B) = \inf_{b\in B}d(a;b)\). A multivalued 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 \(x, y \in X\).

Let \(2 < \alpha\leq3 \), \(1 < p \leq2\), \(t\in J=[0,1]\), \(0< \xi< 1\), and \(\gamma, \eta\in\mathbb{R}\). In this paper, we study the existence of solutions for the following fractional differential inclusion:

$$ {}^{\mathrm{c}}D^{\alpha}u(t)\in F \bigl(t, u(t), u'(t), u''(t), {}^{\mathrm{c}}D^{p}u(t) \bigr) $$
(1.1)

via the integral boundary value conditions

$$ \begin{aligned} &u(0)+u(1)=0, \qquad {}^{\mathrm{c}}D^{p}u( 1)= \gamma \int_{0}^{1} u(\tau)\, d\tau, \\ &u'(0)+u''( \xi)= \eta\int _{0}^{1} u(\tau)\, d\tau, \end{aligned} $$
(1.2)

where \(F: J\times\mathbb{R}^{4} \to{\mathcal{P}}({\mathbb{R}})\) is a compact-valued multifunction.

We mention that the investigated fractional differential inclusion is a generalization of a huge class of classical ordinary differential inclusions which can be found in applications in engineering and physics etc. The boundary conditions (1.2) used in this manuscript, are as general as possible and we recover several cases of fractional nonlinear differential inclusions for many particular cases of the parameters.

We say that \(F: J\times\mathbb{R}^{4} \rightarrow{\mathcal{P}}({\mathbb {R}})\) is a Carathéodory multifunction whenever \(t\mapsto F(t,u,v,z, w)\) is measurable for all \(u,v,z, w\in\mathbb{R}\) and \((u,v,z, w)\mapsto F(t,u,v,z, w)\) is upper semicontinuous for almost all \(t\in J\) ([31] and [29]). Also, a Carathéodory multifunction \(F: J\times \mathbb{R}^{4} \rightarrow{\mathcal{P}}({\mathbb{R}})\) is called \(L^{1}\)-Carathéodory whenever for each \(\rho>0\) there exists \(\phi_{\rho}\in L^{1}(J,\mathbb{R}^{+})\) such that

$$\bigl\Vert F(t,u,v,z, w)\bigr\Vert =\sup_{t\in J}\bigl\{ |s|:s \in F(t,u,v,z, w)\bigr\} \leq\phi _{\rho}(t) $$

for all \(|u|, |v|, |z|, |w|\leq\rho\), and for almost all \(t\in J\) ([31] and [29]). Define the set of selections of F by

$$S_{F,u}:=\bigl\{ v\in L^{1}(J,\mathbb{R}): v(t)\in F\bigl(t, u(t), u'(t), u''(t), {}^{\mathrm{c}}D^{p}u(t) \bigr) \mbox{ for almost all } t\in J\bigr\} . $$

We define the graph of a function G to be the set \(\operatorname{Gr}(G)=\{ (x,y)\in X\times Y, y\in G(x)\}\) and recall two results for closed graphs and upper semicontinuity.

Lemma 1.1

(Proposition 1.2 in [29])

If \(G : X \to\mathcal{P}_{\mathrm{cl}}(Y)\) is u.s.c., then \(\operatorname{Gr}(G)\) is a closed subset of \(X \times Y\); i.e., for every sequence \(\{u_{n}\}_{n \in\mathbb{N}} \subset X\) and \(\{y_{n}\}_{n \in \mathbb{N}} \subset Y\), if when \(n \to\infty\), \(u_{n} \to u_{*}\), \(y_{n} \to y_{*}\), and \(y_{n} \in G(u_{n})\), then \(y_{*} \in G(u_{*})\). Conversely, if G is completely continuous and has a closed graph, then it is upper semicontinuous.

Lemma 1.2

([32])

Let X be a separable Banach space. Let \(F : [0, 1] \times X^{4} \to{\mathcal{P}}_{\mathrm{cp},\mathrm{c}}(X)\) be an \(L^{1}\)-Carathéodory function. Then the operator

$$\Theta\circ S_{F} : C(J,X) \to{\mathcal{P}}_{\mathrm{cp},\mathrm{c}} \bigl(C(J,X)\bigr), \quad x \mapsto(\Theta\circ S_{F}) (x) = \Theta( S_{F,x}) $$

is a closed graph operator.

Now we state some known fixed point theorems which are needed in the sequel.

Lemma 1.3

(Nonlinear alternative for Kakutani maps [33])

Let E be a Banach space, C a closed convex subset of E, U an open subset of C and \(0\in U\). Suppose that \(F: \overline{U}\to{\mathcal{P}}_{\mathrm{cp},\mathrm{c}}(C)\) is a upper semicontinuous compact map. Then either

  1. (i)

    F has a fixed point in \(\overline{U}\), or

  2. (ii)

    there is a \(u\in\partial U\) and \(\lambda\in(0,1)\) with \(u\in \lambda F(u)\).

Lemma 1.4

([34])

Let \((X,d)\) be a complete metric space. If \(N : X \to{\mathcal{P}}_{\mathrm{cl}}(X)\) is a contraction, then \(\operatorname{Fix} N \ne\emptyset\).

2 Main results

Now, we are ready to prove our main results. Let \(X=\{u: u,u', u'', {}^{\mathrm{c}}D^{p} u\in C(J,\mathbb{R})\}\) endowed with the norm \(\|u\|=\sup_{t\in J} |u(t)|+\sup_{t\in J}|u'(t)| + \sup_{t\in J}|u''(t)| + \sup_{t\in J}|{}^{\mathrm{c}}D^{p} u(t)| \). Then \((X, {\Vert\cdot\Vert})\) is a Banach space [35].

Lemma 2.1

Let \(y\in L^{1}(J,{\mathbb{R}})\). Then the integral solution of the linear problem

$$ \left \{ \textstyle\begin{array}{l} {}^{\mathrm{c}}D^{\alpha}u(t)=y(t ), \\ u(0)+u(1)=0, \qquad {}^{\mathrm{c}}D^{p}u( 1)= \gamma\int_{0}^{1} u(\tau )\, d\tau, \\ u'(0)+u''( \xi)= \eta\int_{0}^{1} u(\tau)\, d\tau, \end{array}\displaystyle \right . $$
(2.1)

is given by

$$\begin{aligned} u(t) =& I^{\alpha}y(t) + \biggl(\frac{1}{2}-t \biggr)I^{\alpha-2}y(\xi) + A(t) I^{\alpha}y(1) \\ &{}+ B(t)I^{\alpha-p}y(1) + C(t)I^{\alpha+1}y(1), \end{aligned}$$
(2.2)

where

$$\begin{aligned}& A(t) = -\frac{1}{2}\bigl(1+C(t)\bigr), \\& B(t) = \frac{1}{4} \Gamma(3-p) \bigl(-1+4t-2t^{2}\bigr)+ \frac{\Gamma (3-p)}{12}C(t), \\& C(t) = \frac{3}{12+\gamma\Gamma(3-p)} \bigl[\gamma\Gamma (3-p) \bigl(1-4t+2t^{2} \bigr)+2\eta(2t-1) \bigr]. \end{aligned}$$
(2.3)

