On the fractional perturbed neutral integro-differential systems via deformable derivatives: an existence study

In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-diﬀerential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii’s ﬁxed point theorem


Introduction
On 30 September 1695, L'Hôpital said to Leibniz about a particular notation that he had used in his manuscripts for the n-derivative of the function f (x) = x, i.e., d n f dx n .The question posed to Leibniz by L'Hôpital: What would happen when n = 1 2 ?Leibniz [1] answered as follows: "An apparent paradox, this will have useful effects one day." As predicted by Leibniz, fractional calculus was born with this simple idea.
Fractional calculus was primarily a spark for the best creative minds in mathematics.There have already been several forms of fractional derivative presented.Interestingly, the majority of fractional derivative formulations have an integral form.The readers of this research are advised to read some well-known monographs such as [2][3][4] for a complete knowledge of fractional calculus and also are asked to see the related research publications on fractional differential (or integro-differential) equations (FDEs) in [5][6][7][8][9][10][11][12][13][14][15][16][17].By reading these papers we can easily understand the continual contribution of fractional calculus in the field of dynamic systems and mathematical modeling.These papers, and many others, show the role of fractional derivatives in describing relationships between differential equations.Nowadays, it is clear that the results of fractional models are more accurate and flexible in most cases than the results of classical models, which are based on integer-order derivatives.
As was already mentioned, the integral form is used in the majority of definitions of fractional derivatives.In contrast, Khalil et al. [18] proposed a limit-based definition of a fractional derivative, which characterizes it as a conformable fractional derivative that is comparable to a traditional one.Then Zulfeqarr, Ahuja, and Ujlayan came up with the new concept of the deformable derivative [DD] [19].This new derivative makes use of the same limit method as the conventional derivative.They called it "deformable" as it possesses the inherent quality of constantly deforming functions into derivatives.This idea opens up new research possibilities, such as analyzing qualitative and quantitative behavior in diverse systems.A definition, an integral operator, properties, and an inverse property of deformable derivatives are also provided by Zulfeqarr et al. [19], who then go on to explain how a deformable derivative is used with homogeneous linear FDEs.Recently, Ahuja et al. [20] developed their studies on the deformable Laplace transform and solved some deformable differential equations based on this technique.
A review of the previous works will serve as an introduction to this research.Especially, the authors in [21] discussed the existence of mild solutions of the model where deformable fractional-order derivatives are used.The conclusions are made by the Banach contraction principle and Schauder fixed point theorem in the semigroup theory.Later, Mebrat et al. [22] examined the same properties including the existence of unique solutions for two fractional deformable differential (integro-differential) equations and These existence results were proved by Krasnoselskii's fixed point theorem.Recently, another existence study was conducted by Etefa et al. [23] for the fractional impulsive deformable system By the above ideas the main aim of the present work is to investigate some qualitative properties like the existence of a unique solution for the following fractional perturbed neutral integro-differential system under the deformable derivative: where X is a continuously differentiable function, and w 0 ∈ X, where X is a Banach space (real or complex).The significant findings of the present study are as follows: 1.This is the first effort, to the best of our knowledge, to address the structure of the fractional perturbed neutral integro-differential system that includes the fractional deformable operator in the context of the system introduced in (1.1). 2. By using the fractional derivative of deformable type and its properties we provide the exponential-based solution to the given peturbed system (1.1); see Lemma 2.6.3. The alternative fixed point criteria attributed to Banach and Krasnoselskii are used to obtain the main existence theorems.We provide few instances to highlight how our main results may be used in the final section of our presentation.4. In addition, the results of this work generalized and also upgraded the previous studies published in the literature, including those cited in [22][23][24].This research is organized as follows.The equivalent solution to the mentioned deformable perturbed system is built under the fractional deformable integral equation in Sect. 2. The initial result is based on the Banach contraction principle, and the next one is based on Krasnoselskii's fixed point theorem, which will be proved in Sect.3.Moreover, we consider a nonlocal perturbed integro-differential version of the system and extend our results in this case.Two numerical examples, as some applications of the given deformable perturbed systems, are provided in Sect. 4. The conclusion of this paper is given in Sect. 5 and suggests some future ideas whether we can use these results under artificial intelligence algorithms to simulate dynamics of the models.

