Study of implicit delay fractional differential equations under anti-periodic boundary conditions

*Correspondence: tabdeljawad@psu.edu.sa 2Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia 3Department of Medical Research, China Medical University, Taichung, Taiwan Full list of author information is available at the end of the article Abstract This research work is related to studying a class of special type delay implicit fractional order differential equations under anti-periodic boundary conditions. With the help of classical fixed point theory due to Schauder and Banach, we derive some results about the existence of at least one solution. Further, we also study some results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam Rassias, and generalized Hyers–Ulam–Rassias stability. We provide some test problems for illustrating our analysis.


Introduction
Differential equations have numerous of applications in many applied fields of sciences. Due to these applications, the class of differential equations has remained an interesting area of research. The fractional order derivative is a generalization of the classical derivative, which has been proved to be a strong tool for modeling of many physical, biological, and evolutionary problems. In the recent times this has been the hottest and most interesting area of research in mathematics as well as in other scientific and engineering courses. For some historical and recent work, we refer the readers to [1][2][3][4][5][6][7][8][9]. A comprehensive study in the form of a book has been given by Podlubny [10].
In previous years, the study of nonlinear differential and integral equations has received much attention from mathematicians due to a wide range of their applications. Since using integer order differential operators for modeling various dynamical systems, the hereditary process and memory description cannot be well explained in many situations. Therefore, researchers are applying the fractional differential operators to describe memory and hereditary processes in a more accurate way. This fact motivated researchers to take interest in fractional order differential equations. Various aspects of fractional calculus, such as qualitative theory, stability analysis, optimization, and numerical analysis, have been investigated. In this regard a lot of research work can be found in the literature about existence theory. We refer the readers to [11][12][13][14]. On the other hand, the area devoted to establishing a procedure for numerical solutions has been investigated very well. See [15][16][17][18] and the references therein.
It is necessary for numerical procedure to be stable to produce good results, which is highly acceptable in applications. For this purpose stability analysis is used. This is an important aspect of qualitative analysis. Various kinds of stability, including exponential, Mittag-Leffler, and Lyapunov stability, have been evaluated for a number of problems. In the last few years the mentioned stabilities have been upgraded for linear and nonlinear fractional order differential equations and their systems (for details, see [19][20][21]). Establishing these stabilities for nonlinear systems has merits and de-merits in constructions. Some of them need a pre-defined Lyapunov function, which often is very difficult and time consuming to construct on trail basis. On the other hand, the exponential and Mittag-Leffler stability involving exponential functions often create difficulties in treating during numerical analysis of problems. In 1940-41, Ulam and Hyers introduced the concept of Hyers-Ulam stability. This concept of stability was initially used for functional equations; for details, we refer to [22,23]. Onward the said stability was further modified to a more general form by other researchers for functional equations, ordinary differential equations. Some very fruitful results have been formed in this regard, which can be traced in [24][25][26] and the references therein. In the last two decades the said stability theory has been considered very well for fractional order differential equations and their systems, see [27,28].
The delay differential equations constitute an important class of differential equations. Such equations emphasize the waste analysis of full nonlinear equations or systems in biology and physics, as well as in other applied fields. Among delay differential equations, the pantograph type delay differential equation is a prominent type. Such type of delay differential equations has proportional delay terms. Such type of delay differential equations has applications in electro-dynamic, quantum mechanics, etc. [29]. Therefore, keeping in mind the applications, researchers are devoted to studying different aspects like existence theory and numerical analysis of the mentioned class of differential equations. See for detail [30]. The authors [31] in 2013 studied the following pantograph fractional order differential equation with t ∈ [0, T]: where 0 < α ≤ 1, 0 < λ < 1, and f : [0, T] × R 2 → R. They developed the existence theory for the aforesaid equation by using fixed point theory. Very recently the authors in [32] established qualitative theory for a coupled system of delay fractional order differential equations by using hybrid fixed point theory. Motivated by the above-mentioned work, in this research article we consider the following class of pantograph implicit fractional order differential equations under anti-periodic boundary conditions: where 0 < λ < 1, 0 < p < 1, 1 < q < 2, and f : [0, T] × R 3 → R is a continuous function, C 0 D α t stands for a Caputo derivative of order 2 < α ≤ 3. We investigate qualitative theory as well as different kinds of stability including Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability, and generalized Hyers-Ulam-Rassias stability for the considered problem. For qualitative theory we utilize the usual fixed point theorem due to Schauder and Banach, while for stability theory nonlinear functional analysis is used. Finally, this work is strengthened by providing examples and short conclusion.

Preliminaries
The space M = C([0, T]) is a Banach space with respect to the norm defined by

Definition 1 ([33])
Integral of fractional order for the function z ∈ L 1 ([0, T], R + ) of order α ∈ R + is recalled as such that the integral on right-hand sides is convergent.