Proof

It is well known that the solution of equation \({}^{\mathrm{c}}D^{\alpha}u(t)=y(t)\) can be written as

$$u( t)= I^{\alpha}y(t) + c_{0} + c_{1}t+c_{2}t^{2}, $$

where \(c_{0}, c_{1}, c_{2}\in\mathbb{R}\) are arbitrary constants. Then we get

$$u'( t)=I^{\alpha-1}y(t) +c_{1}+2c_{2}t, \qquad u''(t)=I^{\alpha-2}y(t) +2c_{2} $$

and

$${}^{\mathrm{c}}D^{p} u(t)=I^{\alpha-p}y(t)+c_{2} \frac{2t^{2-p}}{\Gamma(3-p)},\quad 1< p\leq2. $$

By using the boundary conditions \(u(0)+u(1)=0\), \({}^{\mathrm{c}}D^{p}u( 1)= \gamma \int_{0}^{1} u(\tau)\, d\tau\), \(u'(0)+u''( \xi)= \eta\int_{0}^{1} u(\tau)\, d\tau\), we obtain

$$\begin{aligned}& c_{0} = -\frac{1}{2}I^{\alpha}y(1) +I^{\alpha-2}y(\xi) - \frac{\Gamma(3-p)}{4} I^{\alpha-p}y(1) + \frac{\gamma\Gamma (3-p)-2\eta}{4 }\int _{0}^{1} u( \tau)\, d\tau, \\& c_{1} = - I^{\alpha-2}y(\xi) + \Gamma(3-p)I^{\alpha-p}y(1) + \bigl(\eta-\gamma\Gamma(3-p) \bigr)\int_{0}^{1} u( \tau)\, d\tau \end{aligned}$$

and

$$c_{2} = - \frac{\Gamma(3-p)}{2}I^{\alpha-p}y(1) + \frac{\gamma\Gamma (3-p)}{2} \int_{0}^{1} u( \tau)\, d\tau. $$

Then

$$\begin{aligned} u( t ) =& I^{\alpha}y(t) - \frac{1}{2} I^{\alpha}y(1) + \biggl(\frac {1}{2}-t \biggr)I^{\alpha-2}y(\xi) + \frac{\Gamma(3-p)( -1+4t-2t^{2})}{4}I^{\alpha-p}y(1) \\ &{}+\frac{1}{4} \bigl[\gamma\Gamma(3-p) \bigl(1-4t+2t^{2} \bigr)+2\eta(2t-1) \bigr]\int_{0}^{1} u( \tau)\, d \tau. \end{aligned}$$
(2.4)

Letting \(A= \int_{0}^{1} u( \tau)\, d\tau\), we have

$$\begin{aligned} A =& \int_{0}^{1} u( \tau)\, d\tau=\int _{0}^{1}\int_{0}^{t} \frac{(t-s)^{\alpha -1}}{\Gamma(\alpha)}y(s)\, ds\, dt+c_{0}+\frac{1}{2}c_{1}+ \frac{1}{3}c_{2} \\ =&I^{\alpha+1}y(1)+c_{0}+\frac{1}{2}c_{1}+ \frac{1}{3}c_{2}, \end{aligned}$$

or after substituting \(c_{0}\), \(c_{1}\), and \(c_{2}\),

$$A = \frac{12}{12+\gamma\Gamma(3-p)} \biggl[I^{\alpha+1}y(1)-\frac {1}{2}I^{\alpha}y(1) +\frac{\Gamma(3-p)}{12}I^{\alpha-p}y(1) \biggr]. $$

Substituting the value A in (2.4), we get (2.2). The proof is completed. □

Remark 2.2

Throughout this paper, the following relations hold:

$$\begin{aligned}& \bigl\vert A(t)\bigr\vert \le \frac{1}{2}(1+C_{1}),\qquad \bigl\vert B(t)\bigr\vert \le\Gamma(3-p) \biggl( \frac{7}{4}+ \frac{1}{12}C_{1} \biggr), \\& \bigl\vert C(t)\bigr\vert \le \frac{3}{12+\gamma\Gamma(3-p)}\bigl[7\gamma\Gamma(3-p)+2 \eta \bigr]:=C_{1}, \\& \bigl\vert A'(t)\bigr\vert \le \frac{1}{2}C'_{1}, \qquad \bigl\vert B'(t)\bigr\vert \le\Gamma(3-p) \biggl( 1 + \frac{1}{12}C_{1}' \biggr), \\& \bigl\vert C'(t)\bigr\vert \le\frac{12[\gamma\Gamma(3-p)+\eta]}{12+\gamma\Gamma (3-p)}:=C'_{1}, \\& \bigl\vert A''(t)\bigr\vert \le \frac{1}{2}C''_{1},\qquad \bigl\vert B''(t)\bigr\vert \le \Gamma(3-p) \biggl( 1 + \frac{1}{12}C''_{1} \biggr), \\& \bigl\vert C''(t)\bigr\vert \le\frac{12\gamma\Gamma(3-p)}{12+\gamma\Gamma(3-p)}:=C''_{1}, \\& \bigl\vert {}^{\mathrm{c}}D^{p}A(t)\bigr\vert \le D_{1}, \qquad \bigl\vert {}^{\mathrm{c}}D^{p}B(t)\bigr\vert \le1+ \frac{\Gamma(3-p)}{12} D_{1}, \\& \bigl\vert {}^{\mathrm{c}}D^{p}C(t)\bigr\vert \le\frac{12\gamma}{12+\gamma\Gamma(3-p)}:=D_{1}. \end{aligned}$$

Definition 2.3

A function \(u\in C^{2}(J, {\mathbb{R}})\) is called a solution for the problem (1.1)-(1.2) if there exists a function \(v\in L^{1}(J, {\mathbb{R}})\) with \(v(t)\in F(t, u(t), u'(t), u''(t), {}^{\mathrm{c}}D^{p}u(t))\) for almost all \(t\in J\), \(u(0)+u(1)=0\), \({}^{\mathrm{c}}D^{p}u( 1)= \gamma \int_{0}^{1} u(\tau)\, d\tau\), \(u'(0)+u''( \xi)= \eta\int_{0}^{1} u(\tau)\, d\tau\), and

$$ u(t) = I^{\alpha}v(t)+ \biggl(\frac{1}{2}-t \biggr)I^{\alpha-1}v(\xi) + A(t) I^{\alpha}v(1)+ B(t)I^{\alpha-p}v(1) + C(t)I^{\alpha+1}v(1) $$
(2.5)

for all \(t\in J\).

