Stability properties of quadrature-based approximations for integral equations with delay terms

Eleonora Messina; Claudia Panico; Eleonora Messina; Claudia Panico

doi:10.3934/mbe.2026069

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 7: 1886-1907. doi: 10.3934/mbe.2026069

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Research article

Stability properties of quadrature-based approximations for integral equations with delay terms

Eleonora Messina ^,,
Claudia Panico

Department of Mathematics and Applications, University of Naples Federico Ⅱ, via Cintia, I-80126 Naples, Italy

Received: 22 December 2025 Revised: 03 April 2026 Accepted: 15 May 2026 Published: 30 June 2026

We study a class of linear integral equations involving both a discrete delay and a bounded distributed delay. Such equations, as well as their non-linear extensions, arise in the mathematical modelling of biological systems and in control theory. The equations are discretised using quadrature-based methods, and the resulting discrete schemes are analysed with respect to their stability properties. Sufficient conditions are derived under which the discrete formulation preserves the stability characteristics of the underlying continuous problem.
- Volterra integral equations,
- numerical stability,
- distributed delay,
- quadrature methods,
- stability regions,
- integral delay systems
Citation: Eleonora Messina, Claudia Panico. Stability properties of quadrature-based approximations for integral equations with delay terms[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 1886-1907. doi: 10.3934/mbe.2026069

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Abstract

We study a class of linear integral equations involving both a discrete delay and a bounded distributed delay. Such equations, as well as their non-linear extensions, arise in the mathematical modelling of biological systems and in control theory. The equations are discretised using quadrature-based methods, and the resulting discrete schemes are analysed with respect to their stability properties. Sufficient conditions are derived under which the discrete formulation preserves the stability characteristics of the underlying continuous problem.

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