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Theory of Nonlinear Implicit Fractional Differential Equations

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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.

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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.

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Authors and Affiliations

  1. Department of Mathematics, Shivaji University, Kolhapur, Maharashtra, 416 004, India

    Kishor D. Kucche

  2. Department of Mathematical Analysis, University of Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Juan J. Nieto & Venktesh Venktesh

  3. Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Juan J. Nieto

Authors
  1. Kishor D. Kucche
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  2. Juan J. Nieto
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  3. Venktesh Venktesh
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Correspondence to Kishor D. Kucche.

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Kucche, K.D., Nieto, J.J. & Venktesh, V. Theory of Nonlinear Implicit Fractional Differential Equations. Differ Equ Dyn Syst 28, 1–17 (2020). https://doi.org/10.1007/s12591-016-0297-7

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