Fractional-order multivalued problems with non-separated integral-flux boundary conditions

In this paper, we study the existence of solutions for a new kind of boundary value problem of Caputo type fractional differential inclusions with non-separated local and nonlocal integral-flux boundary conditions. We apply appropriate fixed point theorems for multivalued maps to obtain the existence results for the given problems covering convex as well as non-convex cases for multivalued maps. We also include Riemann–Stieltjes integral conditions in our discussion. Some illustrative examples are also presented. The paper concludes with some interesting observations.


Introduction
We investigate existence of solutions for the following fractional differential inclusion: supplemented with non-separated local and nonlocal integral-flux boundary conditions respectively given by and Corresponding author.Email: bashirahmad − qau@yahoo.comwhere c D α , c D β denotes the Caputo fractional derivatives of orders α, β, F : [0, 1] × R → P (R) is a multivalued map, P (R) is the family of all nonempty subsets of R, I γ is the Riemann-Liouville fractional integral of order γ (see Definition 2.1) and a, b are appropriate real constants.
Fractional-order boundary problems involving a variety of boundary conditions have been extensively studied in the recent years.In view of the extensive development of single-valued nonlinear boundary value problems of fractional-order differential equations [1][2][3][4]17,19,26,27,29,31], it is natural to extend this work to the case of fractional-order multi-valued problems.For some recent results on fractional-order inclusions problems, we refer the reader to a series of papers [6-8, 11, 12, 14, 20, 35, 36] and the references cited therein.It is worthwhile to mention that fractional-order differential equations have attracted a great attention due to their widespread applications in applied and technical sciences such as physics, mechanics, chemistry, engineering, biomedical sciences, control theory, etc.One of the reasons for popularity of fractional calculus is that fractional-order operators can describe the hereditary properties of many important materials and processes.Further details can be found in the texts [9,23,28].
The purpose of this paper is to establish some existence results for the problems (1.1)-(1.2) and (1.1)-(1.3)for convex and non-convex values of the multivalued maps involved in the problems.Our main results rely on the well known tools of fixed point theory of multivalued maps such as the nonlinear alternative of Leray-Schauder type and a fixed point theorem for contraction multivalued maps due to Covitz and Nadler.We also discuss the case when the multivalued map is not necessarily convex valued.In this case, we make use of the nonlinear alternative of Leray-Schauder type for single-valued maps and a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values.We emphasize that the tools employed in the present work are well known, however their application in the present framework facilitates to obtain the existence results for the problems (1.1)-(1.2) and (1.1)-(1.3),which is indeed a new development.The paper is organized as follows.Section 2 contains some preliminaries needed for the sequel.In Section 3, we establish the existence results for the problem (1.1)-(1.2) which are well illustrated with the aid of examples.We also discuss the Riemann-Stieltjes integral conditions case in this section.The results for the problem (1.1)-(1.3)are presented in Section 4.

Preliminaries
In this section, we recall some basic concepts of fractional calculus [23,28] and multi-valued analysis [16,21].We also prove an auxiliary lemma which plays a key role in defining a fixed point problem related to the problem (1.1)-(1.2).Definition 2.1.The Riemann-Liouville fractional integral of order q for a continuous function g is defined as provided the integral exists.
Definition 2.2.For an at least n times continuously differentiable function g : [0, ∞) → R, the Caputo derivative of fractional order q is defined as Fractional-order multivalued problems 3 where [q] denotes the integer part of the real number q.
y(s) ds. ( Proof.It is well known that the general solution of the fractional differential equation in (2.1) can be written as where c 0 , c 1 ∈ R are arbitrary constants.Using the boundary condition x (0) = b c D β x(1) in (2.3), we find that In view of the condition x(0) + x(1) = a 1 0 x(s) ds, (2.3) yields which, on inserting the value of c 1 , and using the first relation in part (i) of Lemma 2.3, gives Substituting the values of c 0 , c 1 in (2. Next we recall some basic definitions of multivalued analysis.For a normed space (A, • ),

Existence results for the boundary value problem (1.1)-(1.2)
In this section, we study the existence of solutions for the problem (1.1)-(1.2).

