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

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.

[1]  Michal Fečkan,et al.  On the new concept of solutions and existence results for impulsive fractional evolution equations , 2011 .

[2]  Snezhana G. Hristova,et al.  Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses , 2021, Symmetry.

[3]  Qiankun Song,et al.  Stability analysis of nonlinear fractional-order systems with variable-time impulses , 2017, J. Frankl. Inst..

[4]  Ivanka M. Stamova,et al.  Design of impulsive controllers and impulsive control strategy for the Mittag-Leffler stability behavior of fractional gene regulatory networks , 2020, Neurocomputing.

[5]  H. Alzumi,et al.  Existence results for nonlinear fractional boundary value problem involving generalized proportional derivative , 2019, Advances in Difference Equations.

[6]  M. Abbas Existence results and the Ulam Stability for fractional differential equations with hybrid proportional-Caputo derivatives , 2020, Journal of Nonlinear Functional Analysis.

[7]  Dumitru Baleanu,et al.  Analysis and numerical solution of the generalized proportional fractional Cauchy problem , 2021 .

[8]  Maria Alessandra Ragusa,et al.  On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function , 2021, Symmetry.

[9]  Non‐instantaneous impulsive fractional integro‐differential equations with proportional fractional derivatives with respect to another function , 2021, Mathematical Methods in the Applied Sciences.

[10]  T. Abdeljawad,et al.  Generalized fractional derivatives generated by a class of local proportional derivatives , 2017 .