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Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives

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  • Ravi P. Agarwal

    (Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
    Florida Institute of Technology, Distinguished University Professor of Mathematics, Melbourne, FL 32901, USA)

  • Snezhana Hristova

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria)

  • Donal O’Regan

    (School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 TK33 Galway, Ireland)

  • Ricardo Almeida

    (Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal)

Abstract

The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.

Suggested Citation

  • Ravi P. Agarwal & Snezhana Hristova & Donal O’Regan & Ricardo Almeida, 2021. "Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives," Mathematics, MDPI, vol. 9(16), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1979-:d:617267
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    References listed on IDEAS

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    1. Yang, Xujun & Li, Chuandong & Huang, Tingwen & Song, Qiankun, 2017. "Mittag–Leffler stability analysis of nonlinear fractional-order systems with impulses," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 416-422.
    2. Feifei Wang & Diyi Chen & Xinguang Zhang & Yonghong Wu, 2017. "Finite-time stability of a class of nonlinear fractional-order system with the discrete time delay," International Journal of Systems Science, Taylor & Francis Journals, vol. 48(5), pages 984-993, April.
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