Existence and uniqueness results for Φ-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition

The present paper describes the implicit fractional pantograph differential equation in the context of generalized fractional derivative and anti-periodic conditions. We formulated the Green’s function of the proposed problems. With the aid of a Green’s function, we obtain an analogous integral equation of the proposed problems and demonstrate the existence and uniqueness of solutions using the techniques of the Schaefer and Banach fixed point theorems. Besides, some special cases that show the proposed problems extend the current ones in the literature are presented. Finally, two examples were given as an application to illustrate the results obtained.


Introduction
The fractional calculus, which allows for the integration and differentiation of functions with non-integer orders, is one of the fastest-growing areas of mathematics based on the findings that fractional operators were used in mathematical modeling [10-12, 21, 25, 27, 29]. In [9,19], the authors presented the so-called fractional differential operator of a function with respect to another function. These operators have generalized several wellknown fractional operators dealing with the fractional derivatives of Caputo, Caputo-Hadamard, Caputo-Erdélyi-Kober and Caputo-Katugampola. Also, these operators have been successfully used to solve population growth and other models.
The most favored area of study in the field of fractional differential equations, which has received significant attention from researchers, is the theory of existence and uniqueness of solutions. Many researchers have introduced some interesting results on the existence and uniqueness of solutions of various initial/boundary value problems, using different techniques used in fixed point theory. For more details we refer the reader to [5,6,16,18,26,30,31]. In particular, the pantograph delay equation was used as an effective tool to gain insight into some of the current issues emerging from different fields of knowledge, such as quantum mechanics, probability, number theory, control systems and electrodynamics. However, a significant study has been carried out on the properties of this form of fractional differential equation, both analytical and numerical, and interesting results have been published; see [13,15,23].
To the best of our knowledge, this is the first paper to discuss the existence and uniqueness of solutions for a class of nonlinear implicit fractional pantograph differential equations using -Caputo fractional derivative with a more general anti-periodic boundary condition. Also, this work contributes to an improvement in the qualitative aspects of fractional calculus, boundary value problems, pantograph equations. However, the -Caputo fractional and anti-periodic condition considered in this work is more general than the current ones described in the literature (see the Conclusion section for more details).
The outlined of this paper is as follows: In Sect. 2, we recall the main definitions, the -Caputo fractional derivative and some theoretical results in line with the -fractional operators needed in a later section. In Sect. 3, we establish a relation between the proposed problems and mixed-type integral equation with the help of a Green's function. Besides, we investigate the existence and uniqueness of solutions using the techniques of Schaefer's and Banach's fixed point theorems. Some particular cases and theoretical examples are discussed, as can be seen in this section. Finally, we summarize the theoretical results in Sect. 4.

Preliminaries
This section recalled some basic definitions and lemmas concerning fractional operators that are essential throughout the writing of the paper.
Let E = C([J , R]) be a Banach space equipped with the norm defined by The space of all absolutely continuous real valued function on J is denoted by with the norm defined by ] be a function. Then the fractional operator is referred to as a Riemann-Liouville integral of order r provided the right-hand side of (2.1) exists.

Definition 2.2 ([21])
Let n ∈ N, r, t, a ∈ R + and the function f ∈ C n [a, b]. Then the fractional operator is referred to as Caputo's fractional derivative of order r provided that the right-hand side of (2.2) is point-wise defined on (a, ∞) and n = [r] + 1.
. Then the fractional operator is called a -Riemann-Liouville fractional integral of order r of the function f with respect to another function .
, (t) > 0 and (t) = 0 for all t ∈ J . Then the fractional operator is referred to as the the left-sided -Caputo fractional derivative of a function f of order r with respect to another function .
Theorem 2.8 (Banach's fixed point theorem [36]) Let E be a complete metric space, and N be a contraction on E. Then there exists a unique z ∈ E such that N (z) = z.