Lemma 1 ([34])
If α > 0, the given result holds (1) is Hyers-Ulam stable if there exists a real number C f > 0 such that, for > 0 and for any solutionz ∈ M of the inequality

Definition 3 ([35]) Problem
there is the unique solution z ∈ M of problem (1) such that (1) is generalized Hyers-Ulam stable if there exists ζ ∈ C(R + , R + ), ζ (0) = 0 such that, for any solutionz ∈ M of the inequality (5), there is the unique solution z ∈ M of problem (1) such that

Definition 5 ([35]) Problem (1) is Hyers-Ulam-Rassias stable with respect to ξ ∈ C([0, T], R + ) if there exists a real number C f > 0 such that, for > 0 and for any solution
there exists the unique solution z ∈ M of problem (1) such that Definition 6 ([35]) Problem (1) will be generalized Hyers-Ulam-Rassias stable with respect to ξ ∈ M if there is C f > 0 such that, for any solutionz ∈ M of the inequality (6), there exists the unique solution z ∈ M of problem (1) such that

Theorem 1 Let y ∈ C([0, T], R), then the equivalent integral equation of the following problem
is given by while the Green's function W(t, s) is expressed as Proof Let z be a solution of (9), then by Lemma 1 there exist some real constants c 0 , c 1 , c 2 such that Using the results C 0 . .
Applying the boundary conditions Substituting the values of c 0 , c 1 , and c 2 in (12), one gets the following result: Corollary 1 Problem (1) has the following solution:

Lemma 2
The function W(t, s) in (11) obeys the given relations: . For convenience, we use the notion Proof Proof of (P 1 ) is obvious. To derive (P 2 ), we have . .

Theorem 2 The operator N : M → M defined in
Proof The continuity of f , W(t, s) implies the continuity of operator N . Let B ⊂ M be a bounded set such that By assumption (F 2 ), we have Taking maximum of both sides and simplifying, we have Using property (P 2 ) of the Green's function W(t, s) given in Lemma 2 and inequality (16) in inequality (15), we obtain which shows that N is uniformly bounded. To derive equicontinuity of N , let t 1 , t 2 ∈ [0, T] such that t 1 ≤ t 2 , then From (17), we see that as t 1 → t 2 , the right-hand side tends to zero. Therefore Since N is uniformly bounded, so we can also get that which confirms the equicontinuity of the operator N . Analogously N (B) ⊂ B. Thus, by Arzelá-Ascoli theorem, N is completely continuous.

Theorem 3
Under the complete continuity of operator N and hypotheses (F 1 ), (F 2 ), problem (1) has at least one solution.
Proof We define a set E as The operator N :Ē → M as defined in (14) is completely continuous by Theorem 2. Take z ∈ E . Then, by definition of the set E and (F 2 ), we have Hence the set E is bounded. So the operator N has at least one solution. Consequently, problem (1) maintains at least one solution.

Theorem 4
If hypothesis (F 1 ) and the condition L f 1-K f < 1 hold, where is given in (13), then problem (1) has the unique solution in M.
Proof Here we shall use the Banach theorem to prove the required result. Let z,z ∈ M.

βz(t) = f t,z(t),z(λt), βz(t) .
By (F 1 ), we get which implies Using this result and property (P 2 ), from (19), we have , the operator N is contraction, and thus by the Banach contraction theorem, problem (1) has the unique solution.

Stability results
The present part of our article addresses stability results for the proposed problem.

Lemma 3
For the given problem of pantograph implicit fractional order differential equa- we have the following inequality: where > 0.
Proof Thanks to Corollary 1 the solution of perturbed problem (20) is given bȳ from which we have, by using (i) of Remark 1 and property (P 2 ) of W,

Lemma 4
For the given problem (20), the following inequality holds: Proof Thanks to Corollary 1 the solution of the perturbed problem (20) is given bȳ Using (i) of Remark 2 and property (P 2 ), we have Proof Letz be any solution of inequality (25) and z ∈ M be the unique solution of problem (1). Then By the application of assumption (F 1 ) and Lemma 4, we get Upon simplification (26) gives Thus problem (1) is Hyers-Ulam-Rassias stable.

Lemma 5
The solution of the perturbed problem given in (20) produces the given relation Proof For the proof, follow Lemma 3.

Test problems
To test our theoretical results, we present some problems here.
So the criteria for unique solution have been followed. Further, by using Theorem 5, we observe that K f + 2L f = 11.3048 × 10 -2 < 1.
Hence the solution is Hyers-Ulam stable. Further, it is also generalized Hyers-Ulam stable. For Hyers-Ulam-Rassias stability, we use our Theorem 6 by taking a nondecreasing function ξ (t) = t for t ∈ (0, 1). One has C f = (1-K f ) 1-(K f +2L f ) = 1.38616. Hence we see that, for any solutionz ∈ M and the unique solution z ∈ M, the following relation holds true: zz M ≤ 1.38616 t for all t ∈ [0, 1].