Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.


Introduction
The fractional-order differential equation is the oldest theory in the field of science and engineering. This theory has been used over the years, as the outcomes were found to be important in the field of economics, control theory and material sciences see [1][2][3][4]. Because of the nonlocal property g(0) = g 0 . (1) The pantograph is a device used in electric trains to collects electric current from the overload lines. This equation was modeled by Ockendon and Tayler [50]. Pantograph equation play a vital role in physics, pure and applied mathematics, such as control systems, electrodynamics, probability, number theory, and quantum mechanics. Motivated by their importance, a lot of researchers generalized these equation in to various forms and introduced the solvability aspect of such problems both theoretically and numerically, (for more details see [16,[51][52][53][54][55][56][57] and references therein). However, very few works have been proposed with respect to pantograph fractional differential equations.
As far as we know, to the best of our understanding, results of Ulam-Hyers and generalized Ulam-Hyers stability with respect to the pantograph differential equation are very few and in fact most authors discuss existence and uniqueness, while we study existence, uniqueness and stability analysis for a class of implicit pantograph fractional differential equations with φ-Hilfer derivatives and nonlocal Riemann-Liouville fractional integral condition.
This paper contributes to the growth of qualitative analysis of fractional differential equation in particular pantograph fractional differential equation when φ-Hilfer fractional derivatives involved and the nonlocal initial condition proposed in this paper generalized the following initial conditions:

•
If ρ → 0, the initial condition reduces to multi-point nonlocal condition.

•
If ρ → 1, the initial condition coincide with the nonlocal integral condition.

•
In physical problems, the nonlocal condition yields an excellent results compared with the initial condition z(0) = z 0 [59,60].
Therefore, the paper is organized as follows: In Section 2, it recalls some basic and fundamental definitions and lemmas. In Section 3, we prove existence and uniqueness of the proposed problem (5). Ulam-Hyers and generalized Ulam-Hyers stability for the proposed problem were discussed in Section 4. While in Section 5, two examples were given to illustrate the applicability of our results. Lastly, the conclusion part of the paper is given in Section 6.

Preliminaries
This section will recall some useful prerequisites facts, definitions and some fundamental lemmas with respect to fractional differential equations.
Throughout the paper, we denote C[J , R] the Banach space of all continuous functions from J into R with the norm defined by [1] with the norm Moreover, for each n ∈ N and 0 ≤ q < 1 with q = r + p − rp with the norm Furthermore, we present the following space C ). Let (0, b] be a finite or infinite interval on the half-axis R + , and φ(ξ) ≥ 0 be monotone function on (a, b] whose φ (ξ) is continuous on (0, b). The φ-Hilfer Riemann-Liouville fractional integral of order r ∈ R + of function w is defined by where Γ(·) represent the Gamma function.

Definition 2 ([5]
). Let n − 1 < r < n, 0 ≤ p ≤ 1. The left-sided Hilfer fractional derivative of order r and parameter p of function w is defined by where D n = d dξ n .

Definition 3 ([19]
). Let f , φ ∈ C n [J , R] be two functions such that φ(ξ) ≥ 0 and φ (t) = 0 for all ξ ∈ [J , R] and n − 1 < r < n with n ∈ N. The left-side φ-Hilfer fractional derivative of a function w of order r and type (0 The following lemma shows the semigroup properties of φ-Hilfer fractional integral and derivative.
a.e ξ ∈ J . In particular, if w ∈ C q,φ [J , R] and w ∈ C[J , R], then The composition of the φ-Hilfer fractional integral and derivative operator is given by the following lemmas.

Lemma 4 ([6]
). Let r > 0, 0 ≤ q < 1 and w ∈ C q;φ [J , R]. If r > q, then I r;φ Next, we take into account some important properties of φ-fractional derivative and integral operator as follows: ). Let ξ > 0, r ≥ 0 and s > 0. Then, φ-fractional integral and derivative of a power function are given by Then we have the followings: if and only if z satisfies the following integral equation,

Main Results
In this section, we first adopt some techniques from Lemma 7 in order to establish an important mixed-type integral equation of problem (5). Thus, we need the following auxiliary lemma. (5) if and only if z satisfies the mixed-type integral equation: where δ = 1 For simplicity, we take is a solution to the problem (5), then, we show that z is also a solution of (5). Indeed, from Lemma 7, we have Now, if we substitute t = ξ i and multiply both sides by b i in Equation (12), we obtain Next, by applying I ρ;φ 0 to both sides of Equation (13) and using Lemma 1 and Proposition 1, we get This implies that Inserting the initial condition: which implies that Thus, Hence, the result follows by putting Equation (18) in Equation (12). This implies that z(t) satisfies Equation (9).
Conversely, suppose that z ∈ C q 1−q;φ satisfies the mixed-type integral Equation (9), then, we show that z satisfies Equation (5). Applying D q;φ 0 + to both sides of Equation (9) and using Lemma 2 and Proposition 1, we get Since D r,p;φ 0 + z ∈ C 1−q;φ [J , R], then by definition of C q 1−q;φ [J , R] and make use of Equation (19), For every f ∈ C 1−q;φ [J , R] and Lemma 3, we can see that I Finally, we show that if z ∈ C q 1−q [J , R] satisfies Equation (9), it also satisfies the initial condition. Thus, by applying I 1−q;φ 0 + to both sides of Equation (9) and using Lemma 1 and Proposition 1, we obtain Using Lemma 4 and the fact that 1 − q < 1 − p(1 − r), then taking limit as t → 0 in Equation (21) yields Now, substituting t = ξ i and multiplying through by b i in Equation (9), we get Applying I ρ;φ 0 + to both sides of Equation (23), we obtain which implies and Therefore, in view of Equation (22) and Equation (26), we have

Existence Result Via Schaefer'S Fixed Point Theorem
This subsection will provide the proof of the existence results of Equation (5) using Schaefer's fixed point theorem.