The upper semicontinuous case
For the forthcoming analysis, we need the following lemmas.Lemma 3.2 (Nonlinear alternative for Kakutani maps [22]).Let E be a Banach space, C a closed convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → P cp,c (C) is an upper semicontinuous compact map.Then either (i) F has a fixed point in U, or (ii) there is a u ∈ ∂U and λ ∈ (0, 1) with u ∈ λF(u).Lemma 3.3 ([25]).Let X be a Banach space.Let F : [0, 1] × X → 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 Now we are in a position to prove the existence of the solutions for the boundary value problem (1.1)-(1.2) when the right-hand side is convex valued.Theorem 3.4.Assume that: where Proof.Define the operator for v ∈ S F,x .We will show that Ω F satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.The proof consists of several steps.As a first step, we show that Ω F is convex for each x ∈ C([0, 1], 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 Ω F maps bounded sets (balls) into bounded sets in C([0, 1], R).For a positive number r, let B r = {x ∈ C([0, 1], R) : x ≤ r} be a bounded ball in C([0, 1], R).Then, for each h ∈ Ω F (x), x ∈ B r , there exists v ∈ S F,x such that Then for t ∈ [0, 1] we have

Now we show that
Obviously the right-hand side of the above inequality tends to zero independently of x ∈ B r as t 2 − t 1 → 0. As Ω F satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelà theorem that Fractional-order multivalued problems 7 In our next step, we show that Ω F is upper semicontinuous.It is known [16, Proposition 1.2] that Ω F will be upper semicontinuous if we prove that it has a closed graph, since Ω F is already shown to be completely continuous.Thus we will prove that Ω F has a closed graph.Let x n → x * , h n ∈ Ω F (x n ) and h n → h * .Then we need to show that h * ∈ Ω F (x * ).Associated with h n ∈ Ω F (x n ), there exists v n ∈ S F,x n such that for each t ∈ [0, 1], Thus it suffices to show that there exists v * ∈ S F,x * such that for each t ∈ [0, 1], Let us consider the linear operator Θ : Observe that Thus, it follows by Lemma 3.3 that Θ • S F is a closed graph operator.Further, we have for some v * ∈ S F,x * .Finally, we show there exists an open set U ⊆ C([0, 1], R) with x / ∈ Ω F (x) for any λ ∈ (0, 1) and all x ∈ ∂U.Let λ ∈ (0, 1) and x ∈ λΩ F (x). Then there exists v ∈ L 1 ([0, 1], R) with v ∈ S F,x B. Ahmad and S. K. Ntouyas such that, for t ∈ [0, 1], we have Using the computations of the second step above we have In view of (H 3 ), there exists M such that x = M. Let us set Note that the operator Ω F : U → P (C([0, 1], R)) is upper semicontinuous and completely continuous.From the choice of U, there is no x ∈ ∂U such that x ∈ λΩ F (x) for some λ ∈ (0, 1).Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that Ω F has a fixed point x ∈ U which is a solution of the problem (1.1)-(1.2).This completes the proof.

The Lipschitz case
Now we prove the existence of solutions for the problem (1.1)-(1.2) with a nonconvex valued right hand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler.
Let (X, d) be a metric space induced from the normed space (X; • ).Consider H d : P (X) × P (X) → R ∪ {∞} given by  H d ) is a metric space and (P cl (X), H d ) is a generalized metric space (see [24]).Definition 3.5.A multivalued operator N : X → P cl (X) is called: (a) γ-Lipschitz if and only if there exists γ > 0 such that H d (N(x), N(y)) ≤ γd(x, y) for each x, y ∈ X; (b) a contraction if and only if it is γ-Lipschitz with γ < 1.

