Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

: Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form ∆ β ∗ [ k ]( t ) = w (cid:0) t + β , k ( t + β ) , k ( λ ( t + β )) (cid:1) , with condition k ( 0 ) = p [ k ] for t ∈ N 1 − β , 0 < β ≤ 1, λ ∈ ( 0,1 ) and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach ﬁxed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical ﬁndings.


Introduction
Graphical methods in engineering are very much useful to present clear results, develop reasoning, and spatial thinking. Dependency on computer-based simulations has led to the demise of graphical methods [1]. Though computer simulations with the correct programming convey invariably accurate results, they fail to provide ingenuity, understanding, and conceptual thinking. Graphical methods provide practical knowledge which is more efficient than just going through texts. The science of the mechanisms can be extended beyond its classical limits to include pneumatic, hydraulic, electrical, and electronic links.
A special type of differential equation with delay was discovered when J.R. Ockendon and A.B. Tayler studied motion of pantograph head on an electric locomotive [2]. The equation is of the form x (t) = ax(t) + bx(λt) where x(t) represents the motion of the locomotive and a, b are real constants with 0 < λ < 1. The pantograph is used in locomotion and trams to transfer power from the wire to the traction unit by maintaining electrical contact. They are also used to increase or reduce motion in some definite proportion, as in the indicator rig on an engine where the motion of the crosshead is reduced proportionally to the desired length of the indicator diagram [3]. The pantograph is a four-bar mechanism used to enlarge or reduce drawings for it is evident that similar curves may be traced as well as straight lines. It was originally used in drafting for copying and scale line drawings. Three-dimensional pantograph is used in sculpting to enlarge sculptures by interchanging the positions. Windscreen wipers on pantograph in some vehicles are used to allow blade to cover more windscreen on each wipe. In 1890, the US census made use of keyboard punch which is a pantograph design [4]. Some heavy-duty applications of pantograph include scissor lifts, material handling equipment, stage lifts, etc. During the past few decades, there was a gradual development of the modeling of nonlinear phenomena that occurs in various science and engineering fields [5]. Fractional calculus, which is a generalization of classical integer order calculus, has become popular among the scientists and engineers as it renders new dimension and flexibility in dealing with real-world problems [6]. Increasing interest towards this field is due to non-local behavior and ultimate convergence to the integer order systems. Potential of fractional derivatives has already been widely explored by researchers from different parts of the world by studying its applications in a range of problems in biology, physics, electronics (circuit theory), chemistry, etc. Non-standard Lagrangians have wide range of applications in nonlinear differential equations, dynamical systems, etc. [7][8][9][10][11][12][13]. Fractional action-like variational approach is very useful in giving better description of dissipative system. The fractional non-standard Lagrangians have been effective in various areas of physics like astrophysics, cosmology, quantum and classical dynamical systems. Recent works can be seen in [14][15][16][17][18]. Discrete fractional calculus is gaining its importance in recent years. Recently, Atici and Eloe [19][20][21][22], and Miller and Ross [23], have studied discrete delta fractional calculus. The study of stability is a venerable branch in the qualitative theory of differential equations. Asymptotic stability results for fractional difference equations have been developed by Chen et al. [24][25][26] for both Caputo-and Riemann Liouville-type operators. Other authors studied stability results of nabla fractional equations [27][28][29]. In 2019, the authors investigated the k-dimensional system of Langevin Hadamard-type fractional differential inclusions with 2k different fractional orders and they established existence results for a fraction hybrid differential inclusion with Caputo-Hadamard-type fractional derivative [30,31]. In 2020, Zhou et al. studied a nonlinear non-autonomous model which is composed of two species in a rocky intertidal community and occupy each other by individual organisms, in a rocky intertidal community [32]. One can see some significant applications of fractional differential equations in [33][34][35][36][37][38][39][40][41].
Though a standard pantograph equation is available in literature, the varying design of the pantograph in accordance with its application has inspired us to consider the generalized version of the equation. Motivated by the works in [42][43][44][45][46], we consider the nonlinear discrete fractional pantograph equation for t ∈ N 1−β , where 0 < β ≤ 1, 0 < λ < 1, ∆ β * is a Caputo like difference operator, k represents the motion of the pantograph, w : E → R is continuous with respect to k, and t.
Here, E = [0, ∞) × C × C, N t = {t, t + 1, t + 2, . . .}, and p : C → R is Lipschitz continuous in k where C = C([0, ∞), C). That is, there is a positive constant M ∈ (0, 1) such that for each t ∈ N t and almost all k, l ∈ C. The discretized form of standard pantograph equation can be obtained from Equation (2) when β = 1 and w t + β, k(t + β), k(λ(t + β)) = ak(t + β) + bk(λ(t + β)) for t ∈ N 1−β . By employing fixed point hypotheses based on Krasnoselskii's and generalized Banach fixed point theorems, we investigate the asymptotic stability of solutions of Equation (2). Particular examples are presented to demonstrate the validity of our theoretical findings. Some interesting observations are presented at the end of the paper. This paper is organized as follows. In Section 2, some notations, definitions, and lemmas that are essential in our further analysis are presented. In Section 3, we analyze the asymptotic stability of the problem expressed by (2). Section 4 contains some illustrative examples to show the validity and applicability of our results.

