Existence and Ulam stability results for a class of boundary value problem of neutral pantograph equations with complex

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Existence and Ulam stability results for a class of boundary value problem of neutral pantograph equations with complex order D. Vivek1 · E. M. Elsayed2

· K. Kanagarajan3

Received: 10 June 2019 / Accepted: 31 January 2020 © Sociedad Española de Matemática Aplicada 2020

Abstract The research reported in this paper deals with the existence of solutions of boundary value problem (BVP) of nonlinear neutral pantograph equations by means of complex fractional derivative in weighted spaces. The results were interpreted with the aid of classical fixed point theorems. The Ulam–Hyers–Rassias stability and Ulam–Hyers stability of differential equations are studied by utilizing the complex fractional derivative through the fixed point method. Keywords Neutral pantograph equation · Boundary value problem · Stirling asymptotic formula · Fractional derivative · Existence · Ulam stability Mathematics Subject Classification 26A33 · 34A08 · 34B18

1 Introduction The purpose of this study confines to the problems in the area of fractional calculus. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. In a note to L’Hospital in 1695 Leibniz posed the following question: “Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders ?” The story goes that L’Hospital was fairly interested about that question and responded by another question to Leibniz. “What if the order will be 1 2 Leibniz in a communication dated September 30, 1695 answered: “Itwill lead to a paradox, from which one day useful consequences will be drawn.” The query posed by Leibniz for a


D. Vivek [email protected]


Department of Mathematics, PSG College of Arts & Science, Coimbatore 641014, India


Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah 21589, Saudi Arabia


Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore 641020, India


D. Vivek et al.

fractional derivative was an ongoing problem in the last 300 years. Several mathematicians contributed to this subject over the years. Those like Liouville, Riemann, and Weyl made major roles to the theory of fractional calculus. The story of the fractional calculus maintained with contributions from Fourier, Leibniz, Abel, Letnikov, and Grnwald. At the present time, the fractional calculus attracts various scientists and engineers, for detailed study, the books [16,24,25,28] and references therein. Emerging trends and widespread applications in various fields like physics, chemistry, engineering, finance and other sciences has made fractional differential equations (FDEs) a special area of interest among scholars and researchers. The reformulations and expressions of enormous models in terms of FDEs, has made their physical meaning inclusive in the mathematical models more reasonably. The interdisciplinary applications which are elegantly modelled using fractional derivatives have lai