Existence, uniqueness and controllability results of nonlinear neutral implicit $ \mathcal{ABC} $ fractional integro-differential equations with delay and impulses

Sivaranjani Ramasamy; Thangavelu Senthilprabu; Kulandhaivel Karthikeyan; Palanisamy Geetha; Saowaluck Chasreechai; Thanin Sitthiwirattham; Sivaranjani Ramasamy; Thangavelu Senthilprabu; Kulandhaivel Karthikeyan; Palanisamy Geetha; Saowaluck Chasreechai; Thanin Sitthiwirattham

doi:10.3934/math.2025200

AIMS Mathematics

2025, Volume 10, Issue 2: 4326-4354. doi: 10.3934/math.2025200

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Research article

Existence, uniqueness and controllability results of nonlinear neutral implicit $ \mathcal{ABC} $ fractional integro-differential equations with delay and impulses

1.
Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore 641 407, Tamilnadu, India
2.
Department of Mathematics, KPR College of Arts Science and Research, Coimbatore 641 407, Tamilnadu, India
3.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
4.
Research Group for Fractional Calculus Theory and Applications, Science and Technology Research Institute, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
5.
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand

Received: 07 January 2025 Revised: 18 February 2025 Accepted: 24 February 2025 Published: 28 February 2025
MSC : 34A09, 47H09, 34A37, 93B05, 34K20, 58C30

In this article, the necessary and sufficient conditions for the existence and uniqueness of the mild solutions for nonlinear neutral implicit integro-differential equations of non-integer order $ 0 < \alpha < 1 $ in the sense of $ \mathcal{ABC} $ derivative with impulses, delay, and integro initial conditions were established. The existence results were derived using the semi-group theory, measures of non-compactness, and the fixed-point theory in the sense of Arzel$ \grave{a} $–Ascoli theorem and Schauder's fixed-point theorem. We analyzed the controllability results of the proposed problem by incorporating the ideas of semi-group theory and fixed-point techniques. The Banach contraction principle was used to derive the uniqueness and controllability of the proposed problem. We provide an example to support the theoretical results.
- fractional derivative,
- implicit neutral differential equations,
- controllability,
- semi-group and fixed point theories
Citation: Sivaranjani Ramasamy, Thangavelu Senthilprabu, Kulandhaivel Karthikeyan, Palanisamy Geetha, Saowaluck Chasreechai, Thanin Sitthiwirattham. Existence, uniqueness and controllability results of nonlinear neutral implicit $ \mathcal{ABC} $ fractional integro-differential equations with delay and impulses[J]. AIMS Mathematics, 2025, 10(2): 4326-4354. doi: 10.3934/math.2025200

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Abstract

In this article, the necessary and sufficient conditions for the existence and uniqueness of the mild solutions for nonlinear neutral implicit integro-differential equations of non-integer order $ 0 < \alpha < 1 $ in the sense of $ \mathcal{ABC} $ derivative with impulses, delay, and integro initial conditions were established. The existence results were derived using the semi-group theory, measures of non-compactness, and the fixed-point theory in the sense of Arzel$ \grave{a} $–Ascoli theorem and Schauder's fixed-point theorem. We analyzed the controllability results of the proposed problem by incorporating the ideas of semi-group theory and fixed-point techniques. The Banach contraction principle was used to derive the uniqueness and controllability of the proposed problem. We provide an example to support the theoretical results.

References

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