Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

A NOTE ON GENERALIZED SINGULAR GRONWALL INEQUALITIES

  • Kang, Bowon (Department of Mathematics Chungnam National University) ;
  • Koo, Namjip (Department of Mathematics Chungnam National University)
  • Received : 2018.01.05
  • Accepted : 2018.02.05
  • Published : 2018.02.15
Download PDF
⟨ Previous Next ⟩

Abstract

This paper deals an impulsive fractional integral inequality with singular kernel which can be used in getting the explicit estimate of solutions of impulsive fractional differential equations.

Keywords

References

  1. R. P. Agarwal, S. Deng, and W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput. 165 (2005), 599-612.
  2. D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientitic Publishing Co. Pte Ltd, River Edge(NJ), 1995.
  3. S. K. Choi, B. Kang, and N. Koo, Stability for Caputo fractional differential equations, Proc. Jangjeon Math. Soc. 16 (2013), 165-174.
  4. S. K. Choi and N. Koo, On a Gronwall-type inequality on time scales, J. Chungcheong Math. Soc. 23 (2010), 137-147.
  5. S. K. Choi and N. Koo, The monotonic property and stability of solutions of fractional differential equations, Nonlinear Anal. 74 (2011), 6530-6536.
  6. S. K. Choi and N. Koo, A note on linear impulsive fractional differential equations, J. Chungcheong Math. Soc. 28 (2015), 583-590.
  7. S. K. Choi, N. Koo, and C. Ryu, Impulsive integral inequalities with a non-separable kernel, J. Chungcheong Math. Soc. 27 (2014), 651-659.
  8. Z. Denton and A. S. Vatsala, Fractional integral inequalities and applications, Comput. Math. Appl. 59 (2010), 1087-1094.
  9. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  10. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishing Co. Pte. Ltd. NJ, 1989.
  11. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers Ltd, 2009.
  12. O. Lipovan, A retarded Gronwall-like inquality and its applications, J. Math. Anal. Appl. 252 (2000), 389-401.
  13. B. G. Pachpatte, On some generalizations of Bellman's lemma, J. Math. Anal. Appl. 5(1975), 141-150.
  14. J. Wang, Y. Zhou, and M. Feckan, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (2012), 3389-3405.
  15. H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to fractional differential equations, J. Math. Anal. Appl. 328 (2007), 1075-1081.