The C ˘adariu-Radu Method for Existence, Uniqueness and Gauss Hypergeometric Stability of Ω -Hilfer Fractional Differential Equations

: Using the C˘adariu–Radu method derived from the Diaz–Margolis theorem, we study the existence, uniqueness and Gauss hypergeometric stability of Ω -Hilfer fractional differential equations deﬁned on compact domains. Next, we show the main results for unbounded domains. To illustrate the main result for a fractional system, we present an example.


Introduction
In 1941, Hyers proved that for each ϑ > 0, there exists a σ > 0 such that if h(z + w) − h(z) − h(w) < σ, then there exists an additive mapping h (z) with h(z) − h (z) ≤ ϑ. Next, the Hyers' results has been developed by Th. M. Rassias. In fact, he attempted to weaken the condition as follows: this led to the generalization of what is known as Hyers-Ulam-Rassias stability of functional equations [1,2].
On the other hand, in 1695, the question of the semi-derivative was raised. The first known references can be found in inventing of the concept of the derivative of nth order, belonging to Marquis de l'Hospital and Gottfried Leibniz. This question attracted the interest of many mathematicians like Liouville, Riemann, Euler, Laplace and many others. The theory of fractional calculus (FC), developed rapidly and its application is done very widely nowadays. For more details, see [3][4][5][6][7][8].
be an interval on R, and S 1 > 0.
We assume the following conditions hold: for every τ ∈ (0, p].

Applying Theorem 1 and
we conclude that g(τ) satisfies the initial value problem Equation (1) if and only if g(τ) satisfies the integral Equation (8).
Proof. We are going to show the result for R. By the same way, we can prove the theorem for (−∞, 0] or [0, +∞). For each n ∈ N, we consider P n = [P − n, P + n]. As stated by Theorem 4, we can find a unique mapping g n ∈ C 1−S 3 ;Ω (P n , R), such that H D S 1 ,S 2 ;Ω 0 + g n (τ) = F(τ, g n (τ), g n (µ(τ))), and for every τ ∈ P n . The uniqueness of g n implies that if τ ∈ P n , then For all τ ∈ R, we define n(τ) ∈ N as n(τ) = min{n ∈ N | τ ∈ P n }.