For the sake of brevity, we set

$$\begin{aligned}& \Lambda_{1} =\frac{1}{\Gamma(\alpha+1)}+\frac{1}{2} \frac{\xi ^{\alpha-2}}{\Gamma(\alpha-1)}+\frac{1}{2}(1+C_{1})\frac{1}{\Gamma(\alpha +1)} \\ & \hphantom{\Lambda_{1} =}{}+\Gamma(3-p) \biggl(\frac{7}{4}+\frac{1}{12}C_{1} \biggr)\frac{1}{\Gamma(\alpha-p+1)} +\frac{1}{2} C_{1}\frac{1}{\Gamma(\alpha+2)}, \end{aligned}$$
(2.6)
$$\begin{aligned}& \Lambda_{2} = \frac{1}{\Gamma(\alpha)}+ \frac{\xi^{\alpha -2}}{\Gamma(\alpha-1)}+ \frac{1}{2}C'_{1}\frac{1}{\Gamma(\alpha+1)} \\ & \hphantom{\Lambda_{2} =}{} +\Gamma(3-p) \biggl( 1 +\frac{1}{12}C'_{1} \biggr)\frac {1}{\Gamma(\alpha-p+1)} +\frac{1}{2} C'_{1} \frac{1}{\Gamma(\alpha+2)}, \end{aligned}$$
(2.7)
$$\begin{aligned}& \Lambda_{3} = \frac{1}{\Gamma(\alpha-1)} +C''_{1} \frac{1}{\Gamma(\alpha+1)} \\ & \hphantom{\Lambda_{3} =}{}+\Gamma(3-p) \biggl( 1+\frac{1}{12}C''_{1} \biggr)\frac {1}{\Gamma(\alpha-p+1)} +\frac{1}{2} C''_{1} \frac{1}{\Gamma(\alpha+2)} \end{aligned}$$
(2.8)

and

$$\begin{aligned} \Lambda_{4} =& \frac{1}{\Gamma(\alpha-p +1)}+D_{1} \frac{1}{\Gamma(\alpha+1)} \\ &{}+ \biggl(1+ \frac{\Gamma(3-p)}{12} D_{1} \biggr) \frac{1}{\Gamma(\alpha-p+1)}+\frac{1}{2} D_{1}\frac{1}{\Gamma(\alpha+2)}. \end{aligned}$$
(2.9)

Theorem 2.4

Suppose that:

(H1):

\(F: J\times\mathbb{R}^{4} \to{\mathcal{P}}_{\mathrm{cp},\mathrm{c}}(\mathbb {R})\) is a \(L^{1}\)-Carathéodory multifunction.

(H2):

There exist continuous nondecreasing functions \(\psi_{i} : [0,\infty) \to (0,\infty)\) and functions \(p_{i} \in C(J,\mathbb{R}^{+})\), \(1\le i\le4\), such that

$$\bigl\Vert F(t, x_{1}, x_{2}, x_{3}, x_{4})\bigr\Vert :=\sup\bigl\{ |v|: v\in F(t, x_{1}, x_{2}, x_{3}, x_{4})\bigr\} \le\sum _{i=1}^{4} p_{i}(t)\psi_{i} \bigl(\vert x_{i}\vert \bigr) $$

for each \((t,x_{i}) \in J\times\mathbb{R}\), \(1\le i\le4\).

(H3):

There exists a constant \(M>0\) such that

$$\frac{M}{ \sum_{i=1}^{4}\Lambda_{i} \|p_{i}\|\psi_{i}(M)}> 1, $$

where \(\Lambda_{i}\), \(1\le i\le4\) are defined by (2.6)-(2.9).

Then the inclusion boundary value problem (1.1)-(1.2) has at least one solution.

Proof

To transform the problem (1.1)-(1.2) into a fixed point problem, we define an operator \({\mathcal{N}}:X\rightarrow{\mathcal{P}}(X)\) as

$${\mathcal{N}}(u)=\left \{ h \in X: h(t) = \left \{ \textstyle\begin{array}{l} I^{\alpha}v(t)+ (\frac{1}{2}-t )I^{\alpha-1}v(\xi)+ A(t) I^{\alpha }v(1) \\ \quad {}+ B(t)I^{\alpha-p}v(1) + C(t)I^{\alpha+1}v(1), t\in J, v\in S_{F,u} \end{array}\displaystyle \right \} \right \}. $$

We will show that \({\mathcal{N}}\) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that \({\mathcal{N}}\) is convex for each \(u \in X\). This step is obvious since \(S_{F,u}\) is convex (F has convex values), and therefore we omit the proof.

In the second step, we show that \({\mathcal{N}}\) maps bounded sets (balls) into bounded sets in X. For a positive number r, let \(B_{r} = \{u \in X: \|u\| \le r \}\) be a bounded ball in X. Then, for each \(h \in{\mathcal{N}} (u)\) and \(u \in B_{r}\), there exists \(v \in S_{F,u}\) such that

$$h(t) = I^{\alpha}v(t)+ \biggl(\frac{1}{2}-t \biggr)I^{\alpha -1}v( \xi)+ A(t) I^{\alpha}v(1) + B(t)I^{\alpha-p}v(1) + C(t)I^{\alpha+1}v(1), \quad t\in J. $$

