Abstract :
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.
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