On a Transformation of Integral Equations

摘要

Let E = E(a, b) be some Banach space of measurable functions on (a, b), I be the identity operator, and let (K) over cap be a Fredholm-type regular integral operator acting on E and (K) over cap (+/-) be its triangular parts. We consider the representation I - (K) over cap = (I - (K) over cap (-))(I - (U) over cap)(I - (K) over cap (+)), for some known classes of integral operators. In particular,we show that under certain conditions the operator (U) over cap is positive and its spectral radius satisfies the condition r((U) over cap) 1. Also, we give some possible applications of the representation.