Then we have

$$\begin{aligned} \bigl\vert h(t)\bigr\vert \le& I^{\alpha}\bigl\vert v(t)\bigr\vert +\biggl\vert \frac{1}{2}-t\biggr\vert I^{\alpha -2}\bigl\vert v( \xi)\bigr\vert + \bigl\vert A(t)\bigr\vert I^{\alpha}\bigl\vert v(1) \bigr\vert \\ &{}+ \bigl\vert B(t)\bigr\vert I^{\alpha-p}\bigl\vert v(1) \bigr\vert + \bigl\vert C(t)\bigr\vert I^{\alpha+1}\bigl\vert v(1) \bigr\vert \\ \le& I^{\alpha}\bigl[p_{1}(t)\psi_{1}\bigl(\bigl\vert u(t)\bigr\vert \bigr)+p_{2}(t)\psi _{2}\bigl(\bigl\vert u'(t)\bigr\vert \bigr) \\ &{}+p_{3}(t)\psi_{3} \bigl(\bigl\vert u''(t)\bigr\vert \bigr) +p_{4}(t)\psi_{4}\bigl(\bigl\vert {}^{\mathrm{c}}D^{p}u(t) \bigr\vert \bigr)\bigr] \\ &{} + \frac{1}{2} I^{\alpha-1}\bigl[p_{1}(\xi) \psi_{1}\bigl(\bigl\vert u(\xi)\bigr\vert \bigr)+p_{2}( \xi)\psi _{2}\bigl(\bigl\vert u'(\xi)\bigr\vert \bigr) \\ &{}+p_{3}(\xi)\psi_{3}\bigl(\bigl\vert u''(\xi)\bigr\vert \bigr) +p_{4}(\xi) \psi_{4}\bigl(\bigl\vert {}^{\mathrm{c}}D^{p}u(\xi )\bigr\vert \bigr)\bigr] \\ &{}+ \frac{1}{2}(1+C_{1}) I^{\alpha}\bigl[p_{1}(1) \psi_{1}\bigl(\bigl\vert u(1)\bigr\vert \bigr)+p_{2}(1) \psi _{2}\bigl(\bigl\vert u'(1)\bigr\vert \bigr) \\ &{}+p_{3}(1)\psi_{3}\bigl(\bigl\vert u''(1) \bigr\vert \bigr) +p_{4}(1)\psi_{4}\bigl(\bigl\vert {}^{\mathrm{c}}D^{p}u(1)\bigr\vert \bigr)\bigr] \\ &{} +\Gamma(3-p) \biggl(\frac{7}{4}+\frac{1}{12}C_{1} \biggr)I^{\alpha-p} \bigl[p_{1}(1)\psi_{1}\bigl(\bigl\vert u(1)\bigr\vert \bigr)+p_{2}(1)\psi_{2}\bigl(\bigl\vert u'(1)\bigr\vert \bigr) \\ &{}+p_{3}(1) \psi_{3}\bigl(\bigl\vert u''(1)\bigr\vert \bigr)+p_{4}(1)\psi_{4}\bigl(\bigl\vert {}^{\mathrm{c}}D^{p}u(1) \bigr\vert \bigr)\bigr] \\ &{} + C_{1}I^{\alpha+1}\bigl[p_{1}(1)\psi_{1} \bigl(\bigl\vert u(1)\bigr\vert \bigr)+p_{2}(1)\psi _{2} \bigl(\bigl\vert u'(1)\bigr\vert \bigr) \\ &{}+p_{3}(1) \psi_{3}\bigl(\bigl\vert u''(1)\bigr\vert \bigr) +p_{4}(1)\psi_{4}\bigl(\bigl\vert {}^{\mathrm{c}}D^{p}u(1)\bigr\vert \bigr)\bigr] \\ \le& \bigl[\|p_{1}\|\psi_{1}(r)+\|p_{2}\| \psi_{2}(r)+\|p_{3}\|\psi_{3}(r)+\|p_{4}\| \psi_{4}(r) \bigr] \biggl\{ \frac{1}{\Gamma(\alpha+1)}+\frac{1}{2} \frac{\xi ^{\alpha-2}}{\Gamma(\alpha-1)} \\ &{}+\frac{1}{2}(1+C_{1})\frac{1}{\Gamma(\alpha+1)}+\Gamma(3-p) \biggl( \frac {7}{4}+\frac{1}{12}C_{1} \biggr)\frac{1}{\Gamma(\alpha-p+1)} + \frac{1}{2} C_{1}\frac{1}{\Gamma(\alpha+2)} \biggr\} \\ =&\Lambda_{1}\sum_{i=1}^{4} \|p_{i}\|\psi_{i}(r) \end{aligned}$$

for all \(t\in J\). In a similar manner we obtain

$$\begin{aligned}& \bigl\vert h'(t)\bigr\vert \le \bigl[\|p_{1}\| \psi_{1}(r)+\|p_{2}\|\psi_{2}(r)+\|p_{3}\| \psi_{3}(r)+\|p_{4}\| \psi_{4}(r) \bigr] \biggl\{ \frac{1}{\Gamma(\alpha)}+ \frac{\xi^{\alpha -2}}{\Gamma(\alpha-1)} \\& \hphantom{\bigl\vert h'(t)\bigr\vert \le}{} +\frac{1}{2}C'_{1} \frac{1}{\Gamma(\alpha+1)}+\Gamma(3-p) \biggl( 1 +\frac {1}{12}C'_{1} \biggr)\frac{1}{\Gamma(\alpha-p+1)} +\frac{1}{2} C'_{1} \frac{1}{\Gamma(\alpha+2)} \biggr\} \\& \hphantom{\bigl\vert h'(t)\bigr\vert } = \Lambda_{2}\sum_{i=1}^{4} \|p_{i}\|\psi_{i}(r), \\& \bigl\vert h''(t)\bigr\vert \le \bigl[ \|p_{1}\|\psi_{1}(r)+\|p_{2}\|\psi_{2}(r)+ \|p_{3}\|\psi_{3}(r)+\|p_{4}\| \psi_{4}(r) \bigr] \biggl\{ \frac{1}{\Gamma(\alpha-1)} +C''_{1} \frac{1}{\Gamma (\alpha+1)} \\& \hphantom{\bigl\vert h''(t)\bigr\vert \le}{} +\Gamma(3-p) \biggl( 1 +\frac{1}{12}C''_{1} \biggr)\frac{1}{\Gamma(\alpha-p+1)} +\frac{1}{2} C''_{1} \frac{1}{\Gamma(\alpha+2)} \biggr\} \\& \hphantom{\bigl\vert h''(t)\bigr\vert } = \Lambda_{3}\sum_{i=1}^{4} \|p_{i}\|\psi_{i}(r) \end{aligned}$$

and

$$\begin{aligned} \bigl\vert {}^{\mathrm{c}}D^{p}h(t)\bigr\vert \le& \bigl[ \|p_{1}\|\psi_{1}(r)+\|p_{2}\|\psi_{2}(r)+ \|p_{3}\|\psi_{3}(r)+\|p_{4}\| \psi_{4}(r) \bigr] \biggl\{ \frac{1}{\Gamma(\alpha-p +1)} \\ &{}+D_{1}\frac{1}{\Gamma (\alpha+1)}+\biggl(1+\frac{\Gamma(3-p)}{12} D_{1}\biggr)\frac{1}{\Gamma(\alpha-p+1)} + \frac{1}{2} D_{1}\frac{1}{\Gamma(\alpha+2)} \biggr\} \\ =&\Lambda_{4}\sum_{i=1}^{4} \|p_{i}\|\psi_{i}(r) \end{aligned}$$

for all \(t\in J\). Thus we get

$$\|h\|\le(\Lambda_{1}+\Lambda_{2}+\Lambda_{3}+ \Lambda_{4}) \sum_{i=1}^{4} \|p_{i}\|\psi_{i}(r)=\sum_{i=1}^{4} \Lambda_{i}\|p_{i}\|\psi_{i}(r), $$

which implies that \({\mathcal{N}}\) maps bounded sets into bounded sets in X.

