Applications of new contraction mappings on existence and uniqueness results for implicit ϕ-Hilfer fractional pantograph differential equations

In this paper, we consider initial value problems for two different classes of implicit ϕ-Hilfer fractional pantograph differential equations. We use different approach that is based on α−ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha -\psi $\end{document}-contraction mappings to demonstrate the existence and uniqueness of solutions for the proposed problems. The mappings are defined in appropriate cones of positive functions. The presented examples demonstrate the efficiency of the used method and the consistency of the proposed results.


Introduction
Calculus of arbitrary order integration and differentiation has achieved a remarkable growth over the last few decades due to its applications in a wide range of fields such as engineering, physics, neural networks, control theory, population dynamics, and epidemiology; see for instance [1][2][3][4].
In this context, there have appeared many definitions of fractional derivatives including the well-known types of Caputo, Riemann-Liouville, Hadamard, Katugampola derivatives and many others. Consequently, this has led to several problems defined by different fractional operators. However, it has been realized that the most efficient way to deal with such a variety of fractional operators is to accommodate generalized forms of fractional operators that include other operators. In [5], the Hilfer fractional derivative D υ,ι a + of order α and type ι was introduced. This definition provides a concatenation betwixt the Riemann-Liouville and Caputo fractional derivatives. The type-parameter ι allows some freedom of action in the initial conditions, which produces more kinds of stationary states. Some models based on this definition can be seen in the papers [6][7][8]. Meanwhile, for the sake of in terms of the Riemann-Liouville and Caputo fractional derivatives were introduced, respectively [9,10]. In this perspective, the φ-Hilfer derivative was defined in [11]. For more relevant applications concerning φ-Hilfer derivative, we refer the reader to [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].
On the one hand, the contraction mappings are widely utilized to examine of the existence and uniqueness of fixed points. For this purpose, many contraction mappings have been developed and used by interested researchers who applied these mappings in various disciplines. In [28], the authors presented αψ-contraction mappings. This mapping and its extensions as well as some useful fixed point results can be seen in several papers [29][30][31][32][33][34][35][36][37][38][39].
On the other hand, the pantograph is a mechanical connection set in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. In the study of the motion of the pantograph head on an electric locomotive, the authors of [40] came across the following delay differential equation: which was referred to in the literature as pantograph differential equation. During the last decades, it has been realized that pantograph differential equation has many applications in various disciplines, see the paper [41] and the references cited therein.
Knowing the significant of fractional operators in modeling processes, the consideration of pantograph differential equation in fractional settings has come true. We review some relevant results for the sake of completeness. Nonlinear fractional pantograph differential equation was studied in [42,43] where the existence of solutions is investigated using fractional calculus and fixed point theorems.
The objective of this paper is to provide different approach to investigate the existence and uniqueness of solutions for Eq. (1). To do this, we use a new technique that is based on the application of αψ-contraction mappings which are defined in appropriate cones of positive functions. Furthermore, we extend the proposed results to cover the following modified implicit φ-Hilfer pantograph fractional differential equation: which satisfies the same assumptions with I ι;φ 0 + is φ-Riemann-Liouville fractional integral of order ι > 0 relative to the other function φ with φ (·) = 0, g : J × R 3 → R is continuous, T > 0 and ϑ i ∈ J satisfying 0 < ϑ 1 ≤ ϑ 2 ≤ · · · ≤ ϑ m < T for i = 1, 2, . . . , m. It is to be noted that the structure of the boundary conditions in Eq. (1) and Eq. (2) is visible and allows better interpretations to many physical problems [45]. Reported results in this paper yield new sufficient existence and uniqueness conditions via different approach comparing to the existing results in the literature [43,44,46,47]. These conditions are easily attainable with the feature that they are less restrictive.
The rest of the paper adheres to the following plan: In Sect. 2, we define the norms, notations, and some spaces of functions. Properties of φ-Hilfer fractional derivative along with some necessary results on αψ-contraction mappings are stated for completeness. In Sect. 3, the main results are stated and proved. The examples presented in Sect. 4 demonstrate the visibility and capability of the proposed results. We end the paper with a conclusion in Sect. 5.

