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 X-0(Omega(0), Sigma(0), mu(0); Y-0) of functions x(0): Omega(0) -> Y-0 and having values in the Banach space X-2(Omega(2), Sigma(2), mu(2); Y-2) of functions x2: S22 -* Y2The peculiarity of X-0 is that the convergence of a sequence x(n)(0) epsilon X-0, n = 1, 2,..., to the function x(0) epsilon X-0 in the norm of X-0 implies the convergence x(n)(0)(s) -> x(0)(s), s epsilon Omega(0), mu(0)-almost everywhere. The assumption n on the space X-2 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 X-0 and the direct product X-1(Omega(1), Sigma(1), mu(1); Y-1) x Y-0 (here X-1(Omega(1), Sigma(1), mu(1); Y-1) is the Banach space of measurable functions x(1) : Omega(1) -> Y-1). The representation X-0 = X-1 x Y-0 leads to a decomposition of L: X-0 -> X-2 for the pair of operators Q: X-1 -> X-2 and A: Y-0 -> X-2. A series of basic properties of (0. 1) is implied by the properties of operator Q. (c) 2007 Elsevier Inc. All rights reserved.