Essential Preliminaries
This section is committed to state some notations and essential preliminaries that are acting as necessary prerequisites for the subsequent sections. First, we recall σ−th fractional sum of function k ∈ C. Definition 1 ([19,20]). Let σ > 0. The σ−th fractional sum of k is defined by where k(s) and ∆ −σ [k](t) are defined for s ≡ a mod (1), for t ≡ (a + σ) mod (1), respectively, for t ∈ N a and ∆ −σ maps functions defined on N a to functions defined on N a+σ , which, upon substitution in Equation (4), leads to Figure 1 presents the convergence of t (−σ) in 3-dimensional view, and it is clear that the greater the value of σ, the lesser the time taken for t (−σ) to approach zero. Figure 2, indeed, illustrates the behavior of t (−σ) in (5) whenever a and σ are changed, respectively. These results are presented in Table 1, and they show that the operator t (−σ) is decreasing with respect to both σ and t. Thus, t (−σ) → 0 as t → ∞.    Further, the authors of [20] proved that for σ ∈ R \ {· · · , −2, −1}. At present, suppose that µ > 0 and − 1 < µ < , where denotes a positive integer, = µ , here . denotes the ceiling of number [5]. Set σ = − µ.
where k is defined on N a with a ∈ Z + . In particular, when 0 < µ < 1 and a = 0, we have where k is defined on N 1 and ∆ µ * is defined on N 1−µ .

Definition 2 ([25]). Let k = ϕ(t) be a solution of Equation
(1) The solution k is said to be stable, whenever for any > 0 and t 0 ∈ R + , there exists The solution k is said to be asymptotically stable, whenever it is stable and attractive.

Definition 3 ([48]
). Let k = ϕ(t) be a solution of Equation (2). A set Ψ of sequences in l ∞ n 0 is uniformly Cauchy or equi-Cauchy, if for every > 0, there exists an integer N such that k(i) − k(j) < whenever i, j > N for every k = {k(n)} in Ψ.

Theorem 2 ([49]
Krasnoselskii fixed point theorem). Let Ψ be a nonempty, closed, convex, and bounded subset of the Banach space X and let G : X → X and H : Ψ → X be two operators such that (a) G is a contraction with constant M < 1.
Then the operator equation

Lemma 4 ([50] Generalized Banach Fixed Point Theorem).
Let Ψ be a nonempty, closed subset of a Banach space (X , · ) and ρ n ≥ 0 for every n ∈ N 0 such that ∞ ∑ n=0 ρ n converges. Moreover, let the mapping Q : Ψ → Ψ satisfy the inequality for all n ∈ N 1 and any k, l ∈ Ψ. Then, Q has a uniquely defined fixed point k * . Furthermore, for any k 0 ∈ Ψ, the sequence {Q n [k 0 ]} ∞ n=1 converges to this fixed point k * .

