Open Access

New fractional-order multivalued problems with nonlocal nonlinear flux type integral boundary conditions

  • Bashir Ahmad1Email author,
  • Sotiris K Ntouyas1, 2,
  • Ahmed Alsaedi1 and
  • Faris Alzahrani1
Boundary Value Problems20152015:83

https://doi.org/10.1186/s13661-015-0346-0

Received: 16 February 2015

Accepted: 8 May 2015

Published: 26 May 2015

Abstract

In this paper, we study new fractional-order multivalued problems supplemented with nonlocal nonlinear flux type integral boundary conditions. Some existence results are obtained for convex as well as non-convex multivalued maps by applying standard fixed point theorems for such maps. We also discuss examples for the illustration of our results.

Keywords

fractional differential inclusionsnonlocal conditionsintegral boundary conditionsfixed point theorems

MSC

34A6034A0834B15

1 Introduction

In this paper, we investigate the existence of solutions for a fractional differential inclusion
$$ {}^{c}D^{q} x(t) \in F \bigl(t, x(t) \bigr), \quad t \in[0, 1], 1 < q \le2, $$
(1.1)
supplemented with nonlocal nonlinear flux type integral boundary conditions
$$ x(0)=\beta g \bigl(x'(\eta) \bigr), \qquad x'(1)=\alpha\int_{0}^{\xi}x'(s) \,ds, \quad 0< \xi, \eta< 1, $$
(1.2)
and
$$ x'(0)=\alpha\int_{0}^{\xi}x'(s) \,ds, \qquad x(1)=\beta g \bigl(x'(\eta) \bigr),\quad 0< \xi, \eta< 1, $$
(1.3)
where \(^{c}D^{q}\) denotes the Caputo fractional derivative of order q, \(F: [0,1]\times\mathbb{R}\to \mathcal{P}(\mathbb{R}) \) is a multivalued map, \(\mathcal{P}(\mathbb{R})\) is the family of all nonempty subsets of \(\mathbb{R}\), \(g: {\mathbb{R}}\to{\mathbb{R}}\) is a continuous function and α, β are appropriate real constants with \(\alpha\xi\ne1\).

Fractional differential equations are found to be of great interest in view of their extensive applications in various scientific disciplines such as fluid mechanics, biomathematics, ecology, visco-elastodynamics, aerodynamics, control theory, electro-dynamics of complex medium, thermodynamics, electrical circuits, electron-analytical chemistry, etc. For details, we refer the reader to the books [16].

The subject of boundary value problems has an enriched history, and several kinds of problems have been discussed over the years. In the past few years, fractional-order boundary value problems have received much attention, and the subject has been developed in an extensive manner. In fact it is a hot topic of research in mathematics and its applications. For some recent results on boundary value problems of fractional differential equations, we refer the reader to a series of papers [718] and the references cited therein.

Differential inclusions (multivalued differential equations), regarded as the generalization of single-valued differential equations, are found to be important mathematical modeling tools in certain problems of economics, optimal control, etc. and are widely studied by many authors; for instance, see [1921] and the references therein. Examples and details of some recent works on differential inclusions of fractional order can be found in [2228] and the references cited therein.

In this paper, we extend the study on fractional boundary value problems further by introducing a new class of problems of fractional differential inclusions supplemented with nonlocal nonlinear flux type integral boundary conditions. We obtain some existence results for the given problems for the cases of convex as well as non-convex multivalued maps. Our results are based on some standard theorems dealing with multivalued maps. The methods used here are well known, however their exposition in the context of problems (1.1)-(1.2) and (1.1)-(1.3) is new.

The paper is organized as follows. In Section 2 we present some related preliminary material from fractional calculus and multivalued analysis. We also describe the details of the proposed work. The main existence results for problem (1.1)-(1.2) are obtained in Section 3, and the examples illustrating the results of Section 3 are given in Section 4. We indicate the results for problem (1.1)-(1.3) in Section 5.

2 Preliminaries

This section is devoted to some preliminary concepts of fractional calculus and multivalued analysis. We also describe the work proposed in this paper together with the allied techniques.

Let us first recall some definitions of fractional calculus [1, 2] and prove an auxiliary result needed to define the solution for problem (1.1)-(1.2).

Definition 2.1

For an \((n-1)\)-times absolutely continuous function \(g : [0,\infty) \to \mathbb{R}\), the Caputo derivative of fractional order q is defined as
$${}^{c}D^{q} g(t)=\frac{1}{\Gamma(n-q)}\int_{0}^{t}(t-s)^{n-q-1}g^{(n)}(s) \,ds,\quad n-1 < q < n, n=[q]+1, $$
where \([q]\) denotes the integer part of the real number q.

Definition 2.2

The Riemann-Liouville fractional integral of order q is defined as
$$I^{q} g(t)=\frac{1}{\Gamma(q)}\int_{0}^{t} \frac{g(s)}{(t-s)^{1-q}}\,ds,\quad q>0, $$
provided the integral exists.

Lemma 2.3

Let \(y \in C([0, 1], \mathbb{R})\) and \(x\in C^{2}([0,1], {\mathbb{R}})\) be a solution of the linear boundary value problem
$$ \left \{ \begin{array}{@{}l} {}^{c}D^{q} x(t) = y(t), \quad t\in[0,1], 1< q\leq2,\\ x(0)=\beta g(x'(\eta)),\qquad x'(1)=\alpha\int_{0}^{\xi}x'(s)\,ds,\quad 0\le\xi,\eta\le1. \end{array} \right . $$
(2.1)
Then
$$\begin{aligned} x(t)={}& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}y(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}y(\tau)\,d\tau \,ds- \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}y(s)\,ds \biggr) \\ &{}+\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}y(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}y(\tau)\,d\tau \,ds \\ &{}-\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma (q)}y(s)\,ds \biggr). \end{aligned}$$
(2.2)

Proof

It is well known that the general solution of the fractional differential equation in (2.1) can be written as
$$ x(t)= c_{0}+c_{1}t+\int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}y(s)\,ds, $$
(2.3)
where \(c_{0}, c_{1} \in\mathbb{R}\) are arbitrary constants.
Applying the given boundary conditions, we find that
$$c_{0}=\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma(q-1)}y(s)\,ds+c_{1} \biggr), $$
and
$$c_{1} = \frac{1}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau )^{q-2}}{\Gamma(q-1)}y(\tau)\,d\tau \,ds- \int_{0}^{1}\frac{(1-s)^{q-2}}{\Gamma(q-1)}y(s)\,ds \biggr). $$
Substituting the values of \(c_{0}\), \(c_{1}\) in (2.3), we get (2.2). This completes the proof. □

Now we outline some background material on multivalued maps [29, 30].