Now, we prove that \({\mathcal{N}}\) maps bounded sets into equi-continuous subsets of X. Suppose that \(u\in B_{r}\) and \(t_{1}, t_{2}\in J\) with \(t_{1}< t_{2}\). Then we have

$$\begin{aligned}& \bigl\vert h(t_{2})-h(t_{1})\bigr\vert \\& \quad \le \biggl\vert \frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}} (t_{2}-s)^{\alpha -1}v(\tau)\, d\tau-\frac{1}{\Gamma(\alpha)}\int _{0}^{t_{1}} (t_{1}-s)^{\alpha -1}v(\tau) \, d\tau\biggr\vert \\& \qquad {} +\vert t_{2}-t_{1}\vert I^{\alpha-2}\bigl\vert v(\xi)\bigr\vert + \bigl\vert A(t_{2})-A(t_{1}) \bigr\vert I^{\alpha}\bigl\vert v(1)\bigr\vert + \bigl\vert B(t_{2})-B(t_{1})\bigr\vert I^{\alpha-p}\bigl\vert v(1)\bigr\vert \\& \qquad {}+ \bigl\vert C(t_{2})-C(t_{1})\bigr\vert I^{\alpha+1}\bigl\vert v(1)\bigr\vert \\& \quad \le \frac{1}{\Gamma(\alpha)}\int_{0}^{t_{1}} \bigl[(t_{2}-s)^{\alpha -1}-(t_{1}-s)^{\alpha-1}\bigr] \bigl\vert v( \tau)\bigr\vert \, d\tau+ \frac{1}{\Gamma(\alpha)}\int _{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}\bigl\vert v( \tau)\bigr\vert \, d\tau \\& \qquad {} +\vert t_{2}-t_{1}\vert I^{\alpha-2}\bigl\vert v(\xi)\bigr\vert + \bigl\vert A(t_{2})-A(t_{1}) \bigr\vert I^{\alpha}\bigl\vert v(1)\bigr\vert + \bigl\vert B(t_{2})-B(t_{1})\bigr\vert I^{\alpha-p}\bigl\vert v(1)\bigr\vert \\& \qquad {}+ \bigl\vert C(t_{2})-C(t_{1})\bigr\vert I^{\alpha+1}\bigl\vert v(1)\bigr\vert \\& \quad \le \sum_{i=1}^{4}\|p_{i} \|\psi_{i}(r) \biggl\{ \bigl(t_{2}^{\alpha}-t_{1}^{\alpha } \bigr)\frac{1}{\Gamma(\alpha+1)}+\vert t_{2}-t_{1}\vert \frac{\xi^{\alpha-2}}{\Gamma (\alpha-1)}+ \bigl\vert A(t_{2})-A(t_{1})\bigr\vert \frac{1}{\Gamma(\alpha+1)} \\& \qquad {}+\bigl\vert B(t_{2})-B(t_{1})\bigr\vert \frac{1}{\Gamma(\alpha-p+1)}+ \bigl\vert C(t_{2})-C(t_{1})\bigr\vert \frac {1}{\Gamma(\alpha+2)} \biggr\} . \end{aligned}$$

Proceeding as above we have

$$\begin{aligned}& \bigl\vert h'(t_{2})-h'(t_{1}) \bigr\vert \\& \quad \le \biggl\vert \frac{1}{\Gamma(\alpha-1)}\int_{0}^{t_{2}} (t_{2}-s)^{\alpha -2}v(\tau)\, d\tau-\frac{1}{\Gamma(\alpha-1)}\int _{0}^{t_{1}} (t_{1}-s)^{\alpha -2}v(\tau) \, d\tau\biggr\vert \\& \qquad {}+ \bigl\vert A'(t_{2})-A'(t_{1}) \bigr\vert I^{\alpha}\bigl\vert v(1)\bigr\vert + \bigl\vert B'(t_{2})-B'(t_{1})\bigr\vert I^{\alpha-p}\bigl\vert v(1)\bigr\vert \\& \qquad {}+ \bigl\vert C'(t_{2})-C'(t_{1}) \bigr\vert I^{\alpha+1}\bigl\vert v(1)\bigr\vert \\& \quad \le \sum_{i=1}^{4}\|p_{i} \|\psi_{i}(r) \biggl\{ \bigl(t_{2}^{\alpha-1}-t_{1}^{\alpha -1} \bigr)\frac{1}{\Gamma(\alpha)} + \bigl\vert A'(t_{2})-A'(t_{1}) \bigr\vert \frac{1}{\Gamma(\alpha +1)} \\& \qquad {}+\bigl\vert B'(t_{2})-B'(t_{1}) \bigr\vert \frac{1}{\Gamma(\alpha-p+1)}+ \bigl\vert C'(t_{2})-C'(t_{1}) \bigr\vert \frac{1}{\Gamma(\alpha+2)} \biggr\} , \\& \bigl\vert h''(t_{2})-h''(t_{1}) \bigr\vert \\& \quad \le\sum_{i=1}^{4}\|p_{i}\| \psi_{i}(r) \biggl\{ \bigl(t_{2}^{\alpha-1}-t_{1}^{\alpha -1} \bigr)\frac{1}{\Gamma(\alpha-1)} + \bigl\vert A''(t_{2})-A''(t_{1}) \bigr\vert \frac{1}{\Gamma (\alpha+1)} \\& \qquad {} +\bigl\vert B''(t_{2})-B''(t_{1}) \bigr\vert \frac{1}{\Gamma(\alpha-p+1)}+ \bigl\vert C''(t_{2})-C''(t_{1}) \bigr\vert \frac{1}{\Gamma(\alpha+2)} \biggr\} \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert {}^{\mathrm{c}}D^{p}h(t_{2})-{}^{\mathrm{c}}D^{p}h(t_{1}) \bigr\vert \\& \quad \le \sum_{i=1}^{4}\|p_{i} \|\psi_{i}(r) \biggl\{ \bigl(t_{2}^{\alpha-p}-t_{1}^{\alpha -p} \bigr)\frac{1}{\Gamma(\alpha-p+1)} + |t_{2}-t_{1}|\frac{12\gamma}{12+\gamma \Gamma(3-p)} \frac{1}{\Gamma(\alpha+1)} \\& \qquad {} +|t_{2}-t_{1}| \biggl(1+ \frac{12\gamma}{12+\gamma\Gamma(3-p)} \biggr) \frac {1}{\Gamma(\alpha-p+1)} \\& \qquad {} +|t_{2}-t_{1}| \frac{1}{2} \frac{\Gamma(3-p) \gamma}{12+\gamma\Gamma (3-p)} \frac{1}{\Gamma(\alpha+2)} \biggr\} . \end{aligned}$$

Obviously the right-hand side of the above inequalities tends to zero independently of \(u \in B_{r}\) as \(t_{2}- t_{1} \to0\). Therefore it follows by the Ascoli-Arzelá theorem that \({\mathcal{N}}: X \to {\mathcal{P}}(X)\) is completely continuous.

