Representation formula for solution of a functional equation with Volterra operator

Abstract

The following functional equation is under consideration, (0.1) L x = f with a linear continuous operator L , defined on the Banach space X 0 ( Ω 0 , Σ 0 , μ 0 ; Y 0 ) of functions x 0 : Ω 0 → Y 0 and having values in the Banach space X 2 ( Ω 2 , Σ 2 , μ 2 ; Y 2 ) of functions x 2 : Ω 2 → Y 2 . The peculiarity of X 0 is that the convergence of a sequence x n 0 ∈ X 0 , n = 1 , 2 , … , to the function x 0 ∈ X 0 in the norm of X 0 implies the convergence x n 0 ( s ) → x 0 ( s ) , s ∈ Ω 0 , μ 0 -almost everywhere. The assumption 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 ( Ω 1 , Σ 1 , μ 1 ; Y 1 ) × Y 0 (here X 1 ( Ω 1 , Σ 1 , μ 1 ; Y 1 ) is the Banach space of measurable functions x 1 : Ω 1 → Y 1 ). The representation X 0 = X 1 × 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.