THE TWO-SIDED ESTIMATES OF THE FREDHOLM RADIUS AND COMPACTNESS CONDITIONS FOR THE OPERATOR ASSOCIATED WITH A SECOND-ORDER DIFFERENTIAL EQUATION

Abstract

In this paper we consider the properties of the resolvent of a linear operator corresponding to a degenerate singular second-order differential equation with variable coefficients, considered in the Lebesgue space. The singularity of the specified differential equation means that it is defined in a noncompact domain - on the whole set of real numbers, and its coefficients are unbounded functions. The conditions for the compactness of the resolvent were obtained, as well as a double-sided estimate of its fredgolm radius. The previously known compactness conditions of the resolvent were obtained under the assumption that the intermediate-term of the differential operator either is missing or, in the operator sense, is subordinate to the sum of the extreme terms. In the current paper these conditions are not met due to the rapid growth at infinity of the intermediate coefficient of the differential equation, and the minor coefficient can change sign. The property of compactness of the resolvent allows, in particular, to justify the process of finding an approximate solution of the associated equation. The Fredholm radius of a bounded operator characterizes its closeness to the Fredholm operator. The operator coefficients are assumed to be smooth functions, but we do not impose any constraints on their derivatives. The result on the invertibility of the operator and the estimation of its maximum regularity obtained by the authors earlier is essentially used in this paper.

Keywords

• second-order differential operator
• fredholm radius
• resolvent
• compactness
• differential equation in an unbounded domain
• differential operator with unbounded coefficients