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Generalized Ulam–Hyers–Rassias Stability of Solution for the Caputo Fractional Non-instantaneous Impulsive Integro-differential Equation and Its Application to Fractional RLC Circuit

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Abstract

This article studies the existence and stability of the mild solution of the Caputo fractional non-instantaneous impulsive integro-differential equation in the Ulam–Hyers–Rassias sense. In order to establish a unique solution, we consider some appropriate assumptions and a specific space of continuous functions. By showing a map from it into it to be a contraction map, the uniqueness is established. The same map along with a piecewise continuous function is utilized for proving the generalized Ulam–Hyers–Rassias stability with the help of Banach fixed-point theorem. We use our results to estimate the bound for the difference between the fractional-order and the integer-order non-instantaneous impulsive RLC circuit current. Three tables presented at the end clearly show how the bounds can be estimated and that the bound mainly depends on the bandwidth of the RLC circuit.

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Acknowledgements

The first author expresses his gratitude to Indian Institute of Technology Guwahati, India, for providing him a research fellowship to carry out research toward his PhD. Both authors are immensely grateful to the three learned Reviewers for spending time in going through the manuscript and for their insightful suggestions which have led to a much better version of the manuscript. The Editor-in-Chief Prof. M.N.S. Swamy is profusely thanked for his understanding of the manuscript and for allowing its revision.

Funding

The first author received junior research fellowship from Indian Institute of Technology Guwahati for the period 2019–2020 and senior research fellowship for the period 2021–present.

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  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Matap Shankar & Swaroop Nandan Bora

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  1. Matap Shankar
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  2. Swaroop Nandan Bora
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Matap Shankar contributed to conceptualization and methodology; Matap Shankar and Swaroop Nandan Bora contributed to formal analysis and investigation, writing—original draft preparation, and writing—review and editing.

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Correspondence to Swaroop Nandan Bora.

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Shankar, M., Bora, S.N. Generalized Ulam–Hyers–Rassias Stability of Solution for the Caputo Fractional Non-instantaneous Impulsive Integro-differential Equation and Its Application to Fractional RLC Circuit. Circuits Syst Signal Process 42, 1959–1983 (2023). https://doi.org/10.1007/s00034-022-02217-x

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