In our next step, we show that \({\mathcal{N}}\) is upper semicontinuous. It is well known by Lemma 1.1 that \({\mathcal{N}}\) will be upper semicontinuous if we prove that it has a closed graph, since \({\mathcal{N}}\) is already shown to be completely continuous. Thus we will prove that \({\mathcal{N}}\) has a closed graph. Let \(u_{n} \to u_{*}\), \(h_{n} \in{\mathcal{N}} (u_{n})\), and \(h_{n} \to h_{*}\). Then we need to show that \(h_{*} \in {\mathcal{N}} (u_{*})\). Associated with \(h_{n} \in{\mathcal{N}} (u_{n})\), there exists \(v_{n} \in S_{F,u_{n}}\) such that, for each \(t \in J\),

$$ h_{n}(t) = I^{\alpha}v_{n}(t)+ \biggl( \frac{1}{2}-t \biggr)I^{\alpha-2}v_{n}(\xi)+ A(t) I^{\alpha}v_{n}(1) + B(t)I^{\alpha-p}v_{n}(1) + C(t)I^{\alpha+1}v_{n}(1). $$

Thus it suffices to show that there exists \(v_{*} \in S_{F,u_{*}}\) such that, for each \(t \in J\),

$$ h_{*}(t) = I^{\alpha}v_{*}(t)+ \biggl(\frac{1}{2}-t \biggr)I^{\alpha-2}v_{*}( \xi)+ A(t) I^{\alpha}v_{*}(1) + B(t)I^{\alpha-p}v_{*}(1) + C(t)I^{\alpha+1}v_{*}(1). $$

Let us consider the linear operator \(\Theta: L^{1}(J, \mathbb{R}) \to C(J, \mathbb{R})\) given by

$$ f \mapsto\Theta(v) (t) =I^{\alpha}v(t)+ \biggl(\frac{1}{2}-t \biggr)I^{\alpha-1}v(\xi)+ A(t) I^{\alpha}v(1) + B(t)I^{\alpha-p}v(1) + C(t)I^{\alpha+1}v(1). $$

Observe that

$$\begin{aligned} \bigl\| h_{n}(t)-h_{*}(t)\bigr\| =& \biggl\| I^{\alpha}\bigl(v_{n}(t)-v_{*}(t) \bigr) + \biggl(\frac {1}{2}-t \biggr)I^{\alpha-2}\bigl(v_{n}( \xi)-v_{*}(\xi)\bigr) \\ &{} + A(t) I^{\alpha}\bigl(v_{n}(1)-v_{*}(1)\bigr)+ B(t)I^{\alpha-p} \bigl(v_{n}(1)-v_{*}(1)\bigr) \\ & {}+ C(t)I^{\alpha+1}\bigl(v_{n}(1)-v_{*}(1)\bigr) \biggr\| \to 0,\quad \mbox{as } n\to\infty. \end{aligned}$$

Thus, it follows by Lemma 1.2 that \(\Theta\circ S_{F}\) is a closed graph operator. Further, we have \(h_{n}(t) \in \Theta(S_{F,u_{n}})\). Since \(u_{n} \to u_{*}\), therefore, we have

$$ h_{*}(t) = I^{\alpha}v_{*}(t)+ \biggl(\frac{1}{2}-t \biggr)I^{\alpha-2}v_{*}( \xi)+ A(t) I^{\alpha}v_{*}(1) + B(t)I^{\alpha-p}v_{*}(1) + C(t)I^{\alpha+1}v_{*}(1) $$

for some \(v_{*} \in S_{F,u_{*}}\).

Finally, we show that there exists an open set \(U\subseteq X\) with \(u\notin{\mathcal{N}} (u)\) for any \(\lambda\in (0,1)\) and all \(u\in\partial U\). Let \(\lambda\in(0,1)\) and \(u\in \lambda{\mathcal{N}} (u)\). Then there exists \(v \in L^{1}(J, \mathbb{R})\) with \(v \in S_{F,u}\) such that, for \(t \in J\), we have

$$ u(t) = \lambda I^{\alpha}v(t)+\lambda \biggl(\frac {1}{2}-t \biggr)I^{\alpha-1}v(\xi)+ \lambda A(t) I^{\alpha}v(1) +\lambda B(t)I^{\alpha-p}v(1) + \lambda C(t)I^{\alpha+1}v(1). $$

Using the computations of the second step above we have

$$\begin{aligned} \|u\| \le& (\Lambda_{1}+\Lambda_{2}+\Lambda_{3}+ \Lambda_{4}) \bigl[\|p_{1}\|\psi _{1}\bigl(\Vert u \Vert \bigr)+\|p_{2}\|\psi_{2}\bigl(\Vert u\Vert \bigr) \\ &{}+ \|p_{3}\|\psi_{3}\bigl(\Vert u\Vert \bigr)+ \|p_{4}\|\psi_{4}\bigl(\Vert u\Vert \bigr) \bigr], \end{aligned}$$

which implies that

$$\frac{\|u\|}{ \sum_{i=1}^{4}\Lambda_{i}\|p_{i}\|\psi_{i}(\|u\| )}\le1. $$

In view of (H3), there exists M such that \(\|u\| \ne M\). Let us set

$$U = \bigl\{ u \in X : \Vert x\Vert < M\bigr\} . $$

Note that the operator \({\mathcal{N}} :\overline{U} \to\mathcal{P}(X)\) is upper semicontinuous and completely continuous. From the choice of U, there is no \(u \in\partial U\) such that \(u \in\lambda{\mathcal{N}} (u)\) for some \(\lambda\in(0,1)\). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 1.3), we deduce that \({\mathcal{N}}\) has a fixed point \(u \in\overline{U}\) which is a solution of the problem (1.1)-(1.2). This completes the proof. □

In the next theorem, we prove the existence of solution for the inclusion boundary value problem (1.1)-(1.2) when the multifunction F is non-convex valued.

Theorem 2.5

Assume that:

(H4):

\(F : J \times\mathbb{R}^{4} \to{\mathcal{P}}_{\mathrm{cp}}(\mathbb{R})\) is such that \(F(\cdot,u,v,z,w) : J \to{\mathcal{P}}_{\mathrm{cp}}(\mathbb{R})\) is measurable for each \(u,v,z,w \in\mathbb{R}\).

(H5):

For almost all \(t \in J\) and \(u_{1},u_{2}, u_{3}, u_{4}, w_{1},w_{2}, w_{3}, w_{4}\in \mathbb{R}\) we have

$$\begin{aligned}& H_{d}\bigl(F(t,u_{1},u_{2},u_{3},u_{4}), F(t,w_{1},w_{2},w_{3},w_{4})\bigr) \\& \quad \leq m(t) \bigl(|u_{1}-w_{1}|+|u_{2}-w_{2}|+|u_{3}-w_{3}|+|u_{4}-w_{4}|\bigr) \end{aligned}$$

with \(m \in C(J, \mathbb{R}^{+})\) and \(d(0,F(t,0,0,0,0))\le m(t)\), for almost all \(t \in J\).

Then the boundary value problem (1.1)-(1.2) has at least one solution on J if \(\|m\|\sum_{i=1}^{4} \Lambda_{i} <1\).