Main results
In this section, we transform the proposed problem (1.4) into an equivalent integral equation with the help of the Green's function. Besides, the existence and uniqueness of solutions of problem (1.4) were establish using the techniques of Schaefer's and Banach's fixed point theorems.
where H(t, τ ) is the Green's function described by For the sake of simplicity, we will denote T z (t) = C D r; are arbitrary constants to be determine. Using the facts that C D p; In view of Eq. (3.3) and the boundary condition Also, from the boundary conditions α 1 C D p; 0 + z(ξ ) = 0 and making use of Eqs. (3.4), (3.5), we obtain (3.9) Thus, the result follows. The converse follows directly.

Lemma 3.2 The Green's function
2) fulfills the following relationships: Proof (A 1 ) follows trivially. In order to show (A 2 ), for any t ∈ J , we have

Existence result
In this subsection, we use Theorem 2.7 to establish the existence of at least one solution to the problem (1.4) on J .

Claim 1 N is continuous.
Suppose that z n → z is a convergent sequence in E. Thus, for t ∈ J and using assumption (A 2 ) yields (3.14) Since T z is continuous, we get N z n -N z E → 0, as n → ∞.

Claim 2 N maps bounded sets into bounded sets.
Let γ > 0, and construct a closed convex set z ∈ B γ = {z ∈ E : z E ≤ γ }. Then we show that there exists η > 0 such that N (z) E ≤ η.

Claim 3 N maps a bounded set into an equicontinuous set.
Suppose t 1 , t 2 ∈ J such that t 2 ≥ t 1 and B γ as defined in Claim 2, above. So, for any z ∈ B γ , we obtain (3.19) Hence, as a consequence of the Arzelá-Ascoli theorem and Claims 1-3, the operator N is completely continuous.

Claim 4 N is a priori bounded.
Now, consider the set Then it is enough to show that ϑ is bounded. Indeed, for any z ∈ ϑ, z = ε(N z) where 0 < ε < 1, we have It follows from (A 2 ) and inequality (3.16) that, for each t ∈ J , (3.20) Thus, the set ϑ is bounded. As a result of Theorem 2.7, the operator N has at least one fixed point. Thus, we conclude that problem (1.4) has at least one solution on J .

Uniqueness result
This subsection gives a detail proof of the uniqueness of solution of problem (1.4) by applying the concept of Banach contraction principle.

Theorem 3.4 Suppose that we have the hypotheses:
(A 4 ) There exist constants K 1 , K 2 > 0 such that for any u 1 , u 2 , u 3 ,ū 1 ,ū 2 ,ū 3 ∈ R and t ∈ J . (A 5 ) We assume where 1 is given by (3.12). If there exists a solution of the proposed problem (1.4), it is unique.
Proof Consider the operator N as defined in (3.13). Let z 1 , z 2 ∈ E and t ∈ J , then we obtain and (3.22) By substituting inequality (3.22) in (3.21), we get (3.23) Thus, In view of (A 5 ), this that N is a contraction and hence has a unique fixed point. Therefore, as a consequences of Theorem 2.8, the proposed problem (1.4) has a unique solution on J .
The -Caputo fractional derivative as described Definition 2.4 comprises, as special cases, certain specific forms (definitions) of fractional derivatives. Thus, the nonlinear implicit fractional pantograph differential equation ( . (3.26)

Corollary 3.5 Suppose that the assumption (A 3 ) holds. Then the proposed problem (1.4)
has at least one solution on J .

Lemma 3.10
The Green's function W(t, τ ) defined in (3.30) adhere to the following conditions: Proof Obviously, for any t ∈ J , the Green's function W(t, τ ) is continuous. Thus, (A 6 ) follows.

Theorem 3.12 Assume that hypothesis (A 4 ) holds and
Now we present some special cases of the proposed problem (1.5) based on the choice of the arbitrary function and parameters α 1 , α 2 . Case 3: [2,3,14] Let α 1 = α 2 = 1 and (t) = t. Then the Green's function defined in (3.30) reduces to where * 2 is defined by (3.34). If there exists a solution of (1.5), it is unique on J .