Asymptotics of the solutions to singularly perturbed integral equations (Q1310870)

The authors study the asymptotics of \(h_ \varepsilon\) as \(\varepsilon \to 0\), where \(h_ \varepsilon\) is a solution of the integral equation \((*)\) \(\varepsilon h_ \varepsilon+Rh_ \varepsilon=f\), \(\varepsilon>0\), \(Rh_ \varepsilon(x)=\int^ \beta_ \alpha R(x-y)h_ \varepsilon(y)dy\) with \(R(x)=P(D)G(x)\), \(P(D)=\sum^ p_{j=0}a_ jD^ j\), \(D=d/dx\), \(Q(D)G(x)=\delta(x)\), \(Q(D)=\sum^ q_{j=0} b_ jD^ j\), \(0 \leq p<q\), \(a_ j\) and \(b_ j\) are constants and \(\delta(x)\) is the delta function. It is assumed that the equation \((*)\) admits a unique solution for small \(\varepsilon>0\). A good number of examples, which form the central part of the paper, is presented to illustrate the results.