Existence and uniqueness results for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations

T he concept of fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non integer order. Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1–17] and the references therein. Recently, Ahmad and Ntouyas [3] discussed the existence of solutions for the hybrid Hadamard differential equation { H Dα ( x(t) g(t,x(t)) ) = f (t, x (t)) , t ∈ [1, T] , H Iαx(t) ∣∣ t=1 = η, where H Dα is the Hadamard fractional derivative of order 0 < α ≤ 1. By employing the Dhage fixed point theorem, the authors obtained existence results. In [4], Ardjouni and Djoudi studied the existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions { Dα 1 (x (t)) = f ( t, x (t) ,Dα 1 (x (t)) ) , t ∈ [1, T] , x (1) + g (x) = x0,


Introduction
T he concept of fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non integer order. Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein.
Recently, Ahmad and Ntouyas [3] discussed the existence of solutions for the hybrid Hadamard differential equation where H D α is the Hadamard fractional derivative of order 0 < α ≤ 1. By employing the Dhage fixed point theorem, the authors obtained existence results.
In [4], Ardjouni and Djoudi studied the existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions where f : [1, T] × R × R → R and g : C ([1, T], R) → R are nonlinear continuous functions and D α 1 denotes the Caputo-Hadamard fractional derivative of order 0 < α < 1.
Motivated by these works, we study the existence, interval of existence and uniqueness of solution for the following nonlinear hybrid implicit Caputo-Hadamard fractional differential equation , t ∈ [1, T] ,

Preliminaries
In this section, we present some basic definitions, notations and results of fractional calculus [2,7,12,15] which are used throughout this paper.

Lemma 4 ([12]
). For all µ > 0 and v > −1, then The following generalization of Gronwall's lemma for singular kernels plays an important role in obtaining our main results. Assume that there is a constant a > 0 such that for 0 < α < 1 Then, there exists a constant k = k(α) such that

Main results
In this section, we give the equivalence of the initial value problem (1) and prove the existence,interval of existence, uniqueness and estimate of solution of (1).
The proof of the following lemma is close to the proof Lemma 6.2 given in [7].  (1) is equivalent to the nonlinear fractional Volterra integro-differential equation for all u, v, u * , v * ∈ R and t ∈ [1, T]. Let Then (1) has a unique solution x ∈ C ([1, b] , R).