Proof

Observe that the set \(S_{F,u}\) is nonempty for each \(u \in X\) by the assumption (H4), so F has a measurable selection (see Theorem III.6 in [36]). Now we show that the operator \({\mathcal{N}}\), defined in the beginning of proof of Theorem 2.4, satisfies the assumptions of Lemma 1.4. To show that \({\mathcal{N}}(u) \in{\mathcal{P}}_{\mathrm{cl}}(X)\) for each \(u \in X\), let \(\{u_{n}\}_{n \ge0} \in {\mathcal{N}}(u)\) be such that \(u_{n} \to u\) (\(n \to\infty\)) in X. Then \(u \in X\) and there exists \(v_{n} \in S_{F,u_{n}}\) such that, for each \(t \in J\),

$$ u_{n}(t) = I^{\alpha}v_{n}(t)+ \biggl( \frac{1}{2}-t \biggr)I^{\alpha-2}v_{n}(\xi)+ A(t) I^{\alpha}v_{n}(1) + B(t)I^{\alpha-p}v_{n}(1) + C(t)I^{\alpha+1}v_{n}(1). $$

As F has compact values, we pass onto a subsequence (if necessary) to find that \(v_{n}\) converges to v in \(L^{1} (J,\mathbb{R})\). Thus, \(v \in S_{F,u}\) and for each \(t \in J\), we have

$$ u_{n}(t)\to u(t) = I^{\alpha}v(t)+ \biggl(\frac{1}{2}-t \biggr)I^{\alpha-1}v(\xi )+ A(t) I^{\alpha}v(1) + B(t)I^{\alpha-p}v(1) + C(t)I^{\alpha+1}v(1). $$

Hence, \(u \in{\mathcal{N}}(u)\).

Next we show that \({\mathcal{N}}\) is a contractive multifunction with constant \(\delta=\|m\|\sum_{i=1}^{4}\Lambda_{i} <1\). Let \(u,w \in X\) and \(h_{1} \in{\mathcal{N}}(u)\). Then there exists \(v_{1} \in S_{F,u}\) such that, for each \(t \in J\),

$$ h_{1}(t) = I^{\alpha}v_{1}(t)+ \biggl( \frac{1}{2}-t \biggr) I^{\alpha-2}v_{1}(\xi) + A(t) I^{\alpha}v_{1}(1) + B(t)I^{\alpha-p} v_{1}(1) + C(t)I^{\alpha+1}v_{1}(1). $$

By (H5), we have

$$\begin{aligned}& H_{d}\bigl(F\bigl(t, u(t), u'(t), u''(t), {}^{\mathrm{c}}D^{p} u(t)\bigr), F\bigl(t, w(t), w'(t), w''(t), {}^{\mathrm{c}}D^{p} w(t)\bigr) \bigr) \\& \quad \leq m(t) \bigl(\bigl\vert u(t)-w(t)\bigr\vert +\bigl\vert u'(t)-w'(t)\bigr\vert + \bigl\vert u''(t)-w''(t)\bigr\vert + \bigl\vert {}^{\mathrm{c}}D^{p} u(t)-{}^{\mathrm{c}}D^{p} w(t)\bigr\vert \bigr), \end{aligned}$$

so there exists \(z\in F(t, u(t), u'(t), u''(t), {}^{\mathrm{c}}D^{p} u(t)) \) such that

$$\bigl\vert v_{1}(t)-z\bigr\vert \leq m(t) \bigl(\bigl\vert u(t)-w(t)\bigr\vert +\bigl\vert u'(t)-w'(t)\bigr\vert + \bigl\vert u''(t)-w''(t) \bigr\vert + \bigl\vert ^{\mathrm{c}}D^{p} u(t)-{}^{\mathrm{c}}D^{p} w(t)\bigr\vert \bigr) $$

for almost all \(t\in J\). Define the multifunction \(U: J \rightarrow {\mathcal{P}}(\mathbb{R})\) by

$$\begin{aligned} U(t) =&\bigl\{ z\in\mathbb{R} : \bigl\vert v_{1}(t)-z\bigr\vert \leq m(t) \bigl( \bigl\vert u(t)-w(t)\bigr\vert +\bigl\vert u'(t)-w'(t) \bigr\vert + \bigl\vert u''(t)-w''(t) \bigr\vert \\ &{}+ \bigl\vert {}^{\mathrm{c}}D^{p} u(t)-{}^{\mathrm{c}}D^{p} w(t)\bigr\vert \bigr) \mbox{ for almost all } t\in J\bigr\} . \end{aligned}$$

It is easy to check that the multifunction \(U( \cdot) \cap F(\cdot ,u(\cdot),u'(\cdot), u''(\cdot),{}^{\mathrm{c}}D^{p} u(\cdot )) \) is measurable. Hence, we can choose \(v_{2}\in S_{F, u}\) such that

$$\bigl\vert v_{1}(t)-v_{2}(t)\bigr\vert \leq m(t) \bigl( \bigl\vert u(t)-w(t)\bigr\vert +\bigl\vert u'(t)-w'(t) \bigr\vert + \bigl\vert u''(t)-w''(t) \bigr\vert + \bigl\vert {}^{\mathrm{c}}D^{p} u(t)-{}^{\mathrm{c}}D^{p} w(t)\bigr\vert \bigr) $$

for almost all \(t\in J\). Consider \(h_{2}\in{\mathcal{N}}(u)\) which is defined by

$$ h_{2}(t) = I^{\alpha}v_{2}(t)+ \biggl( \frac{1}{2}-t \biggr)I^{\alpha-2}v_{2}(\xi)+ A(t) I^{\alpha}v_{2}(1) + B(t)I^{\alpha-p} v_{2}(1) + C(t)I^{\alpha+1}v_{2}(1). $$

Thus,

$$\begin{aligned} \bigl\vert h_{1}(t)-h_{2}(t)\bigr\vert =& I^{\alpha}\bigl\vert v_{1}(t)-v_{2}(t)\bigr\vert + \biggl\vert \frac{1}{2}-t \biggr\vert I^{\alpha-2}\bigl\vert v_{1}(1)-v_{2}(1) \bigr\vert + \bigl\vert A(t)\bigr\vert I^{\alpha}\bigl\vert v_{1}(1)-v_{2}(1)\bigr\vert \\ &{} + \bigl\vert B(t) \bigr\vert I^{\alpha-p} \bigl\vert v_{1}(1)-v_{2}(1)\bigr\vert + \bigl\vert C(t)\bigr\vert I^{\alpha+1}\bigl\vert v_{1}(1)-v_{2}(1)\bigr\vert \\ \le& \|m\| \biggl\{ \frac{1}{\Gamma(\alpha+1)}+\frac{1}{2}\frac{\xi ^{\alpha-2}}{\Gamma(\alpha-1)}+ \frac{1}{2}(1+C_{1})\frac{1}{\Gamma(\alpha +1)} \\ &{}+\Gamma(3-p) \biggl(\frac{7}{4}+\frac{1}{12}C_{1} \biggr)\frac{1}{\Gamma (\alpha-p+1)} +\frac{1}{2} C_{1}\frac{1}{\Gamma(\alpha+2)} \biggr\} \|u-w\| \\ =&\|m\|\Lambda_{1}\|u-w\|. \end{aligned}$$

