Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

In this research work, a class of multi-term fractional pantograph differential equations (FODEs) subject to antiperiodic boundary conditions (APBCs) is considered. The ensuing problem involves proportional type delay terms and constitutes a subclass of delay differential equations known as pantograph. On using fixed point theorems due to Banach and Schaefer, some sufficient conditions are developed for the existence and uniqueness of the solution to the problem under investigation. Furthermore, due to the significance of stability analysis from a numerical and optimization point of view Ulam type stability and its various forms are studied. Here we mention different forms of stability: Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam Rassias (HUR) and generalized Hyers–Ulam–Rassias (GHUR). After the demonstration of our results, some pertinent examples are given.


Introduction
FODEs have many applications in modeling various real world processes and phenomena. Therefore in previous several decades it has been given much attention. In fact, FODEs are definite integrals which include classical differential and integral equations as a special case. In recent time this has become a most intensely studied area for research in mathematics and in other applied sciences like physics, dynamics, electrodynamics and fluid mechanics. The mentioned area has a large number of applications also in mathematical modeling of biological models (for details see [1][2][3][4]).
In the past the areas of nonlinear integrals and DEs were given great importance as they have numerous uses in modeling different problems in several fields of technologies and engineering. This is so because using the usual derivative for modeling various real world processes, hereditary and memory descriptions cannot be expressed properly in many situations. So researchers have proved that on using fractional differential operators to describe memory and hereditary processes in many situations has produced very good results as compared to integer order derivatives. This fact motivated researchers to study FODEs from different directions. The said area has been investigated from different aspects including qualitative and stability theory, optimization and approximation results. Hence plenty of work can be traced in the literature about existence theory of solutions, we refer to [5][6][7][8][9][10].
On the other hand, very well performed research has been conducted on the numerical side of FODEs. In this regard plenty of research articles addressing numerical and qualitative analysis have been presented in the past few years (for instance see [11][12][13][14][15][16] and the references therein). Here we remark that stability analysis is also an important aspect of qualitative analysis. This is so because stability results are the important requirements for numerical and optimization purposes during the investigation of solutions to applied problems. Different kinds of stability results, like exponential, Mittag-Leffler and Lyapunov type, have been studied for classical differential and integral equations. In the previous few years the mentioned theory was greatly updated for FODEs (see for details [17][18][19]). Establishing these stabilities for nonlinear systems have merits and de-merits in constructions. Some of them need a "pre-defined Lyapunov function" and such a function usually is very hard to construct. Also we have the "exponential and Mittag-Leffler stability [20] involving exponential functions" which often create difficulties when one is to find numerical solutions to certain problems. In this regard another kind of stability has been given much attention by researchers, known as HU stability. This kind of stability was first pointed by Ulam in 1940 during a talk. After that in 1940 Hyers very nicely gave explanations for functional equations (for details we refer to [21][22][23]). Interesting results have been developed in the last few years (refer to ( [24][25][26][27] and the references therein).
Study of real world problems with respect to delay problems constitutes a huge class of applied analysis. In this regard, proportional type delay problems constitute a subclass known as pantograph differential equations (PDEs). The aforesaid area is increasingly used to model numerous process. The mentioned type of problems arise in large numbers of applications in electro-dynamics [28]. Therefore, keeping in mind the applications of the said area, many researchers have studied the PDEs from many aspects like existence theory and numerical analysis (see for details [29,30]). The authors of [31] in 2004 investigated the given delay problem with t ∈ [0, ] = J as where 0 < ε ≤ 1, 0 < δ j < 1, j = 1, 2, . . . , n and H : J × R m+1 → R. By fixed point theory, they investigated the results. The generalized form of the aforementioned problem was investigated in 2013 in [32]. The concerned fractional order PDEs have been studied recently for existence theory in [33,34]. Also continuous type delay problems of FODEs have been studied recently for theoretical analysis, we refer the reader to [35][36][37]. Furthermore, in [38], the authors have investigated the following problem under APBCs with 0 < δ < 1: They have established qualitative results of stability and existence theory as regards the given problem in (1). Inspired by the aforementioned work, in this research article, we study the given multiterm problem of FODEs involving delay terms where 0 < δ j < 1, 0 < r < 1, 1 < s < 2, j = 1, 2, . . . , m and the nonlinear function H : J × R m+1 → R is continuous, C D +0 is the Caputo fractional derivative. In the present paper, we develop the aforementioned results for (2). In developing the existence criteria of solution we apply the Schauder and Banach theorems. In the end some pertinent problems are given to illustrate the results.

Preliminaries
In this section, we give some related definitions and results from the given literature. The notation X = C(J) is used for a Banach space under the norm

Definition 1
The integral of fractional order for the function h ∈ L 1 (J, R + ) of order σ ∈ R + is recalled as with the integral on the right being point wise on R + .