Preliminaries
This section provides an overview of the essential concepts and properties of the fractional deformable derivatives, which will be used in establishing our main theorems.
Let X be a Banach space with norm • , and let C([0, ξ ], X) be the Banach space of all continuous functions from [0, ξ ] into X endowed with the supremum norm p C = sup δ∈[0,ξ ] p(δ) .Definition 2.1 [20] The fractional deformable derivative of order ϑ ∈ [0, 1] for a function w : (a, b) → R is defined by with ϑ + ρ = 1, provided that the limit exists.Theorem 2.4 [19] Let a function w be differentiable at a point δ ∈ (a, b).Then w is always ϑ-differentiable in the sense of deformable derivative at that point for every ϑ.Furthermore, in such a case, where Dw = d dδ w is an ordinary derivative.
Theorem 2.5 [19] Assume that w is continuous on For more properties and characteristics of deformable operators, we invite the readers to carefully read [19,20].
Before we go on to define the solution structure of system (1.1), let us consider the following linear system and find its solution structure: Here DD D ϑ is the deformable fractional derivative.

continuous functions, and let h be a continuously differentiable function. A function w is the solution of the integral equation
if and only if w is a solution of the linear system (2.2).
Proof Applying D I ϑ 0 to both sides of (2.2), we have Using the linearity of the fractional deformable derivative DD D ϑ , we have .
By using Theorem 2.5 and w(0) = w 0 we get Therefore On the other hand, assume that w satisfies (2.3).Applying DD D ϑ to both sides of (2.3), by Theorem 2.4 we get Since and The proof is complete.
We now can define the solution to the nonlinear fractional deformable perturbed neutral integro-differential system (1.1) using the information provided in the previous lemma.For our convenience, we denote Ew(δ) = δ 0 h(δ, φ, w(φ)) dφ.Definition 2.7 A function w is said to be a solution to the fractional deformable perturbed neutral integro-differential system (1.1) if provided that the above integral is finite.

Existence theorems
In this section, we present and establish the existence results for the given deformable system (1.1) under the Banach contraction principle and Krasnoselskii's fixed point theorems.
To apply these fixed point theorems, we need the following hypotheses: and there is a constant and continuously differentiable, and there are and and and (iii) (ii) There is (iii) There is Moreover, we consider the Banach space Q := C([0, ζ ], X) of all continuous functions with values in X with sup-norm • Q .In this theorem, we prove that a unique solution exists for the mentioned perturbed problem.Proof To use the well-known fixed point theorem, we define ψ :

