Nonlinear boundary value problems for fractional di ﬀ erential inclusions with Caputo-Hadamard derivatives on the half line

: The authors establish su ﬃ cient conditions for the existence of solutions to a boundary value problem for fractional di ﬀ erential inclusions involving the Caputo-Hadamard type derivative of order r ∈ (1 , 2] on inﬁnite intervals. Both cases of convex and nonconvex valued right hand sides are considered. The technique of proof involves ﬁxed point theorems combined with a diagonalization method.


Introduction
This paper deals with the existence of solutions to boundary value problems (BVP for short) for fractional differential inclusions. In particular, we consider the boundary value inclusion on an infinite interval H C D r y(t) ∈ F(t, y(t)), for a.e t ∈ J = [1, ∞), 1 < r ≤ 2, (1.1) y(1) = y 1 , y bounded on [1, ∞), (1.2) where H C D r is the Caputo-Hadamard fractional derivative, P(R) is the family of all nonempty subsets of R, F : J × R → P(R) is a multivalued map, and y 1 ∈ R.
Fractional order differential equations have proven to be effective models of various phenomena in engineering and the sciences such as viscoelasticity, electrochemistry, control theory, flows through porous media, electromagnetism, and others. Recently, they have been applied to problems in biological modeling and social interactions [14,15]. The monographs of Abbas et al. [1][2][3], Hilfer [21], Kilbas et al. [23], Podlubny [26], Momani et al. [25] contain the mathematical background needed to understand the value of this modeling tool. For results on fractional order derivatives in general and Hadamard fractional derivatives in particular, we refer the reader to [5-7, 10, 17, 18, 20, 28].
The Caputo left-sided fractional derivative of order α is defined by where α > 0 and n = [α]+1. This derivative is very useful in many applied problems because it satisfies its initial data which contains y(0), y (0), etc., as well as the same data for boundary conditions.
The fractional derivative as presented by Hadamard in 1892 [19] differs from the well-known Caputo derivative in two significant ways. First, its kernel involves a logarithmic function with an arbitrary exponent, and secondly, the Hadamard derivative of a constant is not 0.
The Caputo-Hadamard fractional derivative was introduced by Jarad et al. [22] is a modification of the Hadamard fractional derivative that maintains the property that the derivative of a constant is 0. In recent years there have been a number of papers examining problems involving the Caputo-Hadamard derivative, and as examples, we refer the reader to Adjabi et al. [4] and Shammack [27].
Here we present two results guaranteeing the existence of solutions to the problem (1.1)-(1.2); one is for the case where the right hand side is convex valued, and the other is for the nonconvex case. The nonlinear alternative of Leray-Schauder type is used in the proof for the convex case, and the Covitz-Nadler fixed point theorem for multivalued contraction maps is used in the nonconvex case. We should mention that each of approaches are then combined with the diagonalization method to obtain the results. It should be pointed out that this paper actually initiates the application of the diagonalization method to such classes of problems. The theorems in the present paper extend current results in the literature to the multivalued case. and let AC 1 (J, R) be the space of absolutely continuous functions y : J → R with an absolutely continuous first derivative.
For any Banach space (X, · ), we set: We say that a multivalued map G : If G is a multivalued map that is completely continuous with nonempty compact values, then G is upper semi-continuous if and only if G has a closed graph (that is, if x n → x * , y n → y * , and y n ∈ G(x n ), then y * ∈ G(x * )). We say that x ∈ X is a fixed point of G if x ∈ G(x). The set of fixed points of the multivalued operator G will be denote by FixG. A multivalued map G : is a measurable function.  (2) a contraction if and only if it is γ-Lipschitz with γ < 1.
provided that the integral exists.
Here, [r] denotes the integer part of r and log(·) = log e (·).
We next recall the nonlinear alternative of Leray-Schauder.
Theorem 2.9. Let X be a Banach space and C a nonempty closed convex subset of X. Let U be a nonempty open subset of C with 0 ∈ U and T : U → P cp,c (C) be a upper semicontinuous compact map. Then either (1) T has fixed points in U, or (2) There exist u ∈ ∂U and λ ∈ (0, 1) with u ∈ λT (u).

Main results
We begin by defining what we mean by the problem (1.1)-(1.2) having a solution.
Definition 3.1. A function y ∈ AC 2 δ (J, R) is said to be a solution of (1.1)-(1.2), if there exists a function v ∈ L 1 (J, R) with v(t) ∈ F(t, y(t)) for a.e. t ∈ J such that H C D r y(t) = v(t) and the function y satisfies the boundary condition (1.2).
if and only if y is a solution of the nonlinear fractional problem Proof. Applying the Hadamard fractional integral of order r to both sides of (3.2) and then using Lemma 2.8, we obtain Hence, such that T m → ∞ as m → ∞.

