Singular perturbations of integral equations with degenerate kernels
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 \(\varepsilon\to 0\).
scientific article; zbMATH DE number 872692
scientific article