Stability analysis through the Bielecki metric to nonlinear fractional integral equations of $ n $-product operators

Supriya Kumar Paul; Lakshmi Narayan Mishra; Supriya Kumar Paul; Lakshmi Narayan Mishra

doi:10.3934/math.2024377

AIMS Mathematics

2024, Volume 9, Issue 4: 7770-7790. doi: 10.3934/math.2024377

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Research article

Stability analysis through the Bielecki metric to nonlinear fractional integral equations of $ n $-product operators

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India

Received: 08 November 2023 Revised: 26 December 2023 Accepted: 04 January 2024 Published: 23 February 2024
MSC : 26A33, 45M10, 47H10

This work is devoted to the analysis of Hyers, Ulam, and Rassias types of stabilities for nonlinear fractional integral equations with $ n $-product operators. In some special cases, our considered integral equation is related to an integral equation which arises in the study of the spread of an infectious disease that does not induce permanent immunity. $ n $-product operators are described here in the sense of Riemann-Liouville fractional integrals of order $ \sigma_i \in (0, 1] $ for $ i\in \{1, 2, \dots, n\} $. Sufficient conditions are provided to ensure Hyers-Ulam, $ \lambda $-semi-Hyers-Ulam, and Hyers-Ulam-Rassias stabilities in the space of continuous real-valued functions defined on the interval $ [0, a] $, where $ 0 < a < \infty $. Those conditions are established by applying the concept of fixed-point arguments within the framework of the Bielecki metric and its generalizations. Two examples are discussed to illustrate the established results.
- Hyers-Ulam stability,
- $ \lambda $-semi-Hyers-Ulam stability,
- Hyers-Ulam-Rassias stability,
- fractional integral equation,
- Bielecki metric
Citation: Supriya Kumar Paul, Lakshmi Narayan Mishra. Stability analysis through the Bielecki metric to nonlinear fractional integral equations of $ n $-product operators[J]. AIMS Mathematics, 2024, 9(4): 7770-7790. doi: 10.3934/math.2024377

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Abstract

This work is devoted to the analysis of Hyers, Ulam, and Rassias types of stabilities for nonlinear fractional integral equations with $ n $-product operators. In some special cases, our considered integral equation is related to an integral equation which arises in the study of the spread of an infectious disease that does not induce permanent immunity. $ n $-product operators are described here in the sense of Riemann-Liouville fractional integrals of order $ \sigma_i \in (0, 1] $ for $ i\in \{1, 2, \dots, n\} $. Sufficient conditions are provided to ensure Hyers-Ulam, $ \lambda $-semi-Hyers-Ulam, and Hyers-Ulam-Rassias stabilities in the space of continuous real-valued functions defined on the interval $ [0, a] $, where $ 0 < a < \infty $. Those conditions are established by applying the concept of fixed-point arguments within the framework of the Bielecki metric and its generalizations. Two examples are discussed to illustrate the established results.

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