Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives

Ravi Agarwal; Snezhana Hristova; Donal O'Regan; Ravi Agarwal; Snezhana Hristova; Donal O'Regan

doi:10.3934/math.2022164

AIMS Mathematics

2022, Volume 7, Issue 2: 2973-2988. doi: 10.3934/math.2022164

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Research article

Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives

1.
Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 78363, USA
2.
Faculty of Mathematics and Informatics, Plovdiv University, Plovdiv 4000, Bulgaria
3.
School of Mathematical and Statistical Sciences, National University of Ireland, Galway, Ireland

Received: 17 September 2021 Accepted: 15 November 2021 Published: 23 November 2021
MSC : 34A08, 34A12, 34A38

Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.
- Riemann-Liouville fractional derivative,
- impulses,
- Riemann-Liouville integral,
- boundary value problem
Citation: Ravi Agarwal, Snezhana Hristova, Donal O'Regan. Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives[J]. AIMS Mathematics, 2022, 7(2): 2973-2988. doi: 10.3934/math.2022164

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Abstract

Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.

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