Existence Results for Langevin Fractional Differential Inclusions Involving Two Fractional Orders with Four-Point Multiterm Fractional Integral Boundary Conditions

and Applied Analysis 3 Proof. As argued in [23], the solution of cDp(cDq + λ)x(t) = h(t) can be written as x (t) = ∫ t

In recent years, the boundary value problems of fractional order differential equations have emerged as an important area of research, since these problems have applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, aerodynamics, viscoelasticity and damping, electrodynamics of complex medium, wave propagation, and blood flow phenomena [1][2][3][4][5].Many researchers have studied the existence theory for nonlinear fractional differential equations with a variety of boundary conditions; for instance, see the papers [6][7][8][9][10][11][12][13][14][15][16][17] and the references therein.
The main objective of this paper is to develop the existence theory for a class of problems of the type (1), when the right-hand side is convex as well as nonconvex valued.We establish three existence results: the first result is obtained by means of the nonlinear alternative of Leray-Schauder type; the second one relies on the nonlinear alternative of Leray-Schauder type for single-valued maps together with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values; and a fixed point theorem due to Covitz and Nadler for contraction multivalued maps is applied to get
Definition 1.For at least -times differentiable function  : [0, ∞) → R, the Caputo derivative of fractional order  is defined as where [] denotes the integer part of the real number .
Definition 2. The Riemann-Liouville fractional integral of order  is defined as provided the integral exists.
In the following,  1 ([0, ], R) will denote the space of functions  : [0, ] → R that are absolutely continuous and whose first derivative is absolutely continuous.
has a unique solution where ) , ) . ( Proof.As argued in [23], the solution of    (    + )() = ℎ() can be written as Using the given conditions in (9) together with (8), we find that where Solving (10) for  0 and  1 , we find that Substituting these values in (9), we find the desired solution.
In order to simplify the computations in the main results, we present a technical lemma, concerning the bounds of the operators J 1 and J 2 defined in the proof of the above lemma.
Proof.By using the following property of beta function we have which completes the proof.
In the following, for convenience, we put For a normed space (, ‖ ⋅ ‖), let A multivalued map  :  → P() is measurable; (vii) has a fixed point if there is  ∈  such that  ∈ ().
The fixed point set of the multivalued operator  will be denoted by Fix.
For each  ∈ ([0, 1], R), define the set of selections of  by We define the graph of  to be the set () = {(, ) ∈ ×,  ∈ ()} and recall two useful results regarding closed graphs and upper semicontinuity.
Lemma 10 (nonlinear alternative for Kakutani maps [31]).Let  be a Banach space,  a closed convex subset of ,  an open subset of , and 0 ∈ .Suppose that  :  → P ,V () is an upper semicontinuous compact map; here P ,V () denotes the family of nonempty, compact convex subsets of .Then either (i)  has a fixed point in , or (ii) there is a  ∈  and  ∈ (0, 1) with  ∈ ().Definition 11.Let  be a subset of [0, 1] × R.  is L ⊗ B measurable if  belongs to the -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.

Definition 12.
A subset A of  1 ([0, 1], R) is decomposable if, for all , V ∈ A, and measurable J ⊂ [0, 1] = , the function  J + V −J ∈ A, where  J stands for the characteristic function of J.

The Carathéodory Case.
In this section, we are concerned with the existence of solutions for the problem (1) when the right-hand side has convex as well as nonconvex values.Initially, we assume that  is a compact and convex valued multivalued map.
Then BVP (1) has at least one solution.
In our next step, we show that  has a closed graph.Let   →  * , ℎ  ∈ (  ), and ℎ  → ℎ * .Then we need to show that ℎ * ∈ ( * ).Associated with ℎ  ∈ (  ), there exists Thus we have to show that there exists Let us consider the continuous linear operator Θ : Observe that which tends to zero as  → ∞.Thus, it follows from Lemma 9 that Θ∘  is a closed graph operator.Further, we have ℎ  () ∈ Θ( ,  ).Since   →  * , it follows that for some V * ∈  , * .Finally, we discuss a priori bounds on solutions.Let  be a solution of (1).Then there exists Using the computations proving that () maps bounded sets into bounded sets and the notations (17), we have Consequently In view of ( 3 ), there exists  such that ‖  ‖ ̸ = .Let us set Note that the operator  :  → P(([0, 1], R)) is upper semicontinuous and completely continuous.From the choice of , there is no  ∈  such that  ∈ () for some  ∈ (0, 1).Consequently, by the nonlinear alternative of Leray-Schauder type [31], we deduce that  has a fixed point  ∈  which is a solution of the problem (1).This completes the proof.Proof.It follows from ( 4 ) and ( 5 ) that  is of l.s.c.type [35].Then from Lemma 13, there exists a continuous function

The
Consider the problem    (    + )  () =  ( ()) , 0 <  < 1, Observe that, if  ∈  1 ([0, 1]) is a solution of (43), then  is a solution to the problem (1).In order to transform the problem (43) into a fixed point problem, we define the operator  as It can easily be shown that  is continuous and completely continuous.The remaining part of the proof is similar to that of Theorem 16.So we omit it.This completes the proof.
Lower Semicontinuous Case.Next, we study the case where  is not necessarily convex valued.Our approach here is based on the nonlinear alternative of Leray-Schauder type combined with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values.