The convex case
Our first existence result is for the case where F is convex valued. (H3) There exists C > 0 such that Proof. Fix m ∈ N and consider the related boundary value problem H C D r y(t) ∈ F(t, y(t)), for a.e t ∈ J m , 1 < r ≤ 2, (3.8) First, we shall show that the BVP (3.8)-(3.9) has a solution y m ∈ C(J m , R) with where M > 0 is a constant. To do this, consider the multivalued operator N : C(J m , R) → P(C(J m , R)) defined by Clearly, from Lemma 3.2, the fixed points of N are solutions to (3.8)-(3.9). We shall show that N satisfies the hypotheses of the nonlinear Leray-Schauder alternative. We give the proof in steps.
Step 1: N(y) is convex for each y ∈ C(J m , E). For h 1 , h 2 ∈ N(y), there exist v 1 , v 2 ∈ S F,y such that for t ∈ J m and i = 1, 2. Letting 0 ≤ d ≤ 1, we see that for each t ∈ J m , Now F has convex values, so S F,y is convex; hence, so N(y) is convex.
Step 2: N maps bounded sets into bounded sets in C(J m , R). Let B µ * = {y ∈ C(J m , R) : y ∞ ≤ µ * } be a bounded set in C(J m , R) and y ∈ B µ * . Then for each h ∈ N(y), there exists v ∈ S F,y such that By (H2), we have, for each t ∈ J m , Thus, Step 3: N maps bounded sets into equicontinuous sets in C(J m , R). Take t 1 , t 2 ∈ J m with t 1 < t 2 , and take B µ * to be a bounded set in C(J m , R) as we did in Step 2. Let y ∈ B µ * and h ∈ N(y). Then, As t 1 → t 2 , the right hand side of the inequality above approaches zero. Therefore, in view of Steps 1 to 3 and the Arzelà-Ascoli theorem, it follows that N is completely continuous.
Step 4: N is upper semicontinuous. We will show this by showing that N has a closed graph. Let y n → y * , h n ∈ N(y n ), and h n → h * . We need to prove that h * ∈ N(y * ). Now h n ∈ N(y n ) implies there exists v n ∈ S F,y n such that for t ∈ J m , We need to show that there is a v * ∈ S F,y * such that, for each t ∈ J m , Now F(t, ·) is upper semi-continuous, so for every > 0 there exists N ∈ N such that for every n > N , we have v n (t) ∈ F(t, y n (t)) ⊂ F(t, y * (t)) + B(0, 1), a.e. t ∈ J m .
Since F has compact values by (H1), there is a subsequence v n k of v n such that v n k → v * as k → ∞ and v * ∈ F(t, y * (t)), a.e. t ∈ J m .
We can obtain an analogous relation by interchanging the roles of v n k and v * , so |v n k (t) − v * (t)| ≤ H d (F(t, y n (t)), F(t, y * (t))) ≤ l(t) y n − y * ∞ by (H4). It is easy to see that which is what we wished to show.
Step 5: A priori bounds on solutions. Let y ∈ λN(y) with λ ∈ (0, 1]. Then there is a v ∈ S F,y so that for each t ∈ J m , This implies by (H2) that, for each t ∈ J m , we have Then by condition (3.5), there exists C > 0 such that y ∞ C. Let U = {y ∈ C(J m , R) : y ∞ < C}. The operator N : U → P(C(J m , R)) is upper semi-continuous and completely continuous. From the choice of U, there is no y ∈ ∂U such that y ∈ λN(y) for some λ ∈ (0, 1]. It then follows from the Leray-Shauder nonlinear alternative that N has a fixed point y ∈ U that in turn is a solution of problem (3.8)-(3.9).
Step 6: A diagonalization process. First let N m = N * − {m}. For each k ∈ N, let y k (t) be the solution of (3.8)-(3.9) whose existence is guaranteed by Steps 1-5 above, and set For m = 1, there exists v 1 k ∈ S F,u such that By the Arzelà-Ascoli Theorem, {u k } has a uniformly convergent subsequence, so there is a subset N 1 of N and a function z 1 ∈ C([1, T 1 ], R) such that Now for k ∈ N 1 and m = 2, we have Also for t 1 , t 2 ∈ J 2 with t 1 < t 2 , there exists v 2 k ∈ S F,u such that Again using the Arzelà-Ascoli Theorem, {u k } has a uniformly convergent subsequence, so there is a subset N 2 of N 1 and a function z 2 ∈ C([1, T 1 ], R) such that Proceeding inductively, we see that for t 1 , t 2 ∈ J m with t 1 < t 2 , there is v m k ∈ S F,u , such that as n → ∞ through N m . Hence, there exists v ∈ S F,u , such that that is, there exists v ∈ S F,y such that We can apply this method for each t ∈ [1, T m ] and each m ∈ N. Thus, H C D r y(t) ∈ F(t, y(t)) for a.e. t ∈ J = [1, T m ], 1 < r ≤ 2, (3.10) for each m ∈ N. This completes the proof of the theorem.

The nonconvex case
We now consider the case where right hand side of problem (1.1)-(1.2) is nonconvex valued. In this case the proof relies on the fixed point result contained in Theorem 2.3.
Theorem 3.6. In addition to condition (H4) assume that: (H5) F : J m × R → P cp (R) has the property that F(·, u) : J m → P cp (R) is measurable for each u ∈ R.
Remark 3.7. By (H5), we can see that S F,y is nonempty for each y ∈ C(J m , R), so F has a measurable selection by [11,Theorem III.6].
Proof. We will show that N satisfies the conditions of Theorem 2.3. Once again our proof will be given in steps.
Step 1: N(y) ∈ P cl (C(J m , R)) for each y ∈ C(J m , R). Let (y n ) n≥0 ⊂ N(y) be such that y n →ȳ. Then, y ∈ C(J m , R) and there exists v n ∈ S F,y , n = 1, 2, . . . such that, for each t ∈ J m , y n (t) = 1 Γ(r) From the fact that F has compact values and condition (H4), passing if necessary to a subsequence, we can conclude that v n converges weakly to v in L 1 w (J m , R) (the space endowed with the weak topology).

Conclusions
In this paper we consider a boundary value problem for a fractional differential inclusion involving the Caputo-Hadamard type derivative of order r ∈ (1, 2] on the infinite interval [1, ∞). We give sufficient conditions for the existence of solutions in case the right hand side of the inclusion is convex valued and where it is not. In the convex valued case, the nonlinear alternative of Leray-Schauder type is used in the proof, and in the nonconvex case, the Covitz-Nadler fixed point theorem for multivalued contractions is applied. Due to the fact that our problem is on an infinite interval, a diagonalization method was needed to complete the proofs. This was the first time the diagonalization method has been applied to such problems.