Theory of Nonlinear Implicit Fractional Differential Equations

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


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.


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 



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.


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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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