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Rigorous results on the asymptotic solutions of singularly perturbed nonlinear Volterra integral equations. (English) Zbl 1041.45007

Summary: The author studies singularly perturbed Volterra integral equations of the form \[ \varepsilon u(t)=f(t;\varepsilon)+ \int^t_0 g\bigl (t,s,u(s)\bigr)ds,\;0\leq t\leq T, \] where \(\varepsilon\) is a small parameter. The function \(f(t; \varepsilon)\) is defined for \(0\leq t\leq T\) and \(g(t,s,u)\) for \(0\leq s\leq t\leq T\). There are many existence and uniqueness results known that ensure that a unique continuous solution \(u(t;\varepsilon)\) exists for all small \(\varepsilon >0\). 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.

MSC:

45G10 Other nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
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