Solution space characterisation of perturbed linear functional and integrodifferential Volterra convolution equations: Ces\`aro limits

TL;DR

This paper characterizes the solution space of perturbed linear Volterra convolution equations, revealing that Cesàro means of solutions can converge even when perturbations diverge, and extends results to functional differential equations.

Contribution

It provides a detailed analysis of solution behavior under perturbations, including Cesàro mean convergence criteria and explicit limit characterization, extending to functional differential equations.

Findings

Cesàro mean of solutions can converge despite divergence of perturbations

Explicit limits of solutions are identified in terms of problem data

Results apply to perturbed linear functional differential equations

Abstract

In this article we discuss the requirements needed in order to characterise the solution space of perturbed linear integro-differential Volterra convolution equations. We highlight in general how the pointwise behaviour of perturbation functions does not necessarily propagate through to the solution which the classical literature seems to suggest. To illustrate this general idea we show the Ces\`aro mean of the solution can converge even in cases when the Ces\`aro mean of the perturbation function diverges. Furthermore we provide a characterisation of when such convergence takes place and explicitly identify the limit in terms of the problem data. Additionally we prove how all results can also be applied to perturbed linear functional differential equations.