For a normed space \((X, \|\cdot\|)\), let \({\mathcal{P}}_{cl}(X)=\{Y \in {\mathcal{P}}(X) : Y \mbox{ is closed}\}\), \({\mathcal{P}}_{b}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is bounded}\}\), \({\mathcal{P}}_{cp}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is compact}\}\), and \({\mathcal{P}}_{cp, c}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is compact and convex}\}\). 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 \(\mathbb{B} \in{\mathcal{P}}_{b}(X)\) (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 \(\mathbb{B} \in{\mathcal{P}}_{b}(X)\). 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., \(x_{n} \to x_{*}\), \(y_{n} \to y_{*}\), \(y_{n} \in G(x_{n})\) imply \(y_{*} \in G(x_{*})\). 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 \(G : [0,1] \to{\mathcal{P}}_{cl}(\mathbb{R})\) is said to be measurable if for every \(y \in \mathbb{R}\), the function
$$t \longmapsto d \bigl(y,G(t) \bigr) = \inf \bigl\{ |y-z|: z \in G(t) \bigr\} $$
is measurable.
For each \(y \in C( [0,1], \mathbb{R})\), define the set of selections of F by
$$S_{F,y} := \bigl\{ v \in L^{1} \bigl( [0,1],\mathbb{R} \bigr) : v (t) \in F \bigl(t, y(t) \bigr) \mbox{ for a.e. } t \in [0,1] \bigr\} . $$
Next we characterize the proposed work and provide the necessary details for achieving the main results.

To discuss the existence of solutions for problem (1.1)-(1.2) with a nonconvex valued right-hand side (the Lipschitz case), we need the following auxiliary material.

Let \((X,d)\) be a metric space induced from the normed space \((X; \|\cdot\|)\). Consider \(H_{d} : {\mathcal{P}}(X) \times{\mathcal{P}}(X) \to\mathbb{R} \cup\{\infty\}\) 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 \(d(a,B) = \inf_{b\in B}d(a;b)\). Then \(({\mathcal{P}}_{b,cl}(X), H_{d})\) is a metric space and \(({\mathcal{P}}_{cl}(X), H_{d})\) is a generalized metric space (see [31]).

Definition 2.4

A multivalued operator \(N : X \to{\mathcal{P}}_{cl}(X)\) is called:
  1. (a)
    γ-Lipschitz if and only if there exists \(\gamma> 0\) such that
    $$H_{d} \bigl(N(x),N(y) \bigr) \le\gamma d(x,y) \quad\mbox{for each } x, y \in X; $$
     
  2. (b)

    a contraction if and only if it is γ-Lipschitz with \(\gamma< 1\).

     

Lemma 2.5

([32])

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

To establish the existence result when the multivalued map F in (1.1) is convex-valued (the upper semicontinuous case), we recall the following definitions and known results.

Definition 2.6

A multivalued map \(F : [0,1] \times\mathbb{R} \to{\mathcal{P}}(\mathbb{R})\) is said to be Carathéodory if
  1. (i)

    \(t \longmapsto F(t,x)\) is measurable for each \(x \in\mathbb{R}\);

     
  2. (ii)

    \(x \longmapsto F(t,x)\) is upper semicontinuous for almost all \(t\in [0,1]\);

     
Further a Carathéodory function F is called \(L^{1}\)-Carathéodory if
  1. (iii)
    for each \(\rho> 0\), there exists \(\varphi_{\rho} \in L^{1}( [0,1],\mathbb{R}^{+})\) such that
    $$\bigl\| F (t, x) \bigr\| = \sup \bigl\{ |v| : v \in F (t, x) \bigr\} \le \varphi_{\rho} (t) $$
    for all \(\|x\| \le\rho\) and for a.e. \(t \in [0,1]\).
     

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

Lemma 2.7

([29], Proposition 1.2)

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

Lemma 2.8

([33])

Let X be a Banach space. Let \(F : [0, 1] \times X \to{\mathcal{P}}_{cp,c}(X)\) be an \(L^{1}\)-Carathéodory multivalued map, and let Θ be a linear continuous mapping from \(L^{1}( [0,1],X)\) to \(C( [0,1],X)\). Then the operator
$$\Theta\circ S_{F} : C \bigl( [0,1],X \bigr) \to{\mathcal{P}}_{cp,c} \bigl(C \bigl( [0,1],X \bigr) \bigr),\quad x \longmapsto(\Theta \circ S_{F}) (x) = \Theta( S_{F,x}) $$
is a closed graph operator in \(C( [0,1],X) \times C( [0,1],X)\).

Lemma 2.9

(Nonlinear alternative for Kakutani maps) [34]

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

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

     
  2. (ii)

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

     

Finally, we study the case when the multivalued map F in (1.1) is not necessarily convex-valued (the lower semicontinuous case). In this case, the background material is outlined as follows.

Let X be a nonempty closed subset of a Banach space E and \(G : X \to{\mathcal{P}}(E)\) be a multivalued operator with nonempty closed values. G is lower semicontinuous (l.s.c.) if the set \(\{y \in X : G(y)\cap B \ne\emptyset\}\) is open for any open set B in E. Let A be a subset of \([0,1]\times\mathbb{R}\). A is \(\mathcal{L}\otimes\mathcal{B}\) measurable if A belongs to the σ-algebra generated by all sets of the form \(\mathcal{J} \times\mathcal{D}\), where \(\mathcal{J}\) is Lebesgue measurable in \([0,1]\) and \(\mathcal{D}\) is Borel measurable in \(\mathbb{R}\). A subset \(\mathcal{A}\) of \(L^{1}( [0,1], \mathbb{R})\) is decomposable if for all \(u, v \in \mathcal{A}\) and measurable \(\mathcal{J} \subset [0,1]=J\), the function \(u \chi_{\mathcal{J}}+v \chi_{J-\mathcal{J}} \in\mathcal{A}\), where \(\chi_{\mathcal{J}}\) stands for the characteristic function of \(\mathcal{J}\).

Definition 2.10

Let Y be a separable metric space and let \(N : Y \to{\mathcal{P}}(L^{1}( [0,1],\mathbb{R}))\) be a multivalued operator. We say N has a property (BC) if N is lower semicontinuous (l.s.c.) and has nonempty closed and decomposable values.

Let \(F : [0,1] \times\mathbb{R} \to{\mathcal{P}}(\mathbb{R})\) be a multivalued map with nonempty compact values. Define a multivalued operator \(\mathcal{F} : C( [0,1] \times\mathbb{R}) \to{\mathcal{P}}(L^{1}( [0,1],\mathbb{R}))\) associated with F as
$$\mathcal{F}(x)= \bigl\{ w \in L^{1} \bigl( [0,1],\mathbb{R} \bigr) : w(t) \in F \bigl(t,x(t) \bigr) \mbox{ for a.e. } t \in [0,1] \bigr\} , $$
which is called the Nemytskii operator associated with F.

