Integral operators in spaces of summable functions

Integrable solutions of Hammerstein and Urysohn integral equations

“…LEMMA 1 (Krasnosel'ski et al [10]). If a sequence {x n } C L 1 converges weakly to x € L l and is compact in measure then it converges in measure to x.…”

“…These equations have been studied in several papers and monographs (see for example Krasnosel'skii et al [10], Zabrejko et al [14], Appell [1,2] and references therein).…”

“…Namely, it was assumed that F is the so-called "improving" operator while K was supposed to be a regular operator. Conditions of this type were first formulated by Zabrejko and Pustyl'nik [15] (see also [10,14,2] Throughout the remainder of this section we shall discuss the solvability of the Urysohn equation (2). Denote by G the operator associated with the right-hand side of this equation a n d by U the Urysohn operator…”

“…Then the continuity of the Urysohn operator U follows from the known majorant principle given in the book in Krasnosel'skii et al [10], for example. Particularly, as the function p we can take a function determining a Hammerstein operator [14].…”

A stationary criticality problem in generalL p -space for energy dependent neutron transport in cylindrical geometry

“…This permits us to interpolate the continuity property of the operator K and the compactness property of the operator K 2 and to prove that K is bounded and K 2 is compact as operators acting from L v to L v, with 1 _< p < 0% ( [13], Th. 3.10, page 57).…”