Requisite preliminaries
Here we state some explanations which are needed throughout this paper. Further, some essential lemmas and theorems are stated as preparations for the main objectives.
Let 0 < a < T < ∞, and C[a, T] be a Banach space of continuous functions y : [a, T] → R with the norm y = max{|y(ς)| : a ≤ ς ≤ T}. The weighted space C 1-μ;φ [a, T] of continuous functions y is defined in [11] as follows: Obviously, C 1-μ;φ [a, T] is a Banach space with the norm Then the left-sided φ-Riemann-Liouville integral is stated by respectively.
T] is continuous, and the following holds: In the context, we use the following lemma.
We define the following set ψ for any ψ ∈ : The following proposition is helpful.
Then φ has a unique fixed point.

Main results
Let E = C(J, R). We define the cone K ⊂ E by Let M = {f : J → R; f is continuous}. We endow M with a metric defined by x(ς)y(ς) . where Theorem 3.2 Suppose that the following conditions hold: ii) The function f : J × R × R → R + satisfies the following condition:
Proof The results are stated in several steps.
Step 1. First we show T : K → K . By condition (i), we have Thus, it is easy to see that T : K → K .
Step 2. We demonstrate that T is an αψ-contraction.
For this purpose, we can write
Step 4. Using Theorem 2.14 and from Step (3), there exists a subsequence {T θ(n) Step 5. By Theorem 2.15, x * is a fixed point of T 1 and hence x * ∈ M is a solution for (1).
Step 6. In order to show the uniqueness of the solution, we let (u, v) ∈ Fix(T 1 ) × Fix(T 1 ) be an arbitrary pair with u = v. Without loss generality, we set u ≤ v, then the conclusion is obvious. Therefore, by Theorem 2.16 the proof is complete.
Next, we investigate the existence and uniqueness of the solutions for Eq. (2).
Proof The results are stated in several steps.
Step 1. First we show T 1 : K → K . By condition (i), we have Thus, T 1 : K → K .
Step 2. We demonstrate that T 1 is an αψ-contraction.
For this purpose, we can write Define α : M × M → R by the following: Then Setting ψ(ς) = 2ς 3 , we obtain Hence T 1 is an αψ-contraction.
Step 4. From step (3), there exists a subsequence {T θ(n) Step 5. By applying Theorem 2.15, we conclude that x * is a fixed point of T 1 , that is, x * ∈ M is a solution to equation (2).
Step 6. To prove the uniqueness, we let (u, v) ∈ Fix(T 1 ) × Fix(T 1 ) be an arbitrary pair with u = v. Without loss generality, we set u ≤ v, then the result is concluded. By Theorem 2.16 the proof is perfect.
where 0 < ε < 1 and ς ∈ (1, 2]. Let us find solution of the problem from the following cone:  2] . Therefore condition (ii) of Theorem 3.4 holds. Further, it is easy to show that item (iii) in Theorem 3.4 is true. Now since T 1 (0) ≥ 0, so by choosing x 0 = 0, we can establish condition (iv) from 3.4. Therefore, all requirements of 3.4 hold which guarantee the existence and uniqueness of solution for (14).

Conclusion
The fractional derivatives involving φ-Hilfer of a function relative to the other function have widely been used due to their tremendous applications in modeling of various phenomena.
Here, we investigated two initial value problems of implicit φ-Hilfer fractional pantograph differential equations. Unlike the methods used in the literature which were based on classical fixed point theorems, we utilized the αψ-contractions to demonstrate the existence and uniqueness of solutions for the proposed problems. The mappings are defined in appropriate cones of positive functions. In spite of the complex structure of φ-Hilfer fractional derivative, which causes some limitations, we proposed different techniques that produced sufficient existence and conditions which are more appropriate than the existing conditions. For the sake of confirmation, we constructed particular examples corresponding to the main theorems that illustrate the applicability of the mentioned assumptions.
Results of this paper provide a new technique that associates the study of theory of contraction mappings with the theory fractional differential equations. We believe that the contents of this paper will be of great significance for enthusiasts in these two theories. In this context, many promising topics could be discussed in the future such as inclusion boundary value problems involving φ-Hilfer fractional pantograph differential equations and system of fractional pantograph differential equations by using common fixed point theorems of some contractions.