Definition 2.11

Let \(F : [0,1] \times\mathbb{R} \to{\mathcal{P}}(\mathbb{R})\) be a multivalued function with nonempty compact values. We say F is of lower semicontinuous type (l.s.c. type) if its associated Nemytskii operator \(\mathcal{F }\) is lower semicontinuous and has nonempty closed and decomposable values.

Lemma 2.12

([35])

Let Y be a separable metric space and let \(N : Y \to{\mathcal{P}}(L^{1}( [0,1],\mathbb{R}))\) be a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) \(g : Y \to L^{1}( [0,1],\mathbb{R})\) such that \(g(x) \in N(x)\) for every \(x \in Y\).

3 Existence results

Definition 3.1

A function \(x\in C^{2}([0,1], {\mathbb{R}})\) is said to be a solution of the boundary value problem (1.1)-(1.2) if \(x(0)=\beta g(x'(\eta))\), \(x'(1)=\alpha\int_{0}^{\xi}x'(s)\,ds\), and there exists a function \(v\in S_{F,x}\) such that
$$\begin{aligned} x(t)={}& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}v(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s} \frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds - \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds \\ &{}-\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr),\quad t\in[0,1]. \end{aligned}$$
(3.1)
For the sake of convenience, we set
$$ \Lambda=\frac{1}{\Gamma(q+1)} \biggl(1+|\beta|q\eta^{q-1}+ \frac{|\alpha |\xi^{q}+1}{|1-\alpha\xi|}\bigl(1+|\beta|\bigr) \biggr). $$
(3.2)

Now we are in a position to present our first existence result for problem (1.1)-(1.2) which deals with a nonconvex valued right-hand side of (1.1).

Theorem 3.2

(The Lipschitz case)

Assume that
(A1): 

\(F : [0,1] \times\mathbb{R} \to{\mathcal{P}}_{cp}(\mathbb{R})\) is such that \(F(\cdot,x) : [0,1] \to{\mathcal{P}}_{cp}(\mathbb{R})\) is measurable for each \(x \in\mathbb{R}\);

(A2): 

\(H_{d}(F(t,x), F(t,\bar{x}))\le m(t)|x-\bar{x}|\) for almost all \(t \in [0,1]\) and \(x, \bar{x} \in\mathbb{R}\) with \(m \in C( [0,1], \mathbb{R}^{+})\) and \(d(0,F(t,0))\le m(t)\) for almost all \(t \in [0,1]\);

(A3): 

\(g: {\mathbb{R}}\to{\mathbb{R}}\) is continuous and \(|g(v)|\le|v|\), \(\forall v\in{\mathbb{R}}\).

Then the boundary value problem (1.1)-(1.2) has at least one solution on \([0,1]\) if \(\|m\|\Lambda<1\), where Λ is given by (3.2).

Proof

Define the operator \(\Omega_{F}: C( [0,1], \mathbb{R})\to{\mathcal{P}}(C( [0,1], \mathbb{R}))\) by
$$\Omega_{F}(x)=\left \{ \begin{array}{@{}l} h \in C( [0,1], \mathbb{R}): \\ h(t) = \left \{ \begin{array}{@{}l} \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}v(s)\,ds\\ \quad{}+\frac{t}{1-\alpha\xi} (\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds -\int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds )\\ \quad{}+\beta g (\int_{0}^{\eta}\frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v(s)\,ds +\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi}\int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds\\ \quad{}-\frac{1}{1-\alpha\xi}\int_{0}^{1}\frac{(1-s)^{q-1}}{\Gamma (q)}v(s)\,ds ) \end{array} \right . \end{array} \right \} $$
for \(v\in S_{F,x}\).
Observe that the set \(S_{F,x}\) is nonempty for each \(x \in C( [0,1],\mathbb{R})\) by the assumption (A1), so F has a measurable selection (see Theorem III.6 [36]). Now we show that the operator \(\Omega_{F}\) satisfies the assumptions of Lemma 2.5. To show that \(\Omega_{F}(x) \in{\mathcal{P}}_{cl}((C [0,1],\mathbb{R}))\) for each \(x \in C( [0,1], \mathbb{R})\), let \(\{u_{n}\}_{n \ge0} \in\Omega_{F}(x)\) be such that \(u_{n} \to u \) (\(n \to\infty\)) in \(C( [0,1],\mathbb{R})\). Then \(u \in C( [0,1],\mathbb{R})\) and there exists \(v_{n} \in S_{F,x_{n}}\) such that, for each \(t \in [0,1]\),
$$\begin{aligned} u_{n}(t) =& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}v_{n}(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v_{n}( \tau)\,d\tau \,ds-\int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v_{n}(s) \,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v_{n}(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int _{0}^{\xi}\int_{0}^{s} \frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v_{n}(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v_{n}(s)\,ds \biggr). \end{aligned}$$
As F has compact values, we pass onto a subsequence (if necessary) to obtain that \(v_{n}\) converges to v in \(L^{1} ( [0,1],\mathbb{R})\). Thus, \(v \in S_{F,x}\) and for each \(t \in [0,1]\), we have
$$\begin{aligned} v_{n}(t) \to& v(t)= \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma (q)}v(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds- \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr). \end{aligned}$$

Hence, \(u \in\Omega_{F}(x)\).

