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Theory of Nonlinear Implicit Fractional Differential Equations

  • Kishor D. Kucche
  • Juan J. Nieto
  • Venktesh Venktesh
Original Research

Abstract

This paper deals with the basic theory of nonlinear implicit fractional differential equations involving Caputo fractional derivative. In particular, we investigate the existence and interval of existence of solutions, uniqueness, continuous dependence of solutions on initial conditions, estimates on solutions and continuous dependence on parameters and functions involved in the equations. Further, we study \(\varepsilon \)-approximate solution of the implicit fractional differential equations.

Keywords

Implicit fractional differential equations Caputo fractional derivative Existence and uniqueness Continuous dependence \(\varepsilon \)-approximate solution Fixed point theorem Integral inequality 

Mathematics Subject Classification

26A33 34A12 34A08 39B12 

Notes

Acknowledgments

We thank anonymous referees for their valuable comments to improve our paper. The first author was supported by UGC, New Delhi, through the Minor Research Project F. No. 42-997/2013(SR). The work of Juan J. Nieto has been partially supported by the Ministerio de Economia y Competitividad of Spain under Grant MTM2013–43014–P, Xunta de Galicia under grants R2014/002 and GRC 2015/004, and co-financed by the European Community fund FEDER.

References

  1. 1.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  2. 2.
    Miller, K.S., Ross, B.: An introduction to the fractional calculus and differential equations. Wiley, New York (1993)zbMATHGoogle Scholar
  3. 3.
    Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  4. 4.
    Samko, S., Kilbas, A., Marichev, O.: Fractional integrals and derivatives. Gordon and Breach, Yverdon (1993)zbMATHGoogle Scholar
  5. 5.
    Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta. Appl. Math. 109, 973–1033 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Agarwal, R. P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative, advances in difference equations 47, (2009) Article ID 981728Google Scholar
  7. 7.
    Diethelm, K.: The analysis of fractional differential equations: an application-oriented exposition using differential operators of caputo type. Springer-Verlag, New York (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear. Anal. 69, 2677–2682 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lakshmikantham, V., Vatsala, A.S.: General uniqueness and mono- tone iterative technique for fractional differential equations. Appl. Math. Lett. 21, 828–834 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293, 511–522 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Daftardar-Gejji, V., Jafari, H.: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. J. Math. Anal. Appl. 328, 1026–1033 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhou, Y.: Existence and uniqueness of solutions for a system of fractional, differential equations. Fract. Calc. Appl. Anal. 12(2), 195–204 (2003)MathSciNetGoogle Scholar
  14. 14.
    Wang, J., Zhou, Y.: Existence of mild solutions for fractional delay evolution systems. Appl. Math. Comput. 218, 357–367 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang, J., Zhou, Y., Lin, Z.: On a new class of impulsive fractional differential equations. Appl. Math. Comput. 242, 649–657 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Nieto, J.J., Ouahab, A., Venktesh, V.: Implicit fractional differential equations via the Liouville Caputo derivative. Mathematics 3, 398–411 (2015)CrossRefzbMATHGoogle Scholar
  17. 17.
    Benchohra, M., Souid, M.: Integrable solutions for implicit fractional order differential equations. TJMM 6(2), 101–107 (2014)MathSciNetGoogle Scholar
  18. 18.
    Benchohraa, M., Bouriaha, S.: Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order. Moroccan. J. Pure. Appl. Anal. 1(1), 22–37 (2015)Google Scholar
  19. 19.
    Tidke, H.L.: Some theorems on fractional semilinear evolution equations. J. Appl. Anal. 18, 209–224 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tidke, H.L., Dhakne, M.B.: Approximate solutions to nonlinear mixed type integrodifferential equation with nonlocal condition. Commun. Appl. Nonlinear. Anal. 17(2), 35–44 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Luchko, Y., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta. Math. Vietnamica. 24(2), 207–233 (1999)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Henry, D.: Geometric theory of semilinear parabolic partial differential equations. Springer, Berlin, Germany (1989)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2016

Authors and Affiliations

  • Kishor D. Kucche
    • 1
  • Juan J. Nieto
    • 2
    • 3
  • Venktesh Venktesh
    • 2
  1. 1.Department of MathematicsShivaji UniversityKolhapurIndia
  2. 2.Department of Mathematical AnalysisUniversity of Santiago de CompostelaSantiago de CompostelaSpain
  3. 3.Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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