Representation formula for solution of a functional equation with Volterra operator

The following functional equation is under consideration, Lx = f (0.1) with a linear continuous operator L, defined on the Banach space X0(Ω0,Σ0,μ0;Y0) of functions x0 :Ω0 → Y0 and having values in the Banach space X2(Ω2,Σ2,μ2;Y2) of functions x2 :Ω2 → Y2. The peculiarity of X0 is that the convergence of a sequence x0 n ∈ X0, n = 1, 2, . . . , to the function x0 ∈ X0 in the norm of X0 implies the convergence x0 n(s)→x0(s), s ∈ Ω0, μ0-almost everywhere. The assumption on the space X2 is that it is an ideal space. The suggested representation of solution to (0.1) is based on a notion of the Volterra property together with a special presentation of the equation using an isomorphism between X0 and the direct product X1(Ω1,Σ1,μ1;Y1) × Y0 (here X1(Ω1,Σ1,μ1;Y1) is the Banach space of measurable functions x1 :Ω1 →Y1). The representation X0 = X1 × Y0 leads to a decomposition of L:X0 →X2 for the pair of operators Q:X1 →X2 and A:Y0 →X2. A series of basic properties of (0.1) is implied by the properties of operator Q. © 2007 Elsevier Inc. All rights reserved