Next we show that there exists \(\delta<1\) such that
$$H_{d} \bigl(\Omega_{F}(x), \Omega_{F}(\bar{x}) \bigr)\le\delta\|x-\bar{x}\| \quad\mbox{for each } x, \bar{x} \in C^{2} \bigl( [0,1], \mathbb{R} \bigr). $$
Let \(x, \bar{x} \in C^{2}( [0,1], \mathbb{R})\) and \(h_{1} \in\Omega_{F}(x)\). Then there exists \(v_{1}(t) \in F(t,x(t))\) such that, for each \(t \in [0,1]\),
$$\begin{aligned} h_{1}(t) =& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}v_{1}(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v_{1}( \tau)\,d\tau \,ds-\int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v_{1}(s) \,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v_{1}(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v_{1}( \tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v_{1}(s)\,ds \biggr). \end{aligned}$$
By (A2), we have
$$H_{d} \bigl(F(t,x), F(t,\bar{x}) \bigr)\le m(t) \bigl|x(t)-\bar{x}(t) \bigr|. $$
So, there exists \(w \in F(t,\bar{x}(t))\) such that
$$\bigl|v_{1}(t)-w \bigr|\le m(t) \bigl|x(t)-\bar{x}(t) \bigr|,\quad t \in [0,1]. $$
Define \(U : [0,1] \to\mathcal{P}(\mathbb{R})\) by
$$U(t)= \bigl\{ w \in\mathbb{R} : \bigl|v_{1}(t)-w \bigr|\le m(t) \bigl|x(t)-\bar{x}(t) \bigr| \bigr\} . $$
Since the multivalued operator \(U(t)\cap F(t,\bar{x}(t))\) is measurable (Proposition III.4 [36]), there exists a function \(v_{2}(t)\) which is a measurable selection for U. So \(v_{2}(t) \in F(t,\bar{x}(t))\) and for each \(t \in [0,1]\), we have \(|v_{1}(t)-v_{2}(t)|\le m(t)|x(t)-\bar{x}(t)|\).
For each \(t \in [0,1]\), let us define
$$\begin{aligned} h_{2}(t) =& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}v_{2}(s)\,ds\\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v_{2}( \tau)\,d\tau \,ds-\int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v_{2}(s) \,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v_{2}(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v_{2}( \tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v_{2}(s)\,ds \biggr). \end{aligned}$$
Thus,
$$\begin{aligned} \bigl|h_{1}(t)-h_{2}(t) \bigr| \le& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)} \bigl|v_{1}(s)-v_{2}(s) \bigr|\,ds \\ &{} +\frac{t}{|1-\alpha\xi|} \biggl(|\alpha|\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)} \bigl|v_{1}( \tau)-v_{2}(\tau) \bigr|\,d\tau \,ds \\ &{} +\int_{0}^{1}\frac{(1-s)^{q-1}}{\Gamma (q)} \bigl|v_{1}(s)-v_{2}(s) \bigr|\,ds \biggr) \\ &{} +|\beta| g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)} \bigl|v_{1}(s)-v_{2}(s) \bigr|(s)\,ds \\ &{} +\frac{|\alpha|}{|1-\alpha\xi|}\int_{0}^{\xi}\int _{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)} \bigl|v_{1}( \tau)-v_{2}(\tau) \bigr|\,d\tau \,ds \\ &{} +\frac{1}{|1-\alpha\xi|}\int_{0}^{1} \frac {(1-s)^{q-1}}{\Gamma(q)}\bigl|v_{1}(s)-v_{2}(s)\bigr|(s)\,ds \biggr) \\ \le&\|m\| \biggl\{ \frac{1}{\Gamma(q+1)}+\frac{1}{|1-\alpha\xi|} \biggl( \frac{|\alpha|\xi^{q}}{\Gamma(q+1)}+\frac{1}{\Gamma(q+1)} \biggr) \\ &{}+|\beta| \biggl(\frac{\eta^{q-1}}{\Gamma(q)}+\frac{|\alpha|}{|1-\alpha \xi|}\frac{\xi^{q}}{\Gamma(q+1)}+ \frac{1}{|1-\alpha\xi|}\frac{1}{{\Gamma (q+1)}} \biggr) \biggr\} \|x-\bar{x}\| \\ =&\|m\|\Lambda\|x-\bar{x}\|. \end{aligned}$$
Hence,
$$\| h_{1}-h_{2}\| \le\|m\|\Lambda\|x-\bar{x}\|. $$
Analogously, interchanging the roles of x and \(\overline{x}\), we obtain
$$H_{d} \bigl(\Omega_{F}(x), \Omega_{F}(\bar{x}) \bigr) \le \|m\|\Lambda\|x-\bar{x}\|, $$
where \(\delta=\|m\|\Lambda <1\). So \(\Omega_{F}\) is a contraction. Hence it follows by Lemma 2.5 that \(\Omega_{F}\) has a fixed point x which is a solution of (1.1)-(1.2). This completes the proof. □

Our next result concerns the existence of the solutions for problem (1.1)-(1.2) with a convex-valued multivalued map.

Theorem 3.3

(The upper semicontinuous case)

Assume that (A3) holds. In addition we suppose that
(H1): 

\(F : [0,1] \times \mathbb{R} \to{\mathcal{P}}(\mathbb{R})\) is Carathéodory and has nonempty compact and convex values;

(H2): 
there exists a continuous nondecreasing function \(\psi: [0,\infty) \to (0,\infty)\) and a function \(\phi\in C( [0,1],\mathbb{R}^{+})\) such that
$$\bigl\| F(t,x) \bigr\| _{\mathcal{P}}:= \sup \bigl\{ |y|: y \in F(t,x) \bigr\} \le\phi(t) \psi \bigl( \|x\| \bigr) \quad\textit{for each } (t,x) \in [0,1] \times\mathbb{R}; $$
(H3): 
there exists a constant \(M>0\) such that
$$\frac{M}{ \psi(M) \|\phi\|\Lambda}> 1, $$
where Λ is defined by (3.2).
Then the boundary value problem (1.1)-(1.2) has at least one solution on \([0,1]\).

Proof

Consider the operator \(\Omega_{F}: C( [0,1], \mathbb {R})\to{\mathcal{P}}(C( [0,1], \mathbb{R}))\) defined in the beginning of the proof of Theorem 3.2. We will show that \(\Omega_{F}\) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As the first step, we show that \(\Omega_{F}\) is convex for each \(x \in C( [0,1], \mathbb{R})\) . This step is obvious since \(S_{F,x}\) is convex (F has convex values), and therefore we omit the proof.