Theorem 3.1 Let F, G, H, h satisfy (M0)-(M1).
Thus, for δ ∈ [0, ζ ] and w ∈ B O , we have This shows that the ball B O is mapped into itself under the operator ψ, that is, ψB O ⊂ B O .Following the proof, for each w, w ∈ B O , we estimate Hence, for each δ ∈ [0, ζ ], we obtain , depends on the system parameters.Now, as < 1, ψ is a contraction.Hence the fractional deformable perturbed neutral integrodifferential system (1.1) has a unique solution on [0, ζ ].Now we will prove that there are solutions for the fractional deformable perturbed neutral integro-differential system (1.1) under the hypotheses of Krasnoselskii's fixed point theorem.
Thus, for all δ ∈ [0, ζ ] and w ∈ B O , it becomes Thus ψ 1 (w) + ψ 2 (w 1 ) ∈ B O .Next, we prove that ψ 1 is a contraction.Due to the continuity of H, G, h, for w, w ∈ B O , using (3.3) and (M1)(ii)(iii), we estimate Thus, for all δ ∈ [0, ζ ] and w ∈ B O , we have Hence ψ 1 is a contraction.Since the function F is continuous, it follows that the operator ψ 2 is continuous as well.Additionally, ψ 2 is uniformly bounded on B O .We have and thus ψ 2 w Q ≤ A.
Proof By (2.5) we define two operators as follows: and We now show that ψB O ⊂ B O , where B O is the ball For all w, w 1 ∈ B O , we may write Thus, for all δ ∈ [0, ζ ] and w ∈ B O , we have Thus ψ 1 (w)+ψ 2 (w 1 ) ∈ B O .In the next step, we prove that ψ 1 is a contraction.We know that H, G, h are continuous.Letting w, w ∈ B O , from (3.6) and (M1)(ii)(iii) we get the estimate Thus, for δ ∈ [0, ζ ] and w ∈ B O , we have Hence ψ 1 is a contraction.Since the function F is continuous, it follows that the operator ψ 2 is also continuous.Additionally, ψ 2 is uniformly bounded on B O , and so Thus the uniform boundedness of the operator ψ 2 implies that it is compact.Now we prove that it is equicontinuous.For all τ 1 , τ 2 ∈ [0, ζ ] with τ 1 < τ 2 and w ∈ B O , by Theorem 3.2 we have From (3.8) we see that if τ 1 → τ 2 , then the right-hand side of (3.8) goes to zero, so (ψ 2 w)(τ 2 ) -(ψ 2 w)(τ 1 ) → 0 as τ 1 → τ 2 .Thus ψ 2 is equicontinuous, and since ψ 2 (X) ⊂ X, ψ 2 is compact by the Arzelà-Ascoli theorem.Therefore ψ has at least one fixed point.Finally, there is at least one solution to the associated fractional deformable perturbed neutral integro-differential system (1.1).

Nonlocal integro-differential system
This subsection contains a generalization of the results discussed in the previous subsection.We focus on a generalized existence theorem in relation to the nonlocal fractional neutral integro-differential system given by where DD D ϑ , F, G, H, E, w 0 ∈ X are similar to those described in Sect.3, and During the year 1990, Byszewski was the first person to do groundbreaking research on the nonlocal problem.In [25], Byszewski turns to this purpose that the nonlocal condition w(δ 0 ) + q(δ 1 , . . ., δ k , w(•)) = w 0 can be applied in physics to provide results that are better than those described by the initial condition w(δ 0 ) = w 0 .We recommend the readers to follow [23,25] for further information regarding nonlocal conditions and related applications.
To study the existence of a unique solution of the nonlocal fractional neutral integrodifferential system (3.9),we consider the following assumption: (M4) The function q : Q → X is continuous, and M = sup w∈Q q(w) < ∞.There is a constant K q > 0 such that q(u)q(v) ≤ K q uv Q for all u, v ∈ Q.
Theorem 3.4 Let F, G, H, h, and q satisfy hypotheses (M0)-( M1) and (M4).If  G φ, w(φ), For all δ ∈ [0, ζ ] and w ∈ B O , we have This proves that the ball B O is mapped into itself under the operation ψ, that is, ψB O ⊂ B O .
Further, for all w, w ∈ B O , we have the estimate

Applications
Now we provide some examples to validate our results.
Example 4.1 Consider the nonlinear fractional deformable perturbed neutral integrodifferential system where where Ew(δ) = Then all the conditions of Theorem 3.3 are also satisfied.Therefore at least one solution exists on [0, 1] for the given fractional neutral integro-differential system (4.1).

Conclusions
In this paper, Theorems 3.1, 3.2, and 3.3 were proved to conclude that we can find unique solutions for the nonlinear fractional deformable perturbed neutral integro-differential system (1.1).In this direction, Krasnoselskii's fixed point theorem along with the Banach contraction principle were used.Furthermore, we have proved Theorem 3.4 utilizing the Banach contraction principle by using two independent cases in conditions for proving the existence of unique solutions of the given fractional neutral integro-differential system (3.9).Two examples showed the applicability of the main results.In the near future, deformable fractional derivative may be applied in computer artificial intelligence algorithms by using a suitable fixed point iteration method.