Sugumaran HARİKRİSHNAN, Rabha IBRAHİM, Kuppusamy KANAGARAJAN

Establishing the existence of Hilfer fractional pantograph equations with impulses

In [1], the authors established the existence of a class of fractional differential equations of a complex order. In this note, we derive some sufficient conditions for the existence of solutions to a class of Hilfer fractional pantograph equations with impulsive effect. Further, using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss various kinds of Ulam-Hyers stability.

Keywords:

Pantograph equation, Impulsive condition, Fixed point stability,

PDF

___

[1] S. Harikrishnan, RabhaW. Ibrahim, K. Kanagarajan, On y-Hilfer Fractional Differential Equation with Complex Order, Universal Journal of Mathematics and Applications(1) (2018) 33-38 Universal Journal of Mathematics and Applications
[2] M. I. Abbas, Ulam stability of fractional impulsive differential equations with Riemann-Liouville integral boundary conditions, J. Contemp. Mathemat. Anal., 50, (2015), 209-219.
[3] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence of solutions of Nonlinear fractional pantograph equations, Acta Math. Sci., 33B,(2013),1-9.
[4] K.M. Furati, M.D. Kassim and N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64, (2012), 1616-1626.
[5] A. Granas, and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
[6] K. Guan, Q. Wang, X. He, Oscillation of a pantograph differential equation with impulsive perturbations, Appl. Math. Comput., 219, (2012), 3147-3153.
[7] S. Harikrishnan, K. Kanagarajan and E. M. Elsayed, Existence and stability results for langevin equations with Hilfer fractional derivative, Res. Fixed Point Theory Appl., (2018), 10 pages.
[8] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, (1941), 222-224.
[9] R. Hilfer, Applications of Eractional Calculus in Physics, World scientific, Singapore, 1999.
[10] A. Iserles, On the generalized pantograph functional differential equation, European J. Appl. Math., 4, (1993), 1-38.
[11] R.W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstract Appl. Anal., (2012), 10 pages.
[12] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23(5), (2012), 9 pages.
[13] A. A. Kilbas, H.M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier,2006.
[14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989.
[15] X. Liu and Y. Li, Some antiperiodic boundary value problem for nonlinear fractional impulsive differential equations, Abst. Appl. Anal., (2014), 10 pages.
[16] Z. Luo and J. Shen, Global existence results for impusive functional differential equation, J. Math. Anal. Appl., 323(1), (2006), 644-653.
[17] I. Podlubny, Fractional Differential equation, Academic Press, San Diego, 1999.
[18] A. M. Samoilenko and N. A. Perestyuk, Impulsive differential equations, World scientific, Singapore(1995).
[19] D. Vivek, K. Kanagarajan and S. Sivasundaram, Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4), (2016),685-698.
[20] D. Vivek, K. Kanagarajan and E. M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15:15, (2018).
[21] D. Vivek, K. Kanagarajan and S. Sivasundaram, Dynamics and stability results for Hilfer fractional type thermistor problem, Fractal Fract., 1(1), (2017), 1-14.
[22] D. Vivek, K. Kanagarajan, S. Harikrishnan, Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal of Nonlinear Analysis and Application, 2, (2017), 105-112.
[23] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63, (2011), 1-10.
[24] J. Wang, Y. Zhou and M. Feckanc, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comp. Math. Appl., 64, (2012), 3389-3405.
[25] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328(2), (2007), 1075-1081.