In the second step, we show that \(\Omega_{F}\) maps bounded sets (balls) into bounded sets in \(C( [0,1], \mathbb{R})\) . For a positive number ρ, let \(B_{\rho}= \{x \in C( [0,1], \mathbb{R}): \|x\| \le\rho\}\) be a bounded ball in \(C( [0,1], \mathbb{R})\). Then, for each \(h \in\Omega_{F} (x)\), \(x \in B_{\rho}\), there exists \(v \in S_{F,x}\) such that
$$\begin{aligned} h(t) =& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}v(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds- \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr),\quad t\in[0,1]. \end{aligned}$$
Then for \(t\in [0,1]\) we have
$$\begin{aligned} \bigl|h(t) \bigr| \le& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)} \bigl|v(s) \bigr|\,ds \\ &{} +\frac{t}{|1-\alpha\xi|} \biggl(|\alpha|\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)} \bigl|v(\tau) \bigr|\,d\tau \,ds+ \int_{0}^{1}\frac{(1-s)^{q-1}}{\Gamma(q)} \bigl|v(s) \bigr|\,ds \biggr) \\ &{} +|\beta| g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)} \bigl|v(s) \bigr|\,ds+\frac{|\alpha|}{|1-\alpha\xi|}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)} \bigl|v(\tau) \bigr|\,d\tau \,ds \\ & +\frac{1}{|1-\alpha\xi|}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)} \bigl|v(s) \bigr|\,ds \biggr) \\ \le& \psi \bigl(\|x\| \bigr) \|\phi\| \biggl\{ \frac{1}{\Gamma(q+1)}+\frac{1}{|1-\alpha\xi|} \biggl( \frac{|\alpha|\xi^{q}}{\Gamma(q+1)}+\frac{1}{\Gamma(q+1)} \biggr) \\ &{}+|\beta| \biggl(\frac{\eta^{q-1}}{\Gamma(q)}+\frac{|\alpha|}{|1-\alpha \xi|}\frac{\xi^{q}}{\Gamma(q+1)}+ \frac{1}{|1-\alpha\xi|}\frac{1}{\Gamma (q+1)} \biggr) \biggr\} \\ =& \psi \bigl(\|x\| \bigr) \|\phi\|\Lambda. \end{aligned}$$
Consequently,
$$\|h\|\le\psi(\rho) \|\phi\|\Lambda. $$
Now we show that \(\Omega_{F}\) maps bounded sets into equicontinuous sets of \(C( [0,1], \mathbb{R})\) . Let \(t_{1}, t_{2} \in[0,1]\) with \(t_{1}< t_{2}\) and \(x \in B_{\rho}\). For each \(h \in\Omega_{F}(x)\), we obtain
$$\begin{aligned} & \bigl|h(t_{2})-h(t_{1}) \bigr| \\ &\quad\le \biggl\vert \frac{1}{\Gamma(\alpha)}\int_{0}^{t_{2}} (t_{2}-s)^{q-1}v(s)\,ds-\frac{1}{\Gamma(\alpha)}\int _{0}^{t_{1}} (t_{1}-s)^{q-1}v(s) \,ds \biggr\vert \\ &\qquad{}+\frac{|t_{2}-t_{1}|}{|1-\alpha\xi|} \biggl(|\alpha|\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}\bigl|v(\tau)\bigr|\,d\tau \,ds+ \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}\bigl|v(s)\bigr|\,ds \biggr) \\ &\quad\le\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{1}} \bigl[(t_{2}-s)^{q-1}-(t_{1}-s)^{q-1} \bigr] \phi(s)\psi(r)\,ds+ \frac{1}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{q-1} \phi(s)\psi(r)\,ds \\ &\qquad{}+\frac{|t_{2}-t_{1}|}{|1-\alpha\xi|} \biggl(|\alpha|\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}\phi(s)\psi(r)\,d\tau \,ds+\int_{0}^{1}\frac{(1-s)^{q-1}}{\Gamma(q)}\phi(s)\psi(r) \,ds \biggr) \\ &\quad\le\frac{\|\phi\|\psi(\rho)}{\Gamma(\alpha)}\int_{0}^{t_{1}} \bigl[(t_{2}-s)^{q-1}-(t_{1}-s)^{q-1} \bigr] \,ds+ \frac{\|\phi\|\psi(\rho)}{\Gamma (\alpha)}\int_{t_{1}}^{t_{2}}(t_{2}-s)^{q-1} \,ds \\ &\qquad{}+\frac{\|\phi\|\psi(\rho)|t_{2}-t_{1}|}{|1-\alpha\xi|} \biggl(\frac{|\alpha |\xi^{q}}{\Gamma(q+1)}+\frac{1}{\Gamma(q+1)} \biggr). \end{aligned}$$

Obviously the right-hand side of the above inequality tends to zero independently of \(x \in B_{\rho}\) as \(t_{2}- t_{1} \to0\). As \(\Omega_{F}\) satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelà theorem that \(\Omega_{F}: C( [0,1], \mathbb{R}) \to{\mathcal{P}}(C( [0,1], \mathbb{R}))\) is completely continuous.

