Dynamics and stability results of fractional pantograph equations with complex order
DOI:
https://doi.org/10.5890/JAND.2018.06.006Abstract
In this paper, we study the existence, uniqueness and stability of solutions for fractional pantograph equations with complex order. The Krasnoselkii’s fixed point theorem and Banach contraction principle are used to obtain the desired results.References
[1] Balachandran, K., Kiruthika, S., and Trujillo, J.J. (2013), Existence of solutions of Nonlinear fractional pantograph equations, Acta Mathematica Scientia, 33B, 1-9.
[2] Derfel, G.A. and Iserles, A. (1997), The pantograph equation in the complex plane, Journal Mathematical Analysis and Applications, 213, 117-132.
[3] Iserles, A. (1993), On the generalized pantograph functional-differential equations, Eur. Journal of Applied Mathematics, 4, 1-38.
[4] Liu, M.Z. and Li, D. (2004), Runge-Kutta methods for the multi-pantograph delay equation, Applications and Mathematical Computations, 155, 853-871.
[5] Hilfer, R. (1999), Application of fractional Calculus in Physics, World Scientific, Singapore.
[6] Podlubny, I. (1999), Fractional differential equations, Academic Press, San Diego.
[7] Andras, S. and Kolumban, J.J. (2013), On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Analysis, 82, 1-11.
[8] Jung, S.M. (2004), Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17, 1135-1140. 218(3), (2011), 860-865.
[9] Muniyappan, P. and Rajan, S. (2015), Hyers-Ulam-Rassias stability of fractional differential equation, International Journal of pure and Applied Mathematics, 102, 631-642.
[10] Ibrahim, R.W. (2012), Generalized Ulam-Hyers stability for fractional differential equations, International Journal of mathematics, 23, doi:10.1142/S0129167X12500565.
[11] Wang, J., Lv, L., and Zhou, Y. (2011), Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic Journal of Qualitative Theory of Differential Equations, 63, 1-10.
[12] Wang, J. and Zhou, Y. (2012), New concepts and results in stability of fractional differential equations, Communications on Nonlinear Science and Numerical Simulations, 17, 2530-2538.
[13] Neamaty, A., Yadollahzadeh, M., and Darzi, R. (2015), On fractional differential equation with complex order, Progress in fractional differential equations and Apllications, 1(3), 223-227.
[14] Balachandran, K. and Trujillo, K.J.J. (2010), The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Analysis Theory Methods and Applications, 72, 4587-493.
[15] Bashir, A. and Sivasundaram, S. (2008), Some existence results for fractional integro-differential equations with nonlocal conditions, Communications in Applied Analysis, 12, 107-112.
[16] Bai, Z. and Lu, H. (2005), Positive solutions for a boundary value problem of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 311, 495-505.
[17] Hyers, D.H., Isac, G., and Rassias, T.M. (1998), Stability of functional equation in several variables, 34, Progress in nonlinea differential equations their applications, Boston (MA): Birkhauser.
[18] Vivek, D., Kanagarajan, K., and Harikrishnan, S. (2007), Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal of Nonlinear Analysis and Applications (ISPACS), Accepted article-2017. Id: jnaa-00370.
[19] Rus, I.A. (2010), Ualm stabilities of ordinary differential equations in a Banach space, Carpathian Journal Mathematics, 26, 103-107.
[20] Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematics and Applications, 328, 1075-1081.
Article Metrics
Usage tracking begins September 1, 2026.