In a similar manner we obtain

$$\begin{aligned}& \bigl\vert h'_{1}(t)-h'_{2}(t) \bigr\vert \le \|m\| \biggl\{ \frac{1}{\Gamma(\alpha)}+ \frac{\xi^{\alpha -2}}{\Gamma(\alpha-1)}+ \frac{1}{2}C'_{1}\frac{1}{\Gamma(\alpha+1)} \\& \hphantom{\bigl\vert h'_{1}(t)-h'_{2}(t) \bigr\vert \le}{} +\Gamma(3-p) \biggl(1+\frac{1}{12}C'_{1} \biggr)\frac{1}{\Gamma(\alpha-p+1)} +\frac{1}{2} C'_{1} \frac{1}{\Gamma(\alpha+2)} \biggr\} \|u-w\| \\& \hphantom{\bigl\vert h'_{1}(t)-h'_{2}(t) \bigr\vert } = \|m\|\Lambda_{2}\|u-w\|, \\& \bigl\vert h''_{1}(t)-h''_{2}(t) \bigr\vert \le \|m\| \biggl\{ \frac{1}{\Gamma(\alpha-1)} +C''_{1} \frac{1}{\Gamma (\alpha+1)}+\Gamma(3-p) \biggl(1+\frac{1}{12}C''_{1} \biggr)\frac{1}{\Gamma (\alpha-p+1)} \\& \hphantom{\bigl\vert h''_{1}(t)-h''_{2}(t) \bigr\vert \le }{}+\frac{1}{2} C''_{1} \frac{1}{\Gamma(\alpha+2)} \biggr\} \|u-w\| \\& \hphantom{\bigl\vert h''_{1}(t)-h''_{2}(t) \bigr\vert } = \|m\|\Lambda_{3}\|u-w\| \end{aligned}$$

and

$$\begin{aligned} \bigl\vert {}^{\mathrm{c}}D^{p}h_{1}(t)-{}^{\mathrm{c}}D^{p}h_{2}(t) \bigr\vert \le& \|m\| \biggl\{ \frac{1}{\Gamma(\alpha-p + 1)}+D_{1} \frac{1}{\Gamma (\alpha+1)} \\ &{}+ \biggl(1+ \frac{\Gamma(3-p)}{12}D_{1} \biggr)\frac{1}{\Gamma(\alpha-p+1)} + \frac{1}{2} D_{1}\frac{1}{\Gamma(\alpha+2)} \biggr\} \|u-w\| \\ =&\|m\|\Lambda_{4}\|u-w\|. \end{aligned}$$

Hence,

$$\| h_{1}-h_{2}\| \le \|m\| \sum _{i=1}^{4}\Lambda_{i}\|u-w\|. $$

Analogously, interchanging the roles of u and w, we obtain

$$H_{d}\bigl({\mathcal{N}}(u), {\mathcal{N}}(w)\bigr) \le \|m\| \sum _{i=1}^{4}\Lambda_{i}\|u-w\|. $$

Since \(\delta= \|m\| \sum_{i=1}^{4}\Lambda_{i}<1\), \({\mathcal{N}}\) is a contraction, it follows by Lemma 1.4 that \({\mathcal{N}}\) has a fixed point u which is a solution of (1.1)-(1.2). This completes the proof. □

Now, we give an illustrative example.

Example 2.6

We consider the following fractional differential inclusion:

$$\begin{aligned} \begin{aligned}[b] {}^{\mathrm{c}}D^{\frac{5}{2}} u(t)\in{}& \biggl[ 0, \frac{\frac{t}{200} \vert\sin \frac{\pi}{2}t \vert\vert u(t)\vert}{1+\vert u(t)\vert} + \frac {t\vert\sin u'(t) \vert^{3}}{200 + 200 \vert\sin u'(t) \vert^{3} } + \frac {\frac{t}{200} \vert\cos u''(t) \vert^{4}}{1+ \vert\cos u''(t)\vert^{4} } \\ &{}+ \frac{t \vert{}^{\mathrm{c}}D^{\frac{5}{3}}u(t)\vert}{200 \vert\cos\pi t \vert( 1+ \vert{}^{\mathrm{c}}D^{\frac{5}{3}}u(t)\vert)} \biggr], \end{aligned} \end{aligned}$$
(2.10)

via the boundary value conditions

$$\begin{aligned}& u(0)+u(1)=0, \qquad {}^{\mathrm{c}}D^{\frac{5}{3}}u( 1)= \frac{1}{1\text{,}000} \int_{0}^{1} u(\tau)\, d\tau, \\& u'(0)+u'' \biggl( \frac{1}{100} \biggr)= \frac{1}{1\text{,}000} \int_{0}^{1} u(\tau)\, d\tau, \end{aligned}$$

where \(t\in J=[0,1]\). By the above inclusion problem, we have \(\alpha= 5/2\), \(p=5/3\), \(\xi= 1/100\), and \(\gamma= \eta= 1/1\text{,}000\). Now, we define an operator \(F: J \times\mathbb{R} \times\mathbb{R} \times \mathbb{R} \times\mathbb{R} \to P_{\mathrm{cp}}(\mathbb{R})\) by

$$\begin{aligned} F(t, x_{1},x_{2},x_{3},x_{4}) =& \biggl[0, \frac{\frac{t}{200} \vert\sin\frac {\pi}{2}t \vert\vert x_{1}\vert}{1+\vert x_{1}\vert} + \frac{t\vert \sin x_{2} \vert^{3}}{200 + 200 \vert\sin x_{2} \vert^{3} } + \frac{\frac {t}{200} \vert\cos x_{3} \vert^{4}}{1+ \vert\cos x_{3}\vert^{4} } \\ &{} + \frac{t \vert x_{4}\vert}{200 \vert\cos\pi t \vert( 1+ \vert x_{4}\vert)} \biggr]. \end{aligned}$$

Also, we define the function \(m: J \to\mathbb{R}^{+}\) by \(m(t)=t/200\). It is clear that m is continuous on J and \(\|m\| = 1/200\). Finally, one can write

$$\begin{aligned}& H_{d}\bigl(F(t,x_{1},x_{2},x_{3},x_{4}), F(t,y_{1},y_{2},y_{3},y_{4})\bigr) \\& \quad \leq m(t) \bigl(\vert x_{1}-y_{1}\vert +\vert x_{2}-y_{2}\vert +\vert x_{3}-y_{3} \vert +\vert x_{4}-y_{4}\vert \bigr), \end{aligned}$$

where \(t\in J\) and \(x_{i}, y_{i} \in\mathbb{R}\) (\(i=1,2, 3, 4\)). On the other hand, there exist the following values:

$$\Lambda_{1} = 2.13208,\qquad \Lambda_{2} = 1.79407, \qquad \Lambda_{3} = 2.057974, \qquad \Lambda_{4} = 2.08023. $$

Then \(\Vert m\Vert(\Lambda_{1} + \Lambda_{2} + \Lambda_{3} + \Lambda_{4} ) = \frac{1}{200}\times8.064354 = 0.04 < 1 \). Consequently, all assumptions and conditions of Theorem 2.5 are satisfied. Hence, Theorem 2.5 implies that the fractional differential inclusion problem (2.10) has at least one solution.