Existence and stability analysis of solutions for fractional differential equations with delay

. In this manuscript, we establish the existence and stability of solutions for fractional differential equations with delay. We utilize the Bielecki Norm and the Ulam-Hyers stability for our results


Introduction
The concept of the Deformable derivative was introduced by F. Zulfeqarr, A. Ujlayan, and P. Ahuja in 2017 [23].It continuously deforms a function to a derivative, hence the name deformable derivative.This derivative is linearly related to the usual derivative.There are a few manuscripts pertaining to this fractional derivative.For more information, the reader could consult manuscripts such as [9,10,[16][17][18]23].In [9], we established the existence and uniqueness of solutions to impulsive Cauchy problems involving the deformable derivative with local and nonlocal conditions.In [10], we studied the existence of solutions for functional differential equations with infinite delay in the sense of the deformable derivative: In this paper, we study the existence, uniqueness and the Ulam-Hyers type stability of solutions for the following fractional order differential equation: where 0 < α < 1, D α is the deformable derivative, The main motivation for this paper was the work of Develi and Duman (see [8]).

Preliminaries
In this section, X := C([−h, T ], R) stands for the Banach space of all continuous functions with the Bielecki norm: . The Deformable derivative of f of order α at t ∈ (a, b) is defined as: where α + β = 1.If the limit exists, we say that f is α-differentiable at t.
Remark 2.2.If α = 1, then β = 0, we recover the usual derivative.This shows that the deformable derivative is more general than the usual derivative.
1. Let f be differentiable at a point t for some α.Then it is continuous there.
9. Linearity : ) ) is a solution of the inequality (2.1) if and only if there exists a function Remark 2.10.It can readily be seen that using Definition 2.3 and Theorem 2.7, a solution θ ∈ C([0, T ], R) of inequality ( 2) is also a solution to the following integral inequality: We derive the following inequality for our subsequent results: Definition 2.11.[21,22]Let (X, d) be a metric space.An operator A : X → X is said to be a Picard operator if there exists x * ∈ X such that (i) (ii) The sequence (A n (x 0 )) n∈N converges to x * for all x 0 ∈ X.

Existence and Uniqueness
In this section, we prove the existence and uniqueness of solutions for problem (1.1).
We investigate problem (1.1) with the following assumptions: Then we find a unique fixed point of F, which is the unique solution.We consider the Banach space X := C([−h, T ], R) endowed with following norm Using Remark 2.10, we show that F is a contraction mapping on (X, ∥ • ∥ B ).For all u(t), θ(t) ∈ X, The proof for the existence and uniqueness of solutions for the above fractional differential equation is obtained using the following three steps.To that end, we introduce the following Lipschitz condition.
Theorem 3.4.Let f : [0, T ] × R 2 → R be a continuous function.Assume that there exists a positive constant L such that And in addition, assume that κ > L α .Then (3.3) has a unique solution.
Proof.Problem (3.3) is equivalent to: We partition the interval [0,T] into n sub-intervals of equal length S.And have the following for 0 < S < τ and nS = T : We see that t ≤ S i+1 =⇒ t − τ ≤ S i using this argument: be a Banach space of continuous functions u : [−τ, S 1 ] → R with the following norm : and u(t) = µ(t) for −τ ≤ t ≤ 0. Define a mapping F 1 : E 1 → E 1 by: For u(t), θ(t Therefore, Since η = L κα < 1, we get that F 1 is a contraction mapping, and so there exists a unique fixed point Step 2: In this step, we extend the interval of step 1 into Based on the the definition of E 2 , we may derive the following inequality: Proof.Let θ be a solution to (2.1) and u be a unique solution to the following problem: Then Observe that we have the following inequality from Remark 2.10: We define an operator for v ∈ C([−h, T ], R + ): given by We show that A is a Picard operator via the contraction mapping principle.For v, ṽ ∈ C([−h, T ], R + ), one estimates α > 0 , we observe that η < 1, and consequently we get that A is a contraction mapping with respect to the Bielecki norm ∥ • ∥ B on C([−h, T ], R + ).Thus, A is a Picard operator such that F A = {v * } and the Banach Contraction principle gives the equality:

3 .
Let f be continuous on [a,b].Then I α a f is αdifferentiable in (a,b), and we have D α (I α a f (t)) = f (t), and