Nasze serwisy używają informacji zapisanych w plikach cookies. Korzystając z serwisu wyrażasz zgodę na używanie plików cookies zgodnie z aktualnymi ustawieniami przeglądarki, które możesz zmienić w dowolnej chwili. Więcej informacji odnośnie plików cookies.

Obowiązek informacyjny wynikający z Ustawy z dnia 16 listopada 2012 r. o zmianie ustawy – Prawo telekomunikacyjne oraz niektórych innych ustaw.

Wyłącz komunikat

 
 

Logowanie

Logowanie za pomocą Centralnej Usługi Uwierzytelniania PRz. Po zakończeniu pracy nie zapomnij zamknąć przeglądarki.

Journal of Mathematics and Applications

Journal of Mathematics and Applications
04/41, DOI: 10.7862/rf.2018.4

Nonlinear Fractional Differential Equations with Non-Instantaneous Impulses in Banach Spaces

Mouffak Benchohra, Mehdi Slimane

DOI: 10.7862/rf.2018.4

Abstract

This paper is devoted to study the existence of solutions for a class of initial value problems for non-instantaneous impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Mönch's fixed point theorem and the technique of measures of noncompactness.

Full text (pdf)

References

  1. S. Abbas, M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses, Appl. Math. Comput. 257 (2015) 190-198.
  2. S. Abbas, M. Benchohra, M.A. Darwish, New stability results for partial fractional differential inclusions with not instantaneous impulses, Frac. Calc. Appl. Anal. 18 (1) (2015) 172-191.
  3. S. Abbas, M. Benchohra, G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.
  4. S. Abbas, M. Benchohra, G.M. N'Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
  5. R.P. Agarwal, S. Hristova, D. O'Regan, Non-Instantaneous Impulses in Differential Equations, Springer, New York, 2017.
  6. R.P. Agarwal, M. Meehan, D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.
  7. R.R. Akhmerov, M.I. Kamenskii, A.S. Patapov, A.E. Rodkina, B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, trans. from the Russian by A. Iacob, Birkhäuser Verlag, Basel, 1992.
  8. J.C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces, Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid 79 (1985) 53-66.
  9. J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.
  10. J. Banaś, B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003) 1-6.
  11. J. Banaś, K. Sadarangani, On some measures of noncompactness in the space of continuous functions, Nonlinear Anal. 68 (2008) 377-383.
  12. D.D. Bainov, P.S. Simeonov, Systems with Impulse Effect, Horwood, Chichester, 1989.
  13. M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Vol 2, New York, 2006.
  14. M. Benchohra, J. Henderson, D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal. 12 (4) (2008) 419-428.
  15. D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers Group, Dordrecht, 1996.
  16. E. Hernández, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013) 1641-1649.
  17. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  18. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
  19. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, Worlds Scientific, Singapore, 1989.
  20. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980) 985-999.
  21. H. Mönch, G.F. Von Harten, On the Cauchy problem for ordinary differential equations in Banach spaces, Archiv. Math. Basel 39 (1982) 153-160.
  22. M. Pierri, D. O'Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput. 219 (2013) 6743-6749.
  23. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  24. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
  25. Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

About this Article

TITLE:
Nonlinear Fractional Differential Equations with Non-Instantaneous Impulses in Banach Spaces

AUTHORS:
Mouffak Benchohra (1)
Mehdi Slimane (2)

AUTHORS AFFILIATIONS:
(1) Laboratory of Mathematics, Djillali Liabes University of Sidi-Bel-Abbès, ALGERIA
(2) Laboratory of Mathematics, Djillali Liabes University of Sidi-Bel-Abbès, ALGERIA

JOURNAL:
Journal of Mathematics and Applications
04/41

KEY WORDS AND PHRASES:
Initial value problem; Impulses; Caputo fractional derivative; Measure of noncompactness; Fixed point; Banach space

FULL TEXT:
http://doi.prz.edu.pl/pl/pdf/jma/67

DOI:
10.7862/rf.2018.4

URL:
http://dx.doi.org/10.7862/rf.2018.4

RECEIVED:
2017-12-28

COPYRIGHT:
Publishing House of Rzeszow University of Technology Powstańców Warszawy 12, 35-959 Rzeszow

POLITECHNIKA RZESZOWSKA im. Ignacego Łukasiewicza; al. Powstańców Warszawy 12, 35-959 Rzeszów
tel.: +48 17 865 11 00, fax.: +48 17 854 12 60
Administrator serwisu: