The pantograph equation with nonlocal conditions via Katugampola fractional derivative

. We study a Pantograph-type equation with Katugampola fractional derivatives. Under nonlocal conditions, we establish some existence and uniqueness results for the problem. Then, some other main results are proved by introducing new definitions related to ULAM stability.


Introduction
It's seen now that technology is a very important matter basis for peoples life, governments systems, specially with the COVID-19 global pandemic happening. As the technology grow faster the need of mathematical modeling grow bigger.
Nowadays, the fractional calculus theory has proven it important use as a tool in modeling many real life problems as energy-saving, national economics growth, Image processing, engineering, biology, physics and fluid dynamics and many other researches area see [9,12,20,26]. The fractional calculus theory is based on the study of partial and ordinary differential equations, where the derivation or the integration operator is of noninteger order α or complex with Re(α) > 0. The most three known approaches of operators of fractional calculus theory were given by Grünwald-Letnikov in 1867; 1868, Riemann-Liouville in 1832; 1847 and Caputo 1967 [15]. The treatment of a fractional differential equation mostly involve the study of the exitance and uniqueness of the solution or only the existence of the solutions also the stability of this solutions is implicated, many scholars has given a widely amount of interesting results in such researches see [2,4,6,8,11,16,22,28].

The pantograph equation with nonlocal conditions via Katugampola fractional derivative
In 1971 Ockendon and Taylor [21] did the research on the way in which the electric current is collected by the pantograph of an electric locomotive using a delay equation which is now called the Pantograph equation. Since that time many researchers studied and used it in different mathematical and scientific areas as number theory, probability, electrodynamics, medicine, see [21,25,27] and the bibliography therein.
A lot of researches have been done on the fractional pantograph equations due to their importance to many areas of research, such as [24] in which K. Balachandran and S. Kiruthika treated the existence of solutions for the following nonlinear fractional pantograph equation: Also in [23] Y. Jalilian and M. Ghasemi considered the following fractional integro-differential equation of Pantograph type connected with appropriate initial condition where c D α is the derivative in the sense of Caputo of order α ∈ (0, 1]. In this paper, we shall study the following nonlinear fractional pantograph problem where c D α,ρ is the Katugampola-type fractional derivative in Caputo sense of order α, 0 < p < 1, ρ > 0, and I β is the integral of order β > 0, and f, g : [0, T ] × R 2 −→ R are two given functions.
To the best of our knowledge, this is the first time where such problem is studied.

Preliminaries
We recall some definitions and lemmas that will be used later. For more details we refer to [17 − −19].
In particular when a = 0 we denote simply Yasmine BAHOUS, Zakaria BEKKOUCHE, Nabil BEDJAOUI and Zoubir DAHMANI Definition 2.2. For a function f ∈ C n ([a, b], R) and n − 1 < α ≤ n, the Caputo fractional derivative of f is defined by: 3. Let f : [a, b] −→ R be an integrable function, α ∈ (0, 1] and γ > 0. The Katugampola integral of order α of f is given by When a = 0 we denote simply Proof. Let f : [a, b] → R be a continues function then for all α > 0, β > 0 we have By changing the variables = x + (t − x)ϱ and using Beta function we get (2.5)

The pantograph equation with nonlocal conditions via Katugampola fractional derivative
In particular when a = 0 we denote simply Definition 2.6. The Caputo-Katugampola fractional derivatives of order α is defined by In particular when a = 0 we denote simply To study (1.1) we need the following lemma Proof. If we set for a fixed t, Thus, we can write: and, by an integration by parts, we have and since u t (t) = 0, we get that corresponds exactly to (2.7). ■ Remark 2.8. Note that we can rewrite (2.7) in the form Proof. Indeed, using the formula (2.8), we can write (2.10)

■
Let us introduce now the following Lemma: admits as a solution the function: provided that T β < Γ(β + 1).

Main Results
We consider the following hypotheses: (P 2) : There are nonnegative constants L f and L g , such that for all t ∈ J , (P 3) : There exist positive constants λ, δ, that satisfy for all t ∈ [0, T ], and for all x, x * ∈ R |f (t, x, x * )| ≤ λ, and |g(t, x, x * )| ≤ δ.
Also, we consider the quantities:

Existence of a unique solution
The first main result deals with the existence of a unique solution for (1.1). We have: Theorem 3.1. Assume that (P 2) is satisfied. Then, the problem (1.1) has a unique solution, provided that A 1 < 1 and Γ(β + 1) > T β .
Proof. Let us introduce the Banach space Then, we define the nonlinear operator H : E → E as follows: Yasmine BAHOUS, Zakaria BEKKOUCHE, Nabil BEDJAOUI and Zoubir DAHMANI × f (s, y(s), y(ps)) + g(s, y(s), y((1 − p)s)) ds du. We shall prove that H is a contraction mapping in E.
For y, x ∈ E and for each t ∈ [0, T ], we have  Then, Hence, a straightforward computation gives Consequently, The pantograph equation with nonlocal conditions via Katugampola fractional derivative

Existence of at least one solution
The second main result deals with the existence of at least one solution.
Proof. We put and consider the ball B r := {x ∈ E, ∥x∥ E ≤ r}. Then, we define the operators M and N on B r as: , y(ps)) + g(s, y(s), y((1 − p)s)) ds (3.5) and  For y, x ∈ B r , we find that Then, we can write ∥M x + N y∥ E ≤ r. Thus, M x + N y ∈ B r . Furthermore, for x, y ∈ B r , we obtain That is to say that N is contractive on B r . Now we prove that M is a compact operator on B r . We have ∥f (s, y n (s), y n (ps)) − f (s, y(s), y(ps))∥ + T ρα ρ α Γ(α + 1) × ∥g(s, y n (s), y n (1 − p)(s)) − g(s, y(s), y(1 − p)(s))∥.
Thanks to (P 1), and since s → y(s) is bounded on [0, T ], and ∥y n − y∥ E → 0, we reduce the continuity of f and g to a compact set of [0, T ] × R 2 , so that we obtain ∥M y n − M y∥ E → 0.
Yasmine BAHOUS, Zakaria BEKKOUCHE, Nabil BEDJAOUI and Zoubir DAHMANI Also, for y ∈ B r , we get Consequently, M is uniformly bounded on B r . Now, we prove that M is equicontinuous. Let t 1 , t 2 ∈ [0, T ], t 1 < t 2 . Then for y ∈ B r , we have × f (s, y(s), y(ps)) + g(s, y(s), y((1 − p)s)) ds . (3.10) Hence, Then, we get The right hand side of (3.12) tends to zero independently of y as t 1 → t 2 . This implies that M is relatively compact, and by the Arzela-Ascoli theorem, we conclude that M is compact on B r . Hence, the existence of the solution of the (1.1) holds by Krasnoselskii fixed point theorem.