AbstractThe aim and objectives of this paper are devoted to study some adequate results for the existence and stability of solutions of nonlinear neutral pantograph equations with Hilfer-Katugampola fractional derivative. The arguments are based upon Schauder fixed point theorem and Banach contraction principle. Further, we also study the Ulam type stability for proposed problem.
Ahmad B, Ntouyas SK. Initial value problems of fractional order Hadamard-type functional differential equations. Electron J Diff Equ. 2015; 77: 1-9.
Furati KM, Tatar NE. An existence results for a nonlocal fractional differetial problem. J Fractional Calculus, 2004; 26: 43-51.
Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, 2006; 204: Elsevier.
Podlubny I. Fractional Differential Equations, in: Mathematics in Science and Engineering, 1999; 198: Acad. Press.
Oliveira DS. E. Capelas de oliveira, Hilfer-Katugampola fractional derivative. Comp Appl Math. 2018; 37: 3672-3690.
Hilfer R. Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.
Furati KM, Kassim MD, Tatar NE. Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput Math Appl. 2012; 64: 1616-1626.
Gu H, Trujillo JJ. Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl Math Comput. 2014; 257: 344-354.
Katugampola UN. Existence and uniqueness results for a class of generalized fractional differential equations Bull Math Anal App. 2014; 6(4): 1-15.
Kamocki R, Obcznnski C. On fractional Cauchy-type problems containing Hilfer derivative. Electron J Qual Theory Differ Equ. 2016; 50: 1-12.
Vivek D, Kanagarajan K, Harikrishnan S. Existence and uniqueness results for pantograph equations with generalized fractional derivative. J Nonlinear Anal Appl. 2017; 2017(1): 1-8.
Vivek D, Kanagarajan K, Harikrishnan S. Existence results for implicit differential equations with generalized fractional derivative. J Nonlinear Anal Appl. 2017; (Accepted article-ID 00371).
Abbas S, Benchohra M, Sivasundaram S. Dynamics and Ulam stability for Hilfer type fractional differential equations. Nonlinear Stud. 2016; 4: 627-637.
Ibrahim RW, Jalab HA. Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy 2015; 17(5): 3172.
Ibrahim RW. Generalized Ulam-Hyers stability for fractional differential equations. Int J Math. 2012; 23: 1-9.
Muniyappan P, Rajan S. Hyers-Ulam-Rassias stability of fractional differential equation. Int J Pure Appl Math. 2015; 102: 631-642.
Wang J, Lv L, Zhou Y. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron J Qual Theory Differ Equ. 2011; 63: 1-10.
Balachandran K, Kiruthika S, Trujillo JJ. Existence of solutions of Nonlinear fractional pantograph equations. Acta Math Sci. 2013; 33B: 712-720.
Iserles A. On the generalized pantograph functional differential equation. Eur J Appl Math. 1993; 4(1): 1-38.
Vivek D, Kanagarajan K, Sivasundaram S. Dynamics and stability of pantograph equations via Hilfer fractional derivative. Nonlinear Stud. 2016; 23(4): 685-698.
Vivek D, Kanagarajan K, Sivasundaram S. Theory and analysis of nonlinear neutral pantograph equation via Hilfer fractional derivative. Nonlinear Stud. 2017; 24(3): 699-712.
All the published articles are licensed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC 4.0) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.