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A Fractional Bihari Inequality and Some Applications to Fractional Differential Equations and Stochastic Equations

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Abstract

The purpose of this paper is to present a new version of the Bihari inequality with singular kernel and give a simple proof of the fractional Gronwall lemma. Our new ideas rest on the use of Young’s and Hölder’s inequalities to simplify the complex inequalities. Based on this new type of Bihari inequality we can relax many results of fractional differential equations and inclusions and stochastic differential equations. Also, the obtained inequalities can be used to analyze a specific class of fractional differential equations, both linear and nonlinear. Using the Caputo fractional derivative, the study of an initial valued problem for a fractional differential equation provides some topological proprieties for the solution set, and shows it is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser’s theorem on the solution structure of the ordinary differential equation and relax some results about the fractional differential equation. Also, we establish existence results for Caputo fractional stochastic differential equations. Finally, we study the existence of solution for fractional differential inclusion in Banach lattice.

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Acknowledgements

The research of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigacin (AEI) of Spain, cofinanced by the European Fund for Regional Development (FEDER) corresponding to the 2014-2020 multiyear financial framework, project MTM2016-75140-P; and by Xunta de Galicia under Grant ED431C 2019/02. The research of A. Ouahab has been partially supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria. The authors would like to thank the anonymous referees for their careful reading of the manuscript and pertinent comments; their constructive suggestions substantially improved the quality of the work.

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Authors and Affiliations

  1. Laboratory of Mathematics, University of Djillali Liabes, PO Box 89, 22000, Sidi-Bel-Abbès, Algeria

    A. Ouaddah

  2. Department of Mathematics, Baylor University, Waco, TX, 76798-7328, USA

    J. Henderson

  3. Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    J. J. Nieto

  4. Laboratory of Mathematics, Sidi-Bel-Abbès University, PO Box 89, 22000, Sidi-Bel-Abbès, Algeria

    A. Ouahab

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Ouaddah, A., Henderson, J., Nieto, J.J. et al. A Fractional Bihari Inequality and Some Applications to Fractional Differential Equations and Stochastic Equations. Mediterr. J. Math. 18, 242 (2021). https://doi.org/10.1007/s00009-021-01917-z

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