Theorem 2 ([61]
). Let A : X → X be a completely continuous operator. Suppose that the set E (A) = {p ∈ X : p = Ap, f or some ∈ [0, 1]} is bounded, then A has a fixed point.
Thus we need the following assumptions: Theorem 3. Let 0 < r < 1, 0 ≤ p ≤ 1 and q = r + p − rp. Suppose that the assumptions (A 1 ) and (A 2 ) are satisfied. Then there exist at least one solution of the problem (5) in the space C r,p then, clearly the operator F is well-defined. The proof is given in the following steps: Step 1: the operator F is continuous. Let z n be a sequence such that z n → z in C 1−q,φ [J , R]. Then for each t ∈ J , we have Since f is continuous, this implies that T z is also continuous. Therefore, we have T z n − T z C 1−q;φ → 0, as n → ∞.
Step 2: F maps bounded sets into bounded sets in C 1−q;φ [J , R]. Indeed, it suffices to show that for any κ > 0, there exist a µ > 0 such that for any For simplicity, we put and It follows from assumption (A 2 ) that Thus, in view of Equations (30)-(32), we get This implies that, Step 3: F maps bounded sets into equicontinuous set of C 1−q;φ [J , R]. Let t 1 , t 2 ∈ J such that t 1 ≥ t 2 and B κ be a bounded set of C 1−q;φ [J , R] as defined in Step 2. Let z ∈ B κ , then Thus, steps 1-3, together with the Arzela-Ascoli theorem, show that the operator F is completely continuous.
Step 4: a priori bounds. It is enough to show that the set It follows from assumption (A 2 ), that for every t ∈ J , This shows that the set χ is bounded. Hence, by the Schaefer's fixed point theorem, problem (5) has at least one solution.

Existence Result Via Banach Contraction Principle
Now, we prove the uniqueness of problem (5) by means of Banach contraction principle. Therefore, the following hypotheses are needed.
(A 3 ) There exist constants K, L > 0 such that where Theorem 4. Let 0 < r < 1, 0 ≤ p ≤ 1 and q = r + p − rp. Suppose that the hypotheses (A 1 ), (A 3 ) and (A 4 ) are satisfied. Then, problem (5) has a unique solution in the space C r,p Proof. Define the operator F : then, clearly the operator F is well-defined. Let z 1 , z 2 ∈ C r,p 1−q;φ [J , R] and t ∈ J , then, we have and Thus, by substituting Equation (39) in Equation (38), we obtain Also, It follows from hypotheses (A 4 ) that F is a contraction map. Therefore, by Banach contraction principle, we can conclude that problem (5) has a unique solution.

Ulam-Hyers Stabilty
Two types of Ulam stability for (5) are discussed in this section, namely Ulam-Hyers and generalized Ulam-Hyers stability.

Remark 1.
A function x ∈ C 1−q;φ [J , R] is a solution of the inequality (42), if and only if there exist a function g ∈ C 1−q;φ [J , R] such that: Lemma 9. Let 0 < r < 1, 0 ≤ p ≤ 1, if a function x ∈ C 1−q;φ [J , R] is a solution of the inequality (42), then x is a solution of the following integral inequality Proof. Clearly it follow from Remark 1 that , and Hence

Conclusions
In our study, Firstly, we established the equivalence between problem (5) and the Volterra integral equation. Secondly, Banach and Schaefer's fixed point theorems were used to establish the existence and uniqueness solutions for implicit fractional pantograph differential equation which involves φ-Hilfer fractional derivatives. Based on φ-Hilfer fractional derivatives, we found that the stability of Ulam-Hyers and generalized Ulam-Hyers allowed on the implicit fractional pantograph differential equation, supplemented with a nonlocal Riemann-Liouville condition. In addition, examples were given to illustrate our main results. Moreover, it worthy to mention the following remarks: • If ρ → 0 and φ(t) = t, we obtain the results of [48] and [52]. Furthermore, if ρ → 0 we obtain the Ulam-Hyers and generalized Ulam-Hyers stability for the implicit fractional pantograph differential equations with φ-Hilfer fractional derivatives [52,58] and if q = 0 we obtain [51].

•
If ρ → 1, the nonlocal Riemann-Liouville integral condition reduces to a nonlocal integral condition which plays an important role in computational fluid dynamics, ill-posed problems and mathematical models [62].

•
If ρ → 0, the initial condition reduces to multi-point nonlocal condition.

•
If t ∈ [a, b] as defined in paper [58], the function f (t, x(t), x(λt)) is not well-defined for some choice of 0 < λ < 1. Thus, our results modify and improve the above cited remarks and can be considered as the development of the qualitative analysis of fractional differential equations. The study of Ulam-Hyers stability in the frame of φ-Hilfer fractional derivative with a generalized nonlocal boundary condition proposed in this paper and other coupled system will be presented in the near future.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.