In our next step, we show that \(\Omega_{F}\) is upper semicontinuous. To this end it is sufficient to show that \(\Omega_{F}\) has a closed graph by Lemma 2.7. Let \(x_{n} \to x_{*}\), \(h_{n} \in\Omega_{F} (x_{n})\) and \(h_{n} \to h_{*}\). Then we need to show that \(h_{*} \in \Omega_{F} (x_{*})\). Associated with \(h_{n} \in \Omega_{F} (x_{n})\), there exists \(v_{n} \in S_{F,x_{n}}\) such that for each \(t \in [0,1]\),
$$\begin{aligned} h_{n}(t) =& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}v_{n}(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v_{n}( \tau)\,d\tau \,ds-\int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v_{n}(s) \,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v_{n}(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int _{0}^{\xi}\int_{0}^{s} \frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v_{n}(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v_{n}(s)\,ds \biggr). \end{aligned}$$
Thus it suffices to show that there exists \(v_{*} \in S_{F,x_{*}}\) such that for each \(t \in [0,1]\),
$$\begin{aligned} h_{*}(t) =& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}v_{*}(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v_{*}(\tau)\,d\tau \,ds- \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v_{*}(s)\,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v_{*}(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v_{*}(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v_{*}(s)\,ds \biggr). \end{aligned}$$
Let us consider the linear operator \(\Theta: L^{1}( [0,1], \mathbb{R}) \to C( [0,1], \mathbb{R})\) given by
$$\begin{aligned} v\longmapsto{}&\Theta(v) (t)\\ ={}& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma (q)}v(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds- \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr). \end{aligned}$$
Observe that
$$\begin{aligned} & \bigl\| h_{n}(t)-h_{*}(t) \bigr\| \\ &\quad= \biggl\| \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma (q)} \bigl(v_{n}(s)-v_{*}(s) \bigr)\,ds \\ &\qquad{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)} \bigl(v_{n}( \tau)-v_{*}(\tau) \bigr)\,d\tau \,ds-\int _{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr) \\ &\qquad{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)} \bigl(v_{n}(s)-v_{*}(s) \bigr)\,ds \\ &\qquad{}+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi}\int _{0}^{s}\frac{(s-\tau )^{q-2}}{\Gamma(q-1)} \bigl(v_{n}( \tau)-v_{*}(\tau) \bigr)\,d\tau \,ds \\ &\qquad{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma (q)} \bigl(v_{n}(s)-v_{*}(s) \bigr)\,ds \biggr) \biggr\| \to 0 \end{aligned}$$
as \(n\to\infty\).
Thus, it follows by Lemma 2.8 that \(\Theta\circ S_{F}\) is a closed graph operator. Further, we have \(h_{n}(t) \in \Theta(S_{F,x_{n}})\). Since \(x_{n} \to x_{*}\), therefore we have
$$\begin{aligned} h_{*}(t) =& \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}v_{*}(s)\,ds \\ &{} +\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v_{*}(\tau)\,d\tau \,ds- \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v_{*}(s)\,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v_{*}(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v_{*}(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v_{*}(s)\,ds \biggr) \end{aligned}$$
for some \(v_{*} \in S_{F,x_{*}}\).
Finally, we show that there exists an open set \(U\subseteq C( [0,1],{\mathbb{R}})\) with \(x\notin\Omega_{F} (x)\) for any \(\lambda\in(0,1)\) and all \(x\in\partial U\). Let \(\lambda\in (0,1)\) and \(x\in\lambda\Omega_{F} (x)\). Then there exists \(v \in L^{1}( [0,1], \mathbb{R})\) with \(v \in S_{F,x}\) such that, for \(t \in [0,1]\), we have
$$\begin{aligned} x(t) =& \lambda\int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma (q)}v(s) \,ds \\ &{} + \lambda\frac{t}{1-\alpha\xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds- \int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr) \\ &{} + \lambda\beta g \biggl(\int_{0}^{\eta} \frac{(\eta -s)^{q-2}}{\Gamma(q-1)}v(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds \biggr). \end{aligned}$$
Using the computations of the second step above we have
$$\begin{aligned} \bigl|x(t) \bigr| \le& \psi \bigl(\|x\| \bigr) \|\phi\| \biggl\{ \frac{1}{\Gamma(q+1)}+ \frac {1}{|1-\alpha\xi|} \biggl(\frac{|\alpha|\xi^{q}}{\Gamma(q+1)}+\frac {1}{\Gamma(q+1)} \biggr) \\ &{}+|\beta| \biggl(\frac{\eta^{q-1}}{\Gamma(q)}+\frac{|\alpha|}{|1-\alpha \xi|}\frac{\xi^{q}}{\Gamma(q+1)}+ \frac{1}{|1-\alpha\xi|}\frac{1}{{\Gamma (q+1)}} \biggr) \biggr\} \\ =& \psi \bigl(\|x\| \bigr) \|\phi\|\Lambda. \end{aligned}$$
Consequently, we have
$$\frac{\|x\|}{ \psi (\|x\| ) \|\phi\|\Lambda}\le1. $$
In view of (H3), there exists M such that \(\|x\| \ne M\). Let us set
$$U = \bigl\{ x \in C \bigl( [0,1], \mathbb{R} \bigr) : \|x\| < M \bigr\} . $$
Note that the operator \(\Omega_{F} :\overline{U} \to\mathcal{P}(C( [0,1], \mathbb{R}))\) is upper semicontinuous and completely continuous. From the choice of U, there is no \(x \in\partial U\) such that \(x \in\lambda\Omega_{F} (x)\) for some \(\lambda\in(0,1)\). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.9), we deduce that \(\Omega_{F}\) has a fixed point \(x \in\overline{U}\) which is a solution of problem (1.1)-(1.2). This completes the proof. □

Finally, we assume that the multivalued map F in (1.1) is not necessarily convex valued and formulate (and prove) the existence result as follows.

Theorem 3.4

(The lower semicontinuous case)

Assume that (A3), (H2), (H3) and the following condition hold:
(H4): 
\(F : [0,1] \times\mathbb{R} \to{\mathcal{P}}(\mathbb{R})\) is a nonempty compact-valued multivalued map such that
  1. (a)

    \((t,x) \longmapsto F(t,x)\) is \(\mathcal{L}\otimes \mathcal{B}\) measurable,

     
  2. (b)

    \(x \longmapsto F(t,x)\) is lower semicontinuous for each \(t \in [0,1]\);

     
then the boundary value problem (1.1)-(1.2) has at least one solution on \([0,1]\).

Proof

It follows from (H2) and (H4) that F is of l.s.c. type. Then, from Lemma 2.12, there exists a continuous function \(f : C^{2}( [0,1],\mathbb{R}) \to L^{1}( [0,1],\mathbb{R})\) such that \(f (x) \in \mathcal{F}(x)\) for all \(x \in C( [0,1],\mathbb{R})\).

Consider the problem
$$ \left \{ \begin{array}{@{}l} {}^{c}D^{q} x(t)= f(x(t)),\quad t \in[0, 1], 1 < q \le2,\\ x(0)=\beta g(x'(\eta)), \qquad x'(1)=\alpha \int_{0}^{\xi}x'(s)\,ds, \quad 0< \xi, \eta< 1. \end{array} \right . $$
(3.3)
Observe that if \(x \in C^{2}( [0,1],\mathbb{R})\) is a solution of (3.3), then x is a solution to problem (1.1)-(1.2). In order to transform problem (3.3) into a fixed point problem, we define the operator \(\overline{\Omega_{F}}\) as
$$\begin{aligned} \overline{\Omega_{F}} x(t) =& \int_{0}^{t} \frac {(t-s)^{q-1}}{\Gamma(q)}f \bigl(x(s) \bigr)\,ds \\ &{} +\frac{t}{1-\alpha \xi} \biggl(\alpha\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma (q-1)}f \bigl(x(\tau) \bigr)\,d\tau \,ds -\int_{0}^{1}\frac{(1-s)^{q-1}}{\Gamma(q)}f \bigl(x(s) \bigr)\,ds \biggr) \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma (q-1)}f \bigl(x(s) \bigr)\,ds +\frac{\alpha}{1-\alpha\xi}\int _{0}^{\xi} \int_{0}^{s} \frac{(s-\tau)^{q-2}}{\Gamma(q-1)}f \bigl(x(\tau) \bigr)\,d\tau \,ds \\ &{} -\frac{1}{1-\alpha\xi}\int_{0}^{1} \frac{(1-s)^{q-1}}{\Gamma (q)}f \bigl(x(s) \bigr)\,ds \biggr),\quad t\in[0,1]. \end{aligned}$$

It can easily be shown that \(\overline{\Omega_{F}}\) is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.3. So we omit it. This completes the proof. □

4 Examples

Consider the problem
$$ \left \{ \begin{array} {@{}l} {}^{c}D^{3/2}x(t)\in F(t,x(t)), \quad 0\le t \le1, \\ x(0)=\frac{1}{3}g(x'(3/4)), \qquad x'(1)= \frac{1}{2}\int_{0}^{1/3}x'(s)\,ds. \end{array} \right . $$
(4.1)
Here, \(q=3/2\), \(\alpha=1/2\), \(\xi=1/3\), \(\beta=1/3\), \(\eta=3/4\) and
$$g(v)=\left \{ \begin{array}{@{}l@{\quad}l} \sqrt{v}, &|v|\ge1,\\ v^{2}, &|v|< 1. \end{array} \right . $$

With the given values, we find that \(\Lambda \approx 2.3980168\).