Main Results
For the purpose of convenience, we set Let l ∞ 1 be the set of all real sequences k = {k(t)} ∞ t=1 with norm where Thus, the operator G is contraction with M < 1. Condition (a) of the Lemma 2 holds and k(t) is a solution of (2) if it is a fixed point of Q. Now, we proof our key lemmas.
Lemma 5. The map k : N 1 → R is a solution of (2) if and only if k(t) is a solution of the fractional Taylor's difference formula given by Proof. Suppose that k(t) is a solution of (2), we have from (9) This implies that (13) holds. Conversely, if k(t) is solution of (13), comparing (9) and (13) yields, for each t ∈ N 1 . If t = 1 then (14) becomes If t = 2 then from (14) it follows that By using (15), the above equation becomes Thus, by induction, we have that ∆ β * [k](t) = W β λ [k](t) for all t ∈ N 1−β and so k(t) is a solution of (2). This completes the proof.
In order to prove the main results, we make the following assumption.
Proof. For t ∈ N 1 , ξ 1 > 0 Clearly the set Ψ defined in (17) is closed, bounded, and convex subset of R. First, we prove the continuity of the operator H. Using Equations (7) and (12) and the condition (W 1 ), we have For t ∈ N 1 , by using Lemma 2 we obtain Using (18), it is clear that H[k](t) ≤ t (−ξ 1 ) . Thus, H[Ψ] ⊂ Ψ for t ∈ N 1 . Let > 0 be given. Then, there exists S 1 ∈ N 1 , such that t > S 1 implies Consider the sequence {k n } such that k n → k. By the continuity of the function f and Lemma (5) for t ∈ {1, 2, . . . S 1 }, we obtain as n → ∞, For t ∈ N S 1 +1 , as n → ∞ for all t ∈ N 1 . Therefore, the operator H is continuous. Let δ 1 , δ 2 ∈ N 1 and δ 1 < δ 2 . Then, we get

It is clear from the definition of uniformly Cauchy that {H[k], k ∈ Ψ} is bounded and uniformly Cauchy subset and from Discrete Arzelà-Ascoli's Theorem stated in Lemma 1, H[Ψ] is relatively compact. This completes the proof.
Lemma 7. Assume that (3) and condition (W 1 ) hold, then for t ∈ N 1 a solution of (2) is in Ψ.
Proof. Condition (c) of Lemma 2 is yet to be proved. If k = G[k] + H[l], l ∈ Ψ for t ∈ N 1 , we have Therefore, Indeed k(t) ≤ t (−ξ 1 ) . Thus, k(t) ∈ Ψ for t ∈ N 1 . By Theorem 2, Q has a fixed point in Ψ which is solution of (2).
Proof. By Lemma 7, the solutions of (2) exist and are in Ψ. Further, the function k(t) in Ψ tends to zero as t → ∞. Then, clearly the solutions of (2) tend to zero with t approaching infinity. The proof is complete.
Before establishing the theorems, we make the following assumption.
Theorem 4. Assume that (3) together with the condition (W 2 ) is satisfied, then the solution of (2) is unique bounded solution in l ∞ provided that Proof. Let the iterates of operator Q be defined as Q 1 = Q and Q n = Q(Q n−1 ) for each n ∈ N 1 . Now, we shall prove that Q is a contraction operator for sufficiently large n. We have that and Therefore, the (23) is true for n = 1. Assuming (23) is true for n, we obtain By the principle of mathematical induction on n, the statement (23) is true for all n ∈ N 1 . The geometric series ∑ ∞ n=0 ρ n converges, as ρ < 1 and so Q has a unique bounded fixed point in Ψ.
The proof is the consequence of Theorems 3 and 5.

Conclusions
Asymptotic stability of the initial value discrete fractional pantograph equation is established using Krasnoselskii theorem, generalized Banach fixed point theorem, and discrete Arzelà-Ascoli theorem. Numerical simulations are carried out for the stability results illustrating the effects of the fractional order on the stability conditions. The values are tabulated and plotted. The 3-dimensional images are presented to analyze the stability of the equation with simultaneous variation of the fractional order and σ 2 ∈ (β, 1).
Author Contributions: All authors declare that the study was realized in collaboration with equal responsibility. All authors have read and agreed to the published version of the manuscript.
Funding: J. Alzabut would like to thank Prince Sultan University for funding this work.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.