Existence, compactness, estimates of eigenvalues and s-numbers of a resolvent for a linear singular operator of the Korteweg-de Vries type

Abstract

In this paper, we consider a linear operator of the Korteweg-de Vries type Lu = ?u ?y + R2(y)?3u ?x3 + R1(y)?u ?x + R0(y)u initially defined on C? 0,?(??), where ?? = {(x, y) : ?? ? x ? ?,?? < y < ?}. C? 0,?(??) is a set of infinitely differentiable compactly supported function with respect to a variable y and satisfying the conditions: u(i) x (??, y) = u(i) x (?, y), i = 0, 1, 2. With respect to the coefficients of the operator L , we assume that these are continuous functions in R(??,+?) and strongly growing functions at infinity. In this paper, we proved that there exists a bounded inverse operator and found a condition that ensures the compactness of the resolvent under some restrictions on the coefficients in addition to the above conditions. Also, two-sided estimates of singular numbers (s-numbers) are obtained and an example is given of how these estimates allow finding estimates of the eigenvalues of the considered operator.