Dynamics and stability of Hilfer-Hadamard type fractional pantograph equations with boundary conditions

This paper is mainly concerned with existence, uniqueness and Ulam stabilites of solutions of Hilfer-Hadamard type fractional pantograph equations with boundary conditions. The existence results are derived by using Schaefer’s fixed point theorem. Further, Ulam stability results are also discussed. An example is presented to illustrate the theory.


Introduction
Fractional differential equations (FDEs) have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, engineering, etc..In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives.For more details on fractional calculus theory, one can see the monographs of Hilfer [23], Kilbas [26] and Podlubny [28].FDEs involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attentions.Recently, A study of Hilfer type of equation has received a significant amount of interests, the papers in [11,12,13,23,24,25,27] and the references therein..There are some works on FDEs with Hadamard fractional derivative, even if it has been studied many years ago (see for example [2,7,8]).Motivation and basic theoretical development for the research related to Ulam-Hyers stability and Ulam-Hyers-Rassias stability problems to various forms of ordinary differential and integral equations of integer orders can be found in [5,14,16,19].We mention here few recent on Ulam stabilities by different researchers [1,14,21,29,30,31,32]; also see the references cited therein.The pantograph type equations arise in the modeling of various problems in sciences and engineering such as economy, biology, control and electrodynamics, reader can refer to [6,15] and references therein.Recently, fractional pantograph differential equations have been studied by many researchers.One of interesting subjects in this area, is the investigation of the existence of solutions by fixed point theorems, we refer to [6].Vivek et.al.[29] studied dynamics and Ulam stability of pantograph equations with Hilfer fractional derivative.Recently, Kassim et.al. [18] investigated well-posedness and stability for differential equations with Hilfer-Hadamard fractional derivative.Many works have been devoted to the study of FDEs with boundary conditions recently treated in the literature in [2,4,9,10,22,20].Motivated by the above approach, the goal in the present paper is to study existence, uniqueness and stability analysis of Hilfer-Hadamard type fractional pantograph equations with boundary conditions of the form where H D α,β 1 + is the Hilfer-Hadamard fractional derivative, 0 < α < 1, 0 ≤ β ≤ 1, 0 < λ < 1 and let X be a Banach space, f : J × X × X → X is given continuous function.We organize the present work as follows.In Section 2, preliminaries and notations concerning the Hilfer-Hadamard fractional derivative are presented.In Section 3, we present our main result by using Schaefer's fixed point theorem.In section 4, we discuss stability analysis.

Basic concepts
In this section we present some basic definitions, notations and preliminaries which are used in the sequel.For more details on Hilfer fractional derivative, readers can refer to [1,11,12,17,23,24,25].

Definition 2.1. Let C[J, X] denotes the Banach space of continuous function on [1, T ] with the norm
We denote L 1 {R + }, the space of Lebesgue integrable functions on J.By C γ,log [J, X] and C 1 γ,log [J, X], we denote the weighted spaces of continuous functions defined by and . Now, we give some results and properties of Hadamards fractional calculus.Definition 2.2.[2,8] The Hadamard fractional integral of order α for a function h is defined as provided the integral exists.
, then the following composition In [23], R. Hilfer studied applications of a generalized fractional operator having the Riemann-Liouville and Caputo derivatives as specific cases (see also [24,17]).
h exists and in L 1 {R + }, then then In order to solve our problem, the following spaces are presented and It is obvious that

so the homogeneous differential equation with Hilfer-Hadamard fractional order H D
α,β Following lemma plays important role to obtain Ulam stabilities results.

Existence and Uniqueness Results
To obtain existence and uniqueness of solution to Hilfer-Hadamard type pantograph equations with boundary conditions (1.1), we use the following Lemma.

.3) if and only if x is a solution of the Hilfer-Hadamard fractional pantograph BVP H D
α,β ) International Scientific Publications and Consulting Services Proof.Assume x satisfies (3.3).Then Lemma 2.1 implies that From (3.5), a simple calculation gives Hence, we get equation (3.3).Conversly, it is clear that if x satisfies equation (3.3), then equations (3.4)-(3.5)hold.
Before stating and proving the existence result for problem (1.1), we introduce the following conditions; for any u, u, v, v ∈ X, and t ∈ J.
(C3) The function f : J × X → X is completely continuous and there exists a function µ(t Next theorem guarantee existence and uniqueness of solution the problem (1.1).The following existence result for the problem (1.1) is based on Schaefer's fixed point theorem.
Theorem 3.1.Suppose that the conditions (C1),(C3) are satisfied.Then the problem (1.1) has at least one solution defined on J.
Proof.Consider the operator P : It is obvious that the operator P is well defined.
Claim 1: P is continuous.Let x n be a sequence such that

International Scientific Publications and Consulting Services
Since f is continuous, then we have Claim 2: P maps bounded sets into bounded sets in C 1−γ,log [J, X].Indeed, it is enough to show that for q > 0, there exists a positive constant l such that Claim 3: P maps bounded sets into equicontinuous set of Step 2, and x ∈ B q .Then, As t 1 → t 2 , the right hand side of the above inequality tends to zero.As a consequence of claim 1 to 3, together with Arzela-Ascoli theorem, we can conclude that P : is continuous and completely continuous.
Claim 4: A priori bounds.Now it remains to show that the set International Scientific Publications and Consulting Services Let x ∈ ω, x = δ (Px) for some 0 < δ < 1.Thus for each t ∈ J.We have, This implies by (C3)that for each t ∈ J, we have This shows that the set ω is bounded.As a consequence of Schaefer's fixed point theorem, we deduce that P has a fixed point which is a solution of problem (1.1).
Definition 4.1.The equation (1.1) is Ulam-Hyers stable if there exists a real number C f > 0 such that for each ε > 0 and for each solution z ∈ C International Scientific Publications and Consulting Services Definition 4.3.The equation (1.1) is Ulam-Hyers-Rassias stable with respect to φ ∈ C 1−γ,log [J, X] if there exists a real number C f > 0 such that for each ε > 0 and for each solution z ∈ C  We ready to prove our stability results for problem (1.1).The arguments are based on the Banach contraction principle.First we list the following condition: (C4) There exists an increasing function φ ∈ C 1−γ,log [J, X] and there exists λ φ > 0 such that for any t ∈ J I α 1 + φ(t) ≤ λ φ φ(t).
then the problem (1.1) has a unique solution.
Proof.Consider the operator P : It is clear that the fixed points of P are solutions of (1.1).Let x, y ∈ C 1−γ,log [J, X] and t ∈ J, then we have Hence, From (4.7), it follows that P has a unique fixed point which is solution of problem (1.1).Proof.Let ε > 0 and let z ∈ C γ 1−γ,log [J, X] be a function which satisfies the inequality: and let x ∈ C γ 1−γ,log [J, X] be the unique solution of the following Hilfer-Hadamard type pantograph BVP where On the other hand, if  Thus, the equation (1.1) is generalized Ulam-Hyers-Rassias stable.
To complete this paper, we give an example to illustrate the usefulness of our main results.Notice that this problem is a particular case of (1.1).Set f (t, u, v) = 1 5 + 1 10 u + 1 10 v, for u, v ∈ X, and t ∈ J. Clearly, the function f satisfies condition of Theorem 3.1.For each u, v, u, v ∈ X and t ∈ J.