(i) Consider the multivalued map \(F: [0,1]\times{\mathbb{R}}\to{\mathcal{P}}({\mathbb{R}})\) given by
$$F(t,x)= \biggl[0, \frac{1}{6}(t+1)\sin x+\frac{1}{6} \biggr]. $$
Then we have
$$\sup \bigl\{ |v|: v\in F(t,x) \bigr\} \le \frac{1}{6}(t+1)+ \frac{1}{6}, $$
and
$$H_{d}\bigl(F(t,x),F(t,\bar{x})\bigr)\le\frac{1}{6}(t+1)|x-\bar{x}|. $$
Let \(m(t)=\frac{1}{6}(t+1)\). Then \(\|m\|=\frac{1}{3}\) and \(\|m\|\Lambda\approx0.7993389<1\). Hence by Theorem 3.2 problem (4.1) has a solution.
(ii) Let \(F: [0,1]\times{\mathbb{R}}\to{\mathcal{P}}({\mathbb{R}})\) be a multivalued map given by
$$F(t,x)= \biggl[e^{-t} \biggl(\frac{|x|^{3}}{|x|^{3}+5}+9 \biggr),2e^{-t} \biggl(\frac{|x|^{3}}{|x|^{3}+3}+1 \biggr) \biggr]. $$
For \(v\in F\), we have
$$\bigl|v(t) \bigr|\le\max \biggl(e^{-t} \biggl(\frac{|x|^{3}}{|x|^{3}+5}+9 \biggr),2e^{-t} \biggl(\frac{|x|^{3}}{|x|^{3}+3}+1 \biggr) \biggr) \le10e^{-t}, \quad x\in {\mathbb{R}}. $$
Thus
$$\bigl\| F(t,x) \bigr\| _{\mathcal{P}}:=\sup \bigl\{ |y|: y \in F(t,x) \bigr\} \le10 e^{-t}=\phi (t)\psi \bigl(\|x\| \bigr), $$
with \(\phi(t)=e^{-t}\), \(\psi(\|x\|)=10\).

By assumption (H3), we find that \(M>23.980168\). It follows by Theorem 3.3 that problem (4.1) has a solution.

5 Existence results for problem (1.1)-(1.3)

To define the solution for problem (1.1)-(1.3), we need the following lemma.

Lemma 5.1

Let \(y \in C([0, 1], \mathbb{R})\) and \(x\in C^{2}([0,1],{\mathbb{R}})\) be a solution of the linear boundary value problem
$$ \left \{ \begin{array}{@{}l} {}^{c}D^{q} x(t) = y(t),\quad t\in[0,1], 1< q\leq2,\\ x'(0)=\alpha\int_{0}^{\xi}x'(s)\,ds, \qquad x(1)=\beta g(x'(\eta)), \quad 0\le\xi,\eta\le1. \end{array} \right . $$
(5.1)
Then
$$\begin{aligned} x(t)={}& \int_{0}^{t} \frac{(t-s)^{q-1}}{\Gamma(q)}y(s)\,ds \\ &{} +(t-1)\frac{\alpha}{1-\alpha\xi} \int_{0}^{\xi} \int _{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}y(\tau)\,d\tau \,ds -\int _{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}y(s)\,ds \\ &{} +\beta g \biggl(\int_{0}^{\eta} \frac{(\eta-s)^{q-2}}{\Gamma(q-1)}y(s)\,ds+\frac {\alpha}{1-\alpha \xi}\int_{0}^{\xi} \int_{0}^{s}\frac{(s-\tau)^{q-2}}{\Gamma(q-1)}y(\tau)\,d\tau \,ds \biggr). \end{aligned}$$
(5.2)

Proof

We omit the proof as it is similar to that of Lemma 2.3. □

In relation to problem (1.1)-(1.3), we define an operator \(G_{F}: C( [0,1], \mathbb{R})\to{\mathcal{P}}(C( [0,1], \mathbb{R}))\) by
$$G_{F}(x)=\left \{ \begin{array}{@{}l} h \in C( [0,1], \mathbb{R}): \\ h(t) = \left \{ \begin{array}{@{}l} \int_{0}^{t}\frac{(t-s)^{q-1}}{\Gamma(q)}v(s)\,ds\\ \quad{}+(t-1)\frac{\alpha}{1-\alpha\xi} \int_{0}^{\xi}\int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds-\int_{0}^{1}\frac {(1-s)^{q-1}}{\Gamma(q)}v(s)\,ds\\ \quad{}+\beta g (\int_{0}^{\eta}\frac{(\eta-s)^{q-2}}{\Gamma (q-1)}v(s)\,ds+\frac{\alpha}{1-\alpha\xi}\int_{0}^{\xi}\int_{0}^{s}\frac {(s-\tau)^{q-2}}{\Gamma(q-1)}v(\tau)\,d\tau \,ds ) \end{array} \right . \end{array} \right \} $$
for \(v\in S_{F,x}\), and set
$$ \bar{\Lambda}=\frac{1}{\Gamma(q+1)} \biggl(2+ q|\beta| \eta^{q-1}+\frac {|\alpha|(1+|\beta|)\xi^{q}}{|1-\alpha\xi|} \biggr). $$
(5.3)
With the above operator and estimate (5.3), we can reproduce all the existence results obtained in Section 3 for the boundary value problem (1.1)-(1.3).

6 Conclusions

We have studied the existence of solutions for fractional-order differential inclusions supplemented with new nonlocal nonlinear flux type integral boundary conditions for different types of multivalued maps involved in the given problems. Precisely, Section 3 contains some existence results for problem (1.1)-(1.2): the first one (Theorem 3.2) deals with nonconvex valued maps (the Lipschitz case) and is obtained by applying a fixed point theorem for multivalued maps due to Covitz and Nadler; the second result (Theorem 3.3) takes into account the convex-valued maps (the upper semicontinuous case) and relies on nonlinear alternative of Leray-Schauder type; and the third one (Theorem 3.4) involves multivalued maps which are not necessarily convex-valued (the lower semicontinuous case) and is obtained by jointly using the nonlinear alternative of Leray-Schauder type and the selection theorem of Bressan and Colombo [37] for lower semicontinuous maps with decomposable values. These results have been illustrated with the aid of examples in Section 4. Finally, we have provided the platform for proving the existence results for problem (1.1)-(1.3). We emphasize that our results are new, and several special results can be obtained by fixing the parameters α, β, ξ, η involved in the problems at hand.