Definition 2
The Caputo derivative of the function h over J is recalled to be with n = [σ ] + 1.
holds. (2) is HU stable if there exists C H > 0 such that, for all > 0 and for any solutionw ∈ X of the inequality

Definition 3 The delay FODEs
there exists at most one solution w ∈ X to problem (2) with Definition 4 The problem of delay FODEs (2) is GHU stable if there exists β ∈ C(R + , R + ), β(0) = 0, and also regarding any solutionw ∈ X of the inequality (6), there is at most one solution w ∈ X of (2) with Definition 5 The delay FODEs (2) is HUR stable w.r.t. ξ ∈ C(J, R + ), if there exists a real number C H > 0, and¯ > 0, and also, for any solutionw ∈ X of the inequality there exists at most one solution w ∈ X of problem (2), such that

Definition 6
The delay AODE (2) will be GHUR stable w.r.t. ξ ∈ X , if for C H > 0 and any solutionw ∈ X of the inequality (7) there exists at most one solution w ∈ X of problem (2), Remark 1 Letw ∈ X be the result of (6); there exists Remark 2 Letw ∈ X be the result of (7); there exists

Criteria for existence of solution
is given by while Green's function K (t, ) may be provided as Proof 1 The proof of this theorem may be similarly obtained to [38,Theorem 1].
Corollary 1 By Theorem 1, the proposed problem (2) is equivalent to the following integral equation:
By assumption (F 2 ), on simplifying (15), we obtain This yields the uniformly boundedness of T. To derive discontinuity of T, let t 2 > t 1 ∈ J such that At t 1 → t 2 , (16) tends to zero in the right hand side. Thus equicontinuity of T is obtained, which also confirms uniform continuity. Analogously T(B) ⊂ B. Therefore the operator T is completely continuous.

Theorem 3
Under the complete continuity of the operator T and the Hypotheses (F 1 ), (F 2 ), the problem of delay FODEs (2) possesses at least one solution.

Proof 4 Let E be the set
The operator T :Ē → X as provided in (14) is completely continuous by Theorem 2. Take w ∈ E on using (F 2 ), one has From this we have Boundedness of E holds and so (2) has at least one solution. (13), the unique solution will be guaranteed for problem (2) in X .

Theorem 4 Under assumption (F 1 ) and the condition [(m + 1)A H ] < 1, where is given in
In view of property (P 2 ), from (18), one has Since [(m + 1)A H ] < 1, the mapping T is a contraction which confirms that (2) has at most one solution.

Stability theory
Here we develop the required results for stability theory.

Lemma 3 The solution of the given equation with t
obeys the given inequality Proof 6 Like Corollary 1, the solution of (19) can be provided as From this one has on using (i) of Remark 1 and property (P 2 ) of K , Theorem 5 If the conditions (m + 1)A H < 1 hold, then the solution of (2) is HU and GUH stable.

Proof 7
If w ∈ X is at most one result of (2) andw ∈ X is any solution of the said problem, then we may consider with t ∈ J w -w X = max By the application of assumption (F 1 ) and Lemma 3, we get Upon simplification (21) yields Hence the problem (2) is HU stable. Let a nondecreasing function β : (0, 1) → (0, ∞) be such that β(¯ ) =¯ with β(0) = 0, then from (21), we can write Thus problem (2) is GHU stable.

Lemma 4
For the given problem (19), the following inequality holds: On using assumption (F 1 ) and Lemma 4, we get Upon simplification (25) gives Thus the solution of (2) is HUR stable.

Lemma 5
The solution of the perturbed problem given in (19) produces the following relation: Proof 10 On using Lemma 3, the proof is simple.
Theorem 7 Under the Hypothesis (F 1 ) and the inequalities (m + 1)A H < 1 holding, the solution of (2) is GHUR stable.

Proof 11
Keeping in mind Theorem 6, one can write Hence the solution of (2) is GHUR stable.

Illustrative problems
Here we address some illustrative problems.
Thus the given problem of FODEs (1) has at most one solution. Furthermore, using Theorem 5, we see that Thus the concerned conditions for HU and GHU stability hold. Upon using Theorem 6 and taking the nondecreasing function ξ (t) = t 2 for t ∈ (0, 1), one has C H = 1-[(m+1)A H ] = 0.7954472. Hence we see that with the results for the unique solutionw ∈ X and any solution w ∈ X the relation w -w X ≤ 0.7954472¯ t 2 , for all t ∈ [0, 1], holds true. Thus the solution of (1) is HUR stable. Consequently it is GHUR stable on using Theorem 7.
Hence the solution is unique. Analogously by Theorem 5, we have 4A H = 0.0201757 < 1.
Hence the solution is HU stable. Furthermore, it is also GHU stable. For HUR stability, in view of Theorem 6 and by considering the nondecreasing function ξ (t) = t for t ∈ (0, 1), one has C H = 1-(4A H ) = 1.2741035. Hence, we see that with the results for any solution w ∈ X and unique solution w ∈ X the relation w -w X ≤ 1.2741035¯ t, for all t ∈ [0, 1], holds true. Hence the solution of (1) is HUR stable. Consequently it is obviously GHUR stable on using Theorem 7.