Boundary Value Problems for Integral Equations With Operator Measures

Abstract

Abstract. We consider integral equations with operator measures on a segment in the infinite-dimensional case. These measures are defined on Borel sets of the segment and take values in the set of linear bounded operators acting in a separable Hilbert space. We prove that these equations have unique solutions and we construct a family of evolution operators. We apply the obtained results to the study of linear relations generated by an integral equation and boundary conditions. In terms of boundary values, we obtain necessary and sufficient conditions under which these relations T possess the properties: T is a closed relation; T is an invertible relation; the kernel of T is finite-dimensional; the range of T is closed; T is a continuously invertible relation and others. We give examples to illustrate the obtained results.

Dissipative Extensions of Linear Relations Generated by Integral Equations with Operator Measures

“…(2.4) Equations (2.3) and (2.4) have unique solutions (see [8]). By W , denote the operator solution of the equation…”

“…Proof. Equation (2.8) has a unique solution (see [8]). It is enough to prove that if we substitute the function from the right-hand side of (2.9) instead of y in equation (2.8), then we get the identity.…”

“…The study of integral equation (1.1) differs essentially from the study of differential equations by the presence of the following features: i) a representation of a solution of equation (1.1) using an evolutional family of operators is possible if the measures p, m do not have common single-point atoms (see [8]);…”

Generalized resolvents of operators generated by integral equations

“…In general, a solution of (3) can be non-extendable to the left (see [7]). However, if the measure p in (3) is continuous, then a solution can be extended to the left up to the point a and this extension is unique.…”

Invertible linear relations generated by integral equations with operator measures

“…The study of integral equation (1) differs essentially from the study of differential equations by the presence of the following features: i) a representation of a solution of equation (1) using an evolutional family of operators is possible if the measures p, m have not common single-point atoms (see [6]); ii) the Lagrange formula contains summands relating to single-point atoms of the measures p, m (see [7]). Note that this work partially corrects the errors made in the article [8].…”