Investigating a Class of Pantograph Differential Equations Under Multi-points Boundary Conditions with Fractional Order

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Investigating a Class of Pantograph Differential Equations Under Multi-points Boundary Conditions with Fractional Order Gauhar Ali1 · Kamal Shah1

· Ghaus ur Rahman2

Accepted: 13 November 2020 © Springer Nature India Private Limited 2020

Abstract Qualitative theory for fractional order pantograph differential equations is given in this manuscript. The concerned theory is developed via using some tools of nonlinear analysis along with fixed point results. Some adequate conditions are established for obtaining at least one solution and its existence to the subject to multi-point boundary conditions for problem under consideration. Further, various kinds of Ulam stability results for the solution for the proposed problem are also investigated. To demonstrate our results, some pertinent examples are given. Keywords Pantograph differential equation · Ulam stability · Qualitative theory Mathematics Subject Classification 26A33 · 34A08 · 35R11

Introduction The area of fractional calculus is one of the fastest growing research field in recent time which has significant applications in the fields of engineering and science [1,2]. Indeed, there are various applications of differential equations of arbitrary order in electromagnetic [3], control theory [4], viscoelasticity [5], electro-chemistry [6] and movement [7] through porous media. Delay differential equation constitute a large class of the concern area. Such type of equation include continuous, discrete and proportional type delay terms. The respective equations [8,9] have significant applications in mathematical modeling of varies process and


Kamal Shah [email protected] Gauhar Ali [email protected] Ghaus ur Rahman [email protected]


Department of Mathematics, University of Malakand, Chakdara Dir(L) 18000, Pakistan


Department of Mathematics and Statistics, University of Swat, Swat, Khyber Pakhtunkhwa, Pakistan 0123456789().: V,-vol



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Int. J. Appl. Comput. Math

(2021) 7:2

phenomenons. Among delay differential equation, Pantograph is the class includes proportional type term. The pantograph equations have many applications in various fields like electrodynamics, astrophysics, non linear dynamical system [10], quantum mechanics [11], cell growth and probability theory on algebraic structures [12]. Balachandran et. al [13], studied the initial value problem for the existence of solution through fixed point theory. Bai [14], Benchohra et al. [15], Abdo [16], Bai and Lü [17], S. S. Redhwan et. al [18] and Salem [19], dealt with the boundary value problem (BVPs) of the nonlinear fractional differential equations. On further study about delay differential equations and existence theory , we refer [20–24] and the references therein. The nonlocal BVPs of differential equations of arbitrary order have significant applications in different disciplines of applied sciences including hydromechanics, dynamics and engineering. The m-point linear BVPs was first studied by Il’in and Moiseev [25] in 1987. Then the same procedure was

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