INTEGRAL BOUNDARY VALUE PROBLEMS FOR IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING HADAMARD AND CAPUTO-HADAMARD FRACTIONAL DERIVATIVES

In this paper, we examine the existence and uniqueness of integral boundary value problem for implicit fractional differential equations (IFDE’s) involving Hadamard and Caputo-Hadamard fractional derivative. We prove the existence and uniqueness results by utilizing Banach and Schauder’s fixed point theorem. Finally, examples are introduced of our results.


Introduction
FDE's are considered to be a different model to integer differential equations. It has been proved by applying importance in the modeling of various fields of physical sciences, medicine, electronics and wave transformation [8,16,21,23,26]. The dominant techniques are the method of introducing a parameter for solving an implicit differential equations. In past three years, the most of research paper to developed existence and uniqueness of implicit FDE's involving various derivatives like the Caputo,Riemann-Liouville, Caputo-Hadamard, Hadamard, Hilfer-Hadamard fractional derivatives etc., (see [4-7, 9, 14, 15, 19, 20, 24]).
Caputo Hadamard fractional derivatives were studied in [12] by the authors F. Jarad, T. Abdeljawad and D. Baleanu, where a Caputo-type modification for Hadamard derivatives was introduced and studied. Later, more properties of Hadamard fractional derivatives were investigated in [1,2,10,13].
The applications of Hadamard fractional differential equations in mathematical physics cuold be found in [11,17,18,22,25]. In [3] the authors have studied Hilfer-Hadamard FDE's with variable-order fractional integral and fractional derivative. Motivated by the above cited work, we studies the solutions of existence and uniqueness results to the following implicit fractional differential equations with integral boundary conditions of the form where H D ϑ is the Hadamard fractional derivative of order 1 < ϑ ≤ 2, where CH D ϑ is the Caputo-Hadamard fractional derivative of order 1 < ϑ ≤ 2 and g : J × R × R → R is a continuous function.
In this paper, Section 2, has definitions and some of the most important basic concepts of the fractional calculus. In Section 3, existence and uniqueness of solutions for integral boundary conditions of implicit fractional differential equations involving Hadamard fractional derivative and Caputo-Hadamard fractional derivatives are proved by utilizing Banach and Schauder's fixed point theorems. In Section 4, an illustrative examples are provided to explain of the results of the problem (1.1)-(1.4).

Basic Results
In this section, the some most important basic concepts, definitions and some supporting results are used in this paper. By C(J, R) we denote the Banach space of all continuous functions form J into R with the norm ||x|| ∞ = sup{|x(t)| : t ∈ J}. Definition 2.1 ([15]). The derivative of fractional order ϑ > 0 of a function g : (0, ∞) → R is given by where n = [ϑ] + 1, provided the right side is pointwise defined on (0, ∞).
Definition 2.4 ([12]). For at least n-times differentiable function g, the Caputo-Hadamard fractional derivative of order ϑ is defined as is equivalent to the integral equation given by is equivalent to the integral equation given by Lemma 2.3 (Nonlinear alternative of Lerary-Schauder type, [7]). Let B be a Banach space, C a closed, convex subset of B, U an open subset of C and 0 ∈ U. Suppose that F : U → C is a continuous, 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).

Main Results
To prove the existence and uniqueness results we need the following assumptions.
Assumption 3.1. The function g : J × R × R → R is a continuous function. Assumption 3.2. There exists constants K g > 0 and 0 < L g < 1 such that The integral boundary conditions for implicit fractional differential equations with Hadamard fractional derivative (1.1)-(1.2) is equivalent to the integral equation ds. The integral boundary conditions for implicit fractional differential equations with Caputo-Hadamard fractional derivative (1.3)-(1.4) is equivalent to the integral equation , Theorem 3.1. Assume that assumptions 3.1 and 3.2 hold. If Proof.
Obviously, the right-hand side of the above inequality tends to zero independently of u, v ∈ B r 1 as µ 2 − µ 1 → 0. As H satisfies the above assumptions, therefore, by the Arzela-Ascoli theorem, it follows that H : Then, for t ∈ [b, T] and following the similar computations as in the first step, we have Consequently, we have There exists M * such that ||x|| = M * . Let us set