Existence and stability results of relaxation fractional di erential equations with Hilfer Katugampola fractional derivative

In this work, we present the existence, uniqueness, and stability result of solution to the nonlinear fractional di erential equations involving Hilfer Katugampola derivative subject to nonlocal fractional integral boundary conditions. The reasoning is mainly based upon properties of Mittag-Le er functions, and xed-point methods such as Banach contraction principle and Krasnoselskii's xed point theorem. Moreover, the generalized Gornwall inequality lemma is used to analyze di erent types of stability. Finally, one example is given to illustrate our theoretical results.


Introduction
Fractional dierential equations is very important since their nonlocal property is appropriate to describe memory phenomena such as nonlocal elasticity, propagation in complex medium, biological tissues, polymers, earth sediments, etc, and they have been emerging as an important area of investigation in recent decades. For details, we refer the reader to monographs of Hilfer [10], Kilbas [15], Samko [19], Podlubny [17], and references therein.
There are various denitions of fractional derivatives, among these denitions, Riemann-Liouville (1832), Riemann (1849), GrunwaldLetnikov (1867), Caputo (1997), Hilfer (2000), as well as Hadamard (1891). At the same context, Kilbas et al. in [15] introduced the properties of fractional integrals and fractional derivatives with respect to another function. O. P. Agrawal et al. in [1,8], presented the generalized variational calculus in terms of multi-parameters fractional derivatives. Some of generalized fractional integral and dierential operators and their properties were introduced by Agrawal in [2]. A Caputo fractional derivative of a function with respect to another function was proposed by R. Almeida in [7]. Recently, Katugampola in [12] introduced a new fractional dierential operator. Moreover, this operator has been compounded with Hilfer fractional dierential operator introduced by Hilfer [10] which called Hilfer-Katugampola fractional dierential operator [16].
Over the last years, the stability results of fractional dierential equations have been strongly developed. Very signicant contributions about this topic were introduced by Ulam [20], Hyers [11] and this type of stability called Ulam-Hyers stability. The concept of Ulam-Hyers Stability was extended via inserting new function variables provided by Rassias [18] in 1978. Ulam-stability, Ulam-Hyers stability, and Ulam-Hyers-Rassias stability, these labels have become famous today in literature. There are many researchers studied generalized Hilfer fractional dierential equations [3,4,5,14].
The paper is organized as follows. In Section 2, we present notations and denitions which are used throughout this paper. In Sect 3, we discuss the existence and uniqueness results for dierential equations with HilferKatugampola fractional derivative involving nonlocal initial condition. In Section 4, we discuss dierent kinds of fractional Ulam stability.

Preliminaries
In this section, we present some denitions and lemmas that we will use throughout this paper. Let 0 < a < b, J = (a, b] and C [J, R] be the Banach space all continuous functions from J into R with supremum norm y ∞ = sup {|y(ς)| : ς ∈ J}. For 0 < γ < 1, we dened the weighted spaces of continuous functions: with the norms In particular, when c = 1 p , the space X p Denition 2.1. [12,13] Let α ∈ R + , c ∈ R and y(ς) ∈ X p c (a, b). The generalized left-sided fractional integral ρ I α a + of order α > 0 is dened by Denition 2.2. [12,13] The Katugampola fractional derivative of order α ∈ R + \ N and ρ > 0 is dened by where n = [α] + 1.

Exestence of solution
Before starting to prove our results, we make the following hypotheses which are needed to prove the existence and unique solutions for our problem. Proof. Dene the operator T f : Note that for any continuous function f, T f is also continuous. Indeed, for all ς, ς 0 ∈ (a, b], we have Next, we show that the operator T f : which implies that Due to (9), the operator T is a contraction mapping on C 1−γ,ρ [a, b]. According to Banach contraction principle, we deduce that the problem (1) has a unique solution xed point y ∈ C 1−γ,ρ [a, b].
Theorem 3.2. Assume that (H 1 ) and (H 2 ) are satised. Then the problem (1) has at least one solution in Proof. Consider the set χ r in C 1−γ,ρ [a, b] dened by with r ≥ σ 1−Ω , Ω < 1 and where f := sup s∈[a,b] |f (s, 0)| . Now we subdivide the operator T f into two operators A and B on χ r as follows and The proof was divided into several steps as following.
i) For operator A. According to Lemma 2.4 and for ς ∈ (a, b], we have
Step (2): We prove that B is a contraction mapping. By Theorem 3.1, we have T f is a contraction mapping on C 1−γ,ρ [a, b] and hence B is a contraction mapping too.
Step (3): We prove that the operator A is compact and continuous. According to Step 1, we know that .
So the operator A is uniformly bounded. Now we prove the equicontinuous of operator A. For any ς 1 ,ς 2 ∈ (a, b], ς 1 < ς 2 , y ∈ χ r and using Lemma 2.4 we get Thus A is equicontinuous. By the Arzelà-Ascoli theorem, we deduce that the operator A is compact on χ r . It follows from Krasnoselskii xed point theorem that the problem (1) has at least one solution in
This proves that the problem (1) is generalized Ulam-Hyers Rassias stable.
We note that L f = e 2 . Furthermore, by simple calculation, we get Ω 0.67 < 1. Then all the assumptions in Theorem 3.1 are satised, the problem (20) has a unique solution in C 1 6 ,1 [0, 1] .

Conclusion
Here the existence, uniqueness and stability of nonlocal boundary value problem for dierential equation with Hilfer-Katugampola fractional derivative is discussed. Krasnoselskii xed point theorem, Banach contraction principle, and Ulam type stability are utilized to obtain results. In conclusion, Hilfer-Katugampola fractional derivative can be used as a powerful tool for studying the dynamical behavior of many real-world problems.