Singular perturbations of integral equations with degenerate kernels
type: Item
instance of: scholarly article
title: Singular perturbations of integral equations with degenerate kernels
zbMATH Open document ID: 0851.45001
DOI: 10.1007/BF01204605
author: C. A. Shubin
published in: Integral Equations and Operator Theory
publication date: 1996-11-26T00:00:00Z
review text: Let \(K\) be a kernel on \([a, b] \times [a, b]\), having the form \[ K(x, y)= K_1 (x, y) \text{ if } y<x \qquad \text{and} \qquad =K_2 (x, y) \text{ if } x< y, \] where \(K_1\) and \(K_2\) are smooth on \([a, b] \times [a, b]\) with \[ K_1 (x, x^-)- K_2 (x, x^+)= A(x), \qquad A\in C^\infty [a, b], \] i.e. \(K\) has a jump discontinuity along the diagonal. Let \(\varepsilon\) be a small positive parameter. With this, the author studies the singularly perturbed Fredholm equation of the second kind \[ \varepsilon u+ \int^b_a K(\cdot, y) u(y) dy=f \] on \((a, b)\). Setting \(\varepsilon =0\), one obtains the unperturbed equation \[ \int^b_a K(\cdot, y) u(y) dy =f \] which is an integral equation of the first kind. The author provides sufficient conditions for existence and uniqueness of solutions \(u_\varepsilon\) to the above perturbed equation. The behavior of these solutions \(u_\varepsilon\) is also considered. In the nondegenerate case when \(A\) does not vanish on \([a, b]\), \(u_\varepsilon\) tends, as \(\varepsilon\to 0\), to the solution \(u_0\) of the unperturbed equation in some Sobolev norm. This is not true in the degenerate case when \(A\) vanishes at some point of \([a, b]\), where the behavior of \(u_\varepsilon\) as \(\varepsilon\to 0\) is quite complicated. However, with the assumption that \(A\) vanishes to order \(k\) at one point of \([a, b]\), the author shows that under suitable hypotheses, the solution \(u_\varepsilon\) of the perturbed equation converges to the solution \(u_0\) of the unperturbed equation in \(L^2 [a, b]\) as \(\varepsilon\to 0\).
Mathematics Subject Classification ID: 45B05
zbMATH DE Number: 872692
zbMATH Keywords: singular perturbations
zbMATH Keywords: degenerate kernel
zbMATH Keywords: Fredholm equation of the second kind
zbMATH Keywords: integral equation of the first kind
reviewed by: Jacob Burbea
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 872692