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On Nonlinear Fractional Neutral Diﬀerential Equation with the ψ − Caputo Fractional Derivative

: In this article, the solvability of fractional neutral diﬀerential equation involving ψ − Caputo fractional operator is considered using a Krasnoselskii’s ﬁxed point approach. Also, we establish the uniqueness of the solution under certain conditions. Ulam stabilities for the proposed problem are discussed. Finally, examples are displayed to aid the applica-bility of the theory results.


Introduction
Fractional calculus is strong tool of mathematical analysis that studies derivatives and integrals of fractional order. Fractional differential equations (FDE's, for short) are used in many fields of engineering and sciences such as dynamical of biological systems [12], economy [33], theory of control [7], automatic systems [36], signal processing [11], hydro-mechanics and non-linear elasticity [14,32].
Various real life problems can be modeled as differential equation. The study of existence of solution of these differential equation is interest object of mathematical analysis. The fixed point theorems are powerful technique to obtain the existence of solution of these problem. There are many of fixed point theorems can be applied to obtain the solution of mathematical models [24,25]. Krasnoselskii's and Banach fixed point theorems play an important role to obtain the existence of solution of a lot of mathematical problems [35].
In 1940, Ulam purposed new role of the stability analysis of the solutions for functional equations [34]. In the next year, Hyer [15] considered another type of stability in the Banach space which was more generalized than the kind of Ulam stability and applied this stability approach to obtain the stability certain conditions of some functional equations. After that, Rassias [27] considered another approach of stability, this approach is more improved than Hyers stabitity. Rassias used this approach to study stability of FDE's [16,28].
There are many definitions are used to define the fractional derivative such as Riemann-Liouville, Caputo, Erdélyi-Kober and Hadamard [23]. More recently, Almeida [4] considers new investigation of the fractional operator and called it ψ−Caputo derivative. This new approach is more generalized than Riemann-Liouville, Caputo, Erdélyi-Kober and Hadamard derivative operator approaches. After one year, Almeida et al. [5] investigated the uniqueness of solution of initial value problem (I.V.P, for short) of FDE in ψ−Caputo sense.

Preliminaries
In this section, we consider some facts and basic results. We recall the following definition [3]. Next, we recall the definitions of ψ−fractional integral and derivative operators [4,5].
Definition 2.2. Let I = [0, 1] and ψ ∈ C n (I, R), be an increasing functions such that ψ (t) = 0 for all t ∈ I. Consider an integrable function u : I → R. The ψ−Riemann-Liouville fractional integral of order α > 0, α ∈ R of the function u is defined as and the ψ−Riemann-Liouville fractional derivative of order α > 0, α ∈ R of the function u is defined as where n = [α] + 1 and [α] denotes the integral part of α.
Definition 2.3. Let ψ ∈ C n (I, R), be an increasing function such that ψ (t) = 0 for all t ∈ I. Consider an integrable function u : where n = [α] + 1, [α] denotes the integral part of α and u We recall the following Lemma which was given in [5].
Now we recall Krasnoselskii's fixed point theorem which was given in [31].
Now, we recall the definitions of these types of Ulam stability. For more details, see [29].
Definition 2.6. The Eq.(1) is said to be Ulam-Hyers stable (UHS for short) if, there exists λ ∈ R + such that for every ε > 0 and each u ∈ C([ρ, 1], R) solution of the inequality Definition 2.7. The Eq.(1) is said to be generalized Ulam-Hyers stable (GUHS for short) if, there exists ϕ ∈ C([ρ, 1], R), ϕ(0) = 0, such that for every ε > 0 and each there exists a unique solution Let H : I × R → R and F : R × R → R. Then we study the Ulam stabilities of the following proposed problem are continues and ψ ∈ C 1 (I, R) be an increasing function such that ψ (t) = 0 for all t ∈ I. Then, we have the following lemma [9].
Lemma 2.10. The solution of Eq.(1) is equivalent to the following nonlinear integral equation On Fractional Differential Equation with the ψ−Caputo Fractional Derivative 103
Let x, y ∈ C([ρ, 1], R) are arbitrary and t ∈ I Thus, T 1 is a contraction.
Step 2. T 2 is completely continuous. First, we will prove that T 2 is continuous. Let {x n } be a sequence in C([ρ, 1], R) such that x n → x ∈ C([ρ, 1], R). Then, we get T. Nabil So, we have that Hence T 2 is continuous operator. Therefore, for each x, y ∈ C([ρ, 1], R) and t ∈ I, we have Therefore, for all t ∈ I. Hence we have Thus T 2 is bounded. Furthermore, if we choose t 1 , t 2 ∈ I such that t 1 < t 2 , then we get Since ψ is continuous, then we have that |( Thus T 2 (Ω) is relatively compact. From Arzela-Ascoli-theorem, we obtain T 2 is compact. Hence T 2 is completely continuous.

On Fractional Differential Equation with the ψ−Caputo Fractional Derivative 105
Step 3. Finding the fixed poind of T . Let x, y ∈ Ω. We get Thus, the operators T 1 and T 2 satisfy all conditions of Theorem 2.5. Hence there exists x * ∈ Ω such that x * is solution of Eq.(1). Proof. We apply Banach contraction theorem to prove T has a unique fixed point. Let x, y ∈ C([ρ, 1], R). Then, we have Thus Eq.(1) has unique solution.

Ulam Stabilities
In this part, various Ulam stability types will be considered.

Applications
The following examples are applications to the previous theoretical results.