Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator

Year 2018, Volume: 1 Issue: 2, 106 - 112, 26.06.2018

S. Harikrishnan , Rabha Ibrahim , K. Kanagarajan

https://doi.org/10.32323/ujma.419363

Cited By: 5

Abstract

In this note, we set up existence, uniqueness as well as the stability of a special class of fractional differential equation (FDE) with Hilfer-Katugampola fractional differential operator (HKFDO). The outcomes are given by employing the Schaefer's fixed point theorem and Banach contraction principle. Moreover, we modify the fractional Ulam stability (FUS) concept utilizing HKFDO.

Keywords

Fractional calculus , fractional differential equations , fractional differential operator

References

• [1] M. I. Abbas, Ulam stability of fractional impulsive differential equations with Riemann-Liouville integral boundary conditions Journal of Contemporary Mathematical Analysis, 2015, (50), 209-219.
• [2] M. Benchohra, B. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations, Electronic Journal of Differential Equations, 10, (2009), 1-11.
• [3] K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving hilfer fractional derivative, Computer and Mathematics with Application, 64, (2012), 1616-1626.
• [4] R. Hilfer, Applications of Fractional Calculus in Physics, World scientific, Singapore, 1999.
• [5] R.W. Ibrahim , H.A. Jalab , Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy 17 (5) (2015) 3172.
• [6] R.W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstract Appl. Anal. (2012) 1.
• [7] R.W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math. 23 (05) (2012) 1.
• [8] R.W. Ibrahim, Ulam stability for fractional differential equation in complex domain, Abstract Appl. Anal. (2012) 1.
• [9] R. Kamocki, C. Obczynski, On fractional Cauchy-type problems containing Hilfer’s derivative, Electronic Journal of Qualitative Theory of Differential Equations, 2016, 50, 1-12.
• [10] R. Kamocki, A new representation formula for the Hilfer fractional derivative and its application, Journal of Computational and Applied Mathematics, 308, (2016), 39-45.
• [11] U.N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, Bulletin of Mathematical Analysis and Applications, arXiv:1411.5229, v1 (2014). https://arxiv.org/abs/1411.5229.
• [12] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore(1989).
• [13] X. Liu, Y. Li, Some Antiperiodic Boundary Value Problem for Nonlinear Fractional Impulsive Differential Equations, Abstract and Applied Analysis, (2014).
• [14] Z. Luo, J. Shen, Global existence results for impulsive functional differential equation, Journal of Kathematical Analysis and Application, 323, (2006), 644-653.
• [15] D. S. Oliveira, E. Capelas de oliveira, Hilfer-Katugampola fractional derivative, arxiv:1705.07733v1, 2017.
• [16] A. Ouahab, Local and global existence and uniqueness results for impulsive differential equations with multiple delay, Journal of Mathematical Analysis and Application, 323, (2006), 456-472.
• [17] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic Journal of Qualitative Theory of Differential Equations, 63, (2011), 1-10.