Dynamical analysis of Hilfer–Hadamard type fractional pantograph equations via successive approximation

ABSTRACT In this note, we consider a nonlinear pantograph equation with Hilfer–Hadamard fractional derivative. We investigate the existence and continuous dependence results by using successive approximations and generalized Gronwall inequality.


Introduction
Fractional differential equations (FDEs) are appeared in mathematical modelling of processes and phenomenons of science and engineering. Hence the theory of FDEs is an area intensively developed during last few decades. The monographs of Hilfer [1], Kilbas et al. [2], Miller and Ross [3] and Abbas et al. [4], include a study of techniques of solving which are an extension of procedures from differential equations theory. Recently, considerable attention has been given to the Hilfer fractional derivative which was introduced by R. Hilfer in [1]. The numerous results on existence and uniqueness of solution of FDEs with Hilfer and Hilfer-Hadamard fractional derivative have been studied in [5][6][7][8][9][10] by different methods.
Functional differential equations with proportional delays are usually referred to as pantograph equations. It arises in rather different fields of pure and applied mathematics, such as electrodynamics, control systems, number theory, probability, and quantum mechanics. Therefore, the problems have attracted a great deal of attention, one can refer to [11][12][13][14]. Recently, Dhaigude and Bhairat [8] used the Hilfer fractional derivative for investigating the existence and uniqueness of solution of Cauchy-type problem for Hilfer FDEs. Inspired by the discussion, our main concern is to prove the existence and continuation dependence of solution to a pantogarph equation with Hilfer-Hadamard fractional derivative.
It is easy to see that Equations (1)-(2) is equivalent to the following integral equation Here we remark that the fixed point technique does not indicate the interval of existence of solution, which is a necessary aspect for application purpose. This draw back of fixed point technique is removed by using Picard's iterative technique and existence of a solution is confirmed on [1, T] as done in [15]. The fundamental results, existence results and the dependence of solutions on order are studied in the subsequent sections.

Preliminaries
Let −∞ < 1 < T < +∞. Let C [1, T], AC [1, T], and C n [1, T] be the spaces of continuous, absolutely continuous, n-times continuous and continuous differentiable functions on [1, T] respectively. Here L p (1, T), p ≥ 1 is the space of Lebesgue integrable functions on (1, T). Further more, we recall the following weighted space Clearly, H D α,β and C 1−γ ,log [1, T] is a complete metric space with the metric d defined by Definition 2.1: [16] The Riemann-Liouville integral of fractional order α ∈ R + of a function H is defined by provided that the integral on the right side exits over (0, ∞).

Definition 2.2 ([16]):
The Caputo fractional order derivative of order α of function H is defined by provided that integral on the right is pointwise defined on (0, ∞), where n = [α] + 1 and [α] denotes the greatest integer which is less than or equal to the real number α.

Definition 2.3 ([16]):
The Riemann-Liouville fractional order derivative of order α of function H is defined by provided that integral on the right is pointwise defined on (0, ∞), where n = [α] + 1 and [α] denotes the greatest integer which is less than or equal to the real number α.
x(μt)) satisfies Lipschitz condition with respect to x, if for all t ∈ (1, T] and for x,x ∈ G , one has where A > 0 is Lipschitz constant.

Existence results
In this section, we prove the existence and uniqueness of solution of Hilfer-Hadamard type pantograph equation (1)-(2) in C α,β 1−γ [1, T]. We need the following lemma.

Lemma 3.1:
where, M is the bound of a bounded function H .
Proof: From Lemma 2.9, the result follows. Now we prove the estimate (11). By the weighted space given in (4), we have using Lemma 2.10, we get Thus the proof is complete.
holds and first we prove the existence of unique solution x ∈ C 1−γ ,log [1, t 1 ]. We proceed by taking Picard's sequence as described by the functions We now show that x m (t) ∈ C 1−γ ,log [1, T]. From Equation (13), it follows that [1, T], m ∈ N and t ∈ (1, T]. By Equations (13) and (14), we have By using Lemma 3.1 (15) Further we obtain Continuing in this way, m-times, we obtain By Equation (12), we get Again by Lemma 3.1, it follows that and hence by Equation (18) → 0 as m → +∞.

Continuous dependence
To present dependence of solution on the order, let us consider the solutions of two initial value problem for pantograph equations with the neighbouring orders. We need the following lemma.  By subsequent relation (14) and (31), the Lipschitz condition (9) and the inequality (32), we obtain H (s, x 0 (s), x 0 (μs)) − H (s,x 0 (s),x 0 (μs)) ds s Thus, Similarly, by using (33), it directly follows that 2A j (log t) αj (αj + γ ) .
(34) Taking limit as m → +∞ in (34), we have which completes the proof of Theorem 4.3.