Research Article
BibTex RIS Cite
Year 2018, , 106 - 112, 26.06.2018
https://doi.org/10.32323/ujma.419363

Abstract

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.

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

Year 2018, , 106 - 112, 26.06.2018
https://doi.org/10.32323/ujma.419363

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.

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.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

S. Harikrishnan

Rabha Ibrahim

K. Kanagarajan

Publication Date June 26, 2018
Submission Date April 28, 2018
Acceptance Date June 10, 2018
Published in Issue Year 2018

Cite

APA Harikrishnan, S., Ibrahim, R., & Kanagarajan, K. (2018). Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator. Universal Journal of Mathematics and Applications, 1(2), 106-112. https://doi.org/10.32323/ujma.419363
AMA Harikrishnan S, Ibrahim R, Kanagarajan K. Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator. Univ. J. Math. Appl. June 2018;1(2):106-112. doi:10.32323/ujma.419363
Chicago Harikrishnan, S., Rabha Ibrahim, and K. Kanagarajan. “Fractional Ulam-Stability of Fractional Impulsive Differential Equation Involving Hilfer-Katugampola Fractional Differential Operator”. Universal Journal of Mathematics and Applications 1, no. 2 (June 2018): 106-12. https://doi.org/10.32323/ujma.419363.
EndNote Harikrishnan S, Ibrahim R, Kanagarajan K (June 1, 2018) Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator. Universal Journal of Mathematics and Applications 1 2 106–112.
IEEE S. Harikrishnan, R. Ibrahim, and K. Kanagarajan, “Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator”, Univ. J. Math. Appl., vol. 1, no. 2, pp. 106–112, 2018, doi: 10.32323/ujma.419363.
ISNAD Harikrishnan, S. et al. “Fractional Ulam-Stability of Fractional Impulsive Differential Equation Involving Hilfer-Katugampola Fractional Differential Operator”. Universal Journal of Mathematics and Applications 1/2 (June 2018), 106-112. https://doi.org/10.32323/ujma.419363.
JAMA Harikrishnan S, Ibrahim R, Kanagarajan K. Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator. Univ. J. Math. Appl. 2018;1:106–112.
MLA Harikrishnan, S. et al. “Fractional Ulam-Stability of Fractional Impulsive Differential Equation Involving Hilfer-Katugampola Fractional Differential Operator”. Universal Journal of Mathematics and Applications, vol. 1, no. 2, 2018, pp. 106-12, doi:10.32323/ujma.419363.
Vancouver Harikrishnan S, Ibrahim R, Kanagarajan K. Fractional Ulam-stability of fractional impulsive differential equation involving Hilfer-Katugampola fractional differential operator. Univ. J. Math. Appl. 2018;1(2):106-12.

 23181

Universal Journal of Mathematics and Applications 

29207              

Creative Commons License  The published articles in UJMA are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.