Then the boundary value problem (1.1)-(1.2) has at least one solution on
Proof.Observe that the set S F,x is nonempty for each x ∈ C([0, 1], R) by the assumption (H 4 ), so F has a measurable selection (see [10,Theorem III.6]).Now we show that the operator Ω F , defined in the beginning of proof of Theorem 3.4, satisfies the assumptions of Lemma 3.6.To show that 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], R).Thus, v ∈ S F,x and for each t ∈ [0, 1], we have Hence, u ∈ Ω(x).
Next we show that there exists δ < 1 such that By (H 5 ), we have So, there exists w ∈ F(t, x(t)) such that Since the multivalued operator U(t) ∩ F(t, x(t)) is measurable [10, Proposition III.4], there exists a function v 2 (t) which is a measurable selection for U.So v 2 (t) ∈ F(t, x(t)) and for each t ∈ [0, 1], we have |v Hence, Analogously, interchanging the roles of x and x, we obtain where δ = m Λ < 1.So Ω F is a contraction.Hence it follows by Lemma 3.6 that Ω F has a fixed point x which is a solution of (1.1)-(1.2).This completes the proof.

The lower semicontinuous case
Here we study the case when F is not necessarily convex valued in the problem (1.1)-(1.2).We apply the nonlinear alternative of Leray-Schauder type and a selection theorem by Bressan and Colombo for lower semi-continuous maps with decomposable values to establish this result.Let us begin with some preliminary concepts.Let X be a nonempty closed subset of a Banach space E and G : X → P (E) be a multivalued operator with nonempty closed values.G is lower semi-continuous (l.s.c.) if the set {y ∈ X : belongs to the σ−algebra generated by all sets of the form J × D, where J is Lebesgue measurable in [0, 1] and D is Borel measurable in R. A subset A of L 1 ([0, 1], R) is decomposable if for all u, v ∈ A and measurable J ⊂ [0, 1] = J, the function uχ J + vχ J−J ∈ A, where χ J stands for the characteristic function of J .Definition 3.8.Let Y be a separable metric space and let N : Y → P (L 1 ([0, 1], R)) be a multivalued operator.We say N has a property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.
Fractional-order multivalued problems 11 Let F : [0, 1] × R → P (R) be a multivalued map with nonempty compact values.Define a multivalued operator which is called the Nemytskii operator associated with F. Definition 3.9.Let F : [0, 1] × R → P (R) be a multivalued function with nonempty compact values.We say F is of lower semi-continuous type (l.s.c.type) if its associated Nemytskii operator F is lower semi-continuous and has nonempty closed and decomposable values.

Lemma 3.10 ([18]
).Let Y be a separable metric space and let N : Y → P (L 1 ([0, 1], 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 → L 1 ([0, 1], R) such that g(x) ∈ N(x) for every x ∈ Y. Theorem 3.11.Assume that (H 2 ), (H 3 ) and the following condition holds: Consider the problem ) is a solution of (3.2), then x is a solution to the problem (1.1)-(1.2).In order to transform the problem (3.2) into a fixed point problem, we define the operator Ω F as It can easily be shown that Ω F is continuous and completely continuous.The remaining part of the proof is similar to that of Theorem 3.4.So we omit it.This completes the proof.

Examples
Consider the problem

Extension to Riemann-Stieltjes integral conditions case
The concept of Riemann-Stieltjes integral conditions is quite old, see the reviews by Whyburn [33] and Conti [13].It provides a unified approach for dealing with a variety of boundary conditions such as multipoint and integral boundary conditions.For some recent works involving Riemann-Stieltjes integral conditions, we refer the reader to the papers [5,30,32,34] and the references cited therein.
Let us now consider fractional differential inclusion (1.1) supplemented with the boundary data involving Riemann-Stieltjes integral condition given by x(0) + x(1) = a
H d (A, B) = max sup a∈A d(a, B), sup b∈B d(A, b) , where d(A, b) = inf a∈A d(a; b) and d(a, B) = inf b∈B d(a; b).Then (P b,cl (X),