Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives
Abstract
:1. Introduction
2. Main Results
2.1. Statement of the Problem
2.2. Explicit Solution of the Impulsive Linear Fractional Equation
2.3. Mild Lower/Upper Solutions
2.4. Monotone-Iterative Technique
- 1.
- The functions are a mild lower solution and a mild upper solution of the PIVP (1), respectively, such that for ;
- 2.
- The function and there exist constants such that, for any , if then the inequality
- 3.
- The functions and there exist constants , such that, for any , if then the inequalities
- [a]
- The sequences and are defined by andand
- [b]
- For any the functions and are mild lower and mild upper solutions of PIVP (1), respectively;
- [c]
- The sequence is increasing, i.e., , for ;
- [d]
- The sequence is decreasing , i.e., , for , ;
- [e]
- The inequalityholds.
- [f]
- For any , the sequences and converge uniformly on to and , respectively, whereand for
- [g]
- The functions and are mild solutions of the PIVP (1) on and , where
2.5. Example
3. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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Agarwal, R.P.; Hristova, S.; O’Regan, D.; Almeida, R. Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives. Mathematics 2021, 9, 1979. https://doi.org/10.3390/math9161979
Agarwal RP, Hristova S, O’Regan D, Almeida R. Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives. Mathematics. 2021; 9(16):1979. https://doi.org/10.3390/math9161979
Chicago/Turabian StyleAgarwal, Ravi P., Snezhana Hristova, Donal O’Regan, and Ricardo Almeida. 2021. "Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives" Mathematics 9, no. 16: 1979. https://doi.org/10.3390/math9161979
APA StyleAgarwal, R. P., Hristova, S., O’Regan, D., & Almeida, R. (2021). Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives. Mathematics, 9(16), 1979. https://doi.org/10.3390/math9161979