Analysisof NonlinearNeutral PantographDifferentialEquationswith ψ -fractional Derivative
DOI:
https://doi.org/10.5890/JVTSD.2018.03.004Abstract
In this paper, we discuss the existence, uniqueness and stability of nonlinear neutral pantograph equation with ψ -fractional derivative. The arguments are based upon Schauder fixed point theorem and Banach contraction principle. Moreover, we discuss the Ulam-Hyers type stability.References
[1] Ahmad, B. and Ntouyas, S. K. (2015), Initial value problems of fractional order Hadamard-type functional differential equations, Electron. J. Differ. Eq , 77, 1-9.
[2] Furati, K.M. and Tatar, N.E. (2004), An existence results for a nonlocal fractional deferential problem, Journal of Fractional Calculus, 26, 43-51.
[3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, in: Mathematics Studies, 204, Elsevier.
[4] Podlubny, I. (1999), Fractional Differential Equations, in: Mathematics in Science and Engineering, vol. 198, Acad. Press.
[5] Balachandran, K., Kiruthika, S. and Trujillo, J.J. (2013), Existence of solutions of Nonlinear fractional pantograph equations, Acta Math. Sin, 33B, 1-9.
[6] Guan, K., Wang, Q. and He, X. (2012), Oscillation of a pantograph differential equation with impulsive perturbations, Appl. Math. Comput., 219, 3147-3153.
[7] Iserles, A. (1993), On the generalized pantograph functional differential equation, European Journal of Applied Mathematics, 4, 1-38.
[8] Vivek, D., Kanagarajan, K. and Sivasundaram, S. (2016), Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4), 685-698.
[9] Vivek, D., Kanagarajan, K. and Harikrishnan, S. (2017), Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal of Nonlinear Analysis and Application, (Accepted article-ID 00370).
[10] Ricardo Almeida, (2017), A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, 44, 460-481.
[11] Ibrahim, R. W. (2012), Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23, 1-9.
[12] Ibrahim, R.W. and Jalab, H.A. (2015), Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy, 17, 3172.
[13] Muniyappan, P. and Rajan, S. (2015), Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math., 102, 631-642.
[14] Wang, J., Lv, L. and Zhou, Y. (2011), Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory. Differ. Equ., 63, 1-10.
[15] Wang, J. and Zhou, Y. (2012), New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 17, 2530-2538.
[16] Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 1075-1081.
[17] Granas, A. and Dugundji, J. 2003, Fixed point theory, Springer-verlag, New York.
Article Metrics
Usage tracking begins September 1, 2026.