RIGOROUS RESULTS ON THE
ASYMPTOTIC SOLUTIONS OF
SINGULARLY PERTURBED
NONLINEAR VOLTERRA
INTEGRAL EQUATIONS

A.M. BIJURA

Abstract:

This paper studies singularly perturbed Volterra integral equations of the form

$$\epsilon u(t)=f(t;\epsilon )+\int^t_0g(t,s,u(s))\,ds,\quad 0\le t\le T,$$

where $\epsilon$ is a small parameter. The function $f(t;\epsilon )$ is defined for $0\le t\le T$ and $g(t,s,u)$ for $0\le s\le t\le T$. There are many existence and uniqueness results known that ensure that a unique continuous solution $u(t;\epsilon )$ exists for all small $\epsilon >0$. The aim is to find asymptotic approximations to these solutions and rigorously prove the asymptotic correctness. This work is restricted to problems where there is an initial layer; various hypotheses are placed on $g$ to exclude other behaviors.