Uniqueness of Solutionfor Nonlinear Implicit Fractional Differential Equation

We study the uniquenessof solutionfor nonlinear implicit fractional differential equation with initial condition involving Caputo fractional derivative. The technique used in our analysis is based on an application of Bihari and Medved inequalities.


INTRODUCTION
Fractional calculus is generalization of ordinary differential equations and integrations to arbitrary non-integerorders.One can describe many phusical phenomena arising in engineering, physics, economics and science more accurately through the fractional derivative formulation, see [4,5,7,9,11,18,19].
Recently, Pachpatte D. B. and Chinchane V. L. [1] have discussed the uniqueness of solution of fractional differential equation with the Riemann-Liouville derivative.
By motivation of the recent works [1,15], we extends the results presented by Pachpatte D. B. and Chinchane V. L. for nonlinear implicit fractional differential equation with Caputo fractional derivative. In this paper we consider the initial valve problem of the type c D α x t = f t, x t , c D α x t , (1.2) where c D α (0 < α < 1) denotes the Caputo fractional derivative operator and f is real conitnuous valued function on J × ℝ × ℝ into ℝ; ℝ denotes the real space.
The paper is organized as follows. In Section 2, some definitions, lemmas and preliminary results are intoduced to be used in the sequel. Section 3 will deals the results and proofs of uniqueness problm of (1.1).
In [15], the author have been studied the existence and uniqueness of theinitial value problem (1.1)-(1.2), first, let us reduce the problem (1.1)-(1.2) into equivalent fractional integral equation, we obtain The results of our work in the present paper differ substantially from the work of [8,15] in terms of assumptions and methods of proof.
Also, in [17] Medved defined a special class of nonlinear function and developeda method to estimate solution for nonlinear integral inequalities with singularkernel. The class of function defined as follows: Definition 2.6. [17] Let q > 0 be a real number and 0 < b ≤ ∞. The function w : ℝ + → ℝ satisfies the following condition e −qt w u q ≤ R t w e −qt u q ,(2.5) for all u ∈ ℝ + , t ∈ [0, b] , where R(t) is a continuous, nonnegative function.
In [17], Medved introduced the following inequalities which are the best widely and known in the study of many qualitative properties of solution of differential equations. for t ∈ 0, T 1 , where , v 0 > 0, Ω −1 is the inverse of Ω, and t ∈ ℝ + is such that Ω 2a t 2 + g 1 t ∈ Dom(Ω −1 ) for all t ∈ 0, T 1 .
For detail proof of above two theorems see [16].

RESULTS
In this section, we establish the uniqueness of solution of the following initialvalue problem. Now an application of Lemma 2.3 to (3.7) which yields where ( ) is primitive for 1 . We shall prove that the right-hand side of(3.8) tends toward zero as → 0. As (3.11) Therefore, using these (3.10), (3.11) and hypothesis (3.9), we have for some z ≥ 1. Then by Lemma 2.2(ii) to (3.14),again we have, x t − y t < 2 q−1 ) 2 t + t , (3.16) for t ∈ [0, T) where K z is defined by (2.12). Since ε was arbitrary in (3.16), impliesthat x t = y(t) as ϵ → 0. This completes the proof of the theorem.