Declarations

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also acknowledge the reviewers for their useful comments.

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

(1)
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Department of Mathematics, University of Ioannina

References

  1. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  2. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar
  3. Sabatier, J, Agrawal, OP, Machado, JAT (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007) MATHGoogle Scholar
  4. Tomovski, Z, Hilfer, R, Srivastava, HM: Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct. 21, 797-814 (2010) View ArticleMATHMathSciNetGoogle Scholar
  5. Konjik, S, Oparnica, L, Zorica, D: Waves in viscoelastic media described by a linear fractional model. Integral Transforms Spec. Funct. 22, 283-291 (2011) View ArticleMATHMathSciNetGoogle Scholar
  6. Keyantuo, V, Lizama, C: A characterization of periodic solutions for time-fractional differential equations in UMD spaces and applications. Math. Nachr. 284, 494-506 (2011) View ArticleMATHMathSciNetGoogle Scholar
  7. Ahmad, B, Nieto, JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011, Article ID 36 (2011) View ArticleMathSciNetMATHGoogle Scholar
  8. Liang, S, Zhang, J: Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval. Math. Comput. Model. 54, 1334-1346 (2011) View ArticleMATHMathSciNetGoogle Scholar
  9. Su, X: Solutions to boundary value problem of fractional order on unbounded domains in a Banach space. Nonlinear Anal. 74, 2844-2852 (2011) View ArticleMATHMathSciNetGoogle Scholar
  10. Bai, ZB, Sun, W: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 63, 1369-1381 (2012) View ArticleMATHMathSciNetGoogle Scholar
  11. Agarwal, RP, O’Regan, D, Stanek, S: Positive solutions for mixed problems of singular fractional differential equations. Math. Nachr. 285, 27-41 (2012) View ArticleMATHMathSciNetGoogle Scholar
  12. Cabada, A, Wang, G: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403-411 (2012) View ArticleMATHMathSciNetGoogle Scholar
  13. Ahmad, B, Ntouyas, SK, Alsaedi, A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013, Article ID 320415 (2013) MathSciNetMATHGoogle Scholar
  14. O’Regan, D, Stanek, S: Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 71, 641-652 (2013) View ArticleMATHMathSciNetGoogle Scholar
  15. Graef, JR, Kong, L, Wang, M: Existence and uniqueness of solutions for a fractional boundary value problem on a graph. Fract. Calc. Appl. Anal. 17, 499-510 (2014) View ArticleMATHMathSciNetGoogle Scholar
  16. Wang, G, Liu, S, Zhang, L: Eigenvalue problem for nonlinear fractional differential equations with integral boundary conditions. Abstr. Appl. Anal. 2014, Article ID 916260 (2014) MathSciNetGoogle Scholar
  17. Ahmad, B, Nieto, JJ: Sequential differential equations of fractional order with multi-point boundary conditions. Georgian Math. J. 21, 243-248 (2014) View ArticleMATHMathSciNetGoogle Scholar
  18. Ahmad, B, Agarwal, RP: Some new versions of fractional boundary value problems with slit-strips conditions. Bound. Value Probl. 2014, Article ID 175 (2014) View ArticleMathSciNetMATHGoogle Scholar
  19. Smirnov, GV: Introduction to the Theory of Differential Inclusions. Graduate Studies in Mathematics, vol. 41. Am. Math. Soc., Providence (2002) MATHGoogle Scholar
  20. Chang, Y-K, Li, WT, Nieto, JJ: Controllability of evolution differential inclusions in Banach spaces. Nonlinear Anal. 67, 623-632 (2007) View ArticleMATHMathSciNetGoogle Scholar
  21. Li, WS, Chang, YK, Nieto, JJ: Solvability of impulsive neutral evolution differential inclusions with state-dependent delay. Math. Comput. Model. 49, 1920-1927 (2009) View ArticleMATHMathSciNetGoogle Scholar
  22. Henderson, J, Ouahab, A: Fractional functional differential inclusions with finite delay. Nonlinear Anal. 70, 2091-2105 (2009) View ArticleMATHMathSciNetGoogle Scholar
  23. Ahmad, B, Nieto, JJ, Pimentel, J: Some boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1238-1250 (2011) View ArticleMATHMathSciNetGoogle Scholar
  24. Ahmad, B, Ntouyas, SK: Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions. Bound. Value Probl. 2012, Article ID 55 (2012) View ArticleMathSciNetMATHGoogle Scholar
  25. Ahmad, B, Ntouyas, SK: Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2013, 20 (2013) View ArticleMathSciNetMATHGoogle Scholar
  26. Dhage, BC, Ntouyas, SK: Existence results for boundary value problems for fractional hybrid differential inclusions. Topol. Methods Nonlinear Anal. 44, 229-238 (2014) MathSciNetGoogle Scholar
  27. Agarwal, RP, Baleanu, D, Hedayati, V, Rezapour, S: Two fractional derivative inclusion problems via integral boundary condition. Appl. Math. Comput. 257, 205-212 (2015) View ArticleMathSciNetGoogle Scholar
  28. Cernea, A: Filippov lemma for a class of Hadamard-type fractional differential inclusions. Fract. Calc. Appl. Anal. 18, 163-171 (2015) MATHMathSciNetView ArticleGoogle Scholar
  29. Deimling, K: Multivalued Differential Equations. de Gruyter, Berlin (1992) View ArticleMATHGoogle Scholar
  30. Hu, S, Papageorgiou, N: Handbook of Multivalued Analysis, Theory I. Kluwer Academic, Dordrecht (1997) View ArticleMATHGoogle Scholar
  31. Kisielewicz, M: Differential Inclusions and Optimal Control. Kluwer Academic, Dordrecht (1991) MATHGoogle Scholar
  32. Covitz, H, Nadler, SB Jr.: Multivalued contraction mappings in generalized metric spaces. Isr. J. Math. 8, 5-11 (1970) View ArticleMATHMathSciNetGoogle Scholar
  33. Lasota, A, Opial, Z: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13, 781-786 (1965) MATHMathSciNetGoogle Scholar
  34. Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2005) MATHGoogle Scholar
  35. Frigon, M: Théorèmes d’existence de solutions d’inclusions différentielles. In: Granas, A, Frigon, M (eds.) Topological Methods in Differential Equations and Inclusions. NATO ASI Series C, vol. 472, pp. 51-87. Kluwer Academic, Dordrecht (1995) View ArticleGoogle Scholar
  36. Castaing, C, Valadier, M: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977) MATHGoogle Scholar
  37. Bressan, A, Colombo, G: Extensions and selections of maps with decomposable values. Stud. Math. 90, 69-86 (1988) MATHMathSciNetGoogle Scholar

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