A study on the existence results of boundary value problems of fractional relaxation integro-differential equations with impulsive and delay conditions in Banach spaces

Saowaluck Chasreechai; Sadhasivam Poornima; Panjaiyan Karthikeyann; Kulandhaivel Karthikeyan; Anoop Kumar; Kirti Kaushik; Thanin Sitthiwirattham; Saowaluck Chasreechai; Sadhasivam Poornima; Panjaiyan Karthikeyann; Kulandhaivel Karthikeyan; Anoop Kumar; Kirti Kaushik; Thanin Sitthiwirattham

doi:10.3934/math.2024563

AIMS Mathematics

2024, Volume 9, Issue 5: 11468-11485. doi: 10.3934/math.2024563

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A study on the existence results of boundary value problems of fractional relaxation integro-differential equations with impulsive and delay conditions in Banach spaces

1.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
2.
Research Group for Fractional Calculus Theory and Applications, Science and Technology Research Institute, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
3.
Department of Mathematics, Sri Vasavi College, Erode 638136, India
4.
Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore 641407, Tamil Nadu, India
5.
Department of Mathematics and Statistics, Central Univeristy of Punjab, Bathinda, Punjab, India
6.
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand

Received: 10 January 2024 Revised: 10 March 2024 Accepted: 13 March 2024 Published: 25 March 2024
MSC : 26A33, 32C25, 34A12, 34K45

The aim of this paper was to provide systematic approaches to study the existence of results for the system fractional relaxation integro-differential equations. Applied problems require definitions of fractional derivatives, allowing the utilization of physically interpretable boundary conditions. Impulsive conditions serve as basic conditions to study the dynamic processes that are subject to sudden changes in their state. In the process, we converted the given fractional differential equations into an equivalent integral equation. We constructed appropriate mappings and employed the Schaefer's fixed-point theorem and the Banach fixed-point theorem to show the existence of a unique solution. We presented an example to show the applicability of our results.
- Riemann-Liouville fractional derivative,
- fractional relaxation impulsive integro differential equations,
- Liouville-Caputo fractional derivative,
- existence,
- uniqueness,
- delay,
- fixed point
Citation: Saowaluck Chasreechai, Sadhasivam Poornima, Panjaiyan Karthikeyann, Kulandhaivel Karthikeyan, Anoop Kumar, Kirti Kaushik, Thanin Sitthiwirattham. A study on the existence results of boundary value problems of fractional relaxation integro-differential equations with impulsive and delay conditions in Banach spaces[J]. AIMS Mathematics, 2024, 9(5): 11468-11485. doi: 10.3934/math.2024563

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Abstract

The aim of this paper was to provide systematic approaches to study the existence of results for the system fractional relaxation integro-differential equations. Applied problems require definitions of fractional derivatives, allowing the utilization of physically interpretable boundary conditions. Impulsive conditions serve as basic conditions to study the dynamic processes that are subject to sudden changes in their state. In the process, we converted the given fractional differential equations into an equivalent integral equation. We constructed appropriate mappings and employed the Schaefer's fixed-point theorem and the Banach fixed-point theorem to show the existence of a unique solution. We presented an example to show the applicability of our results.

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