ABSTRACT
We point out that the equations (16.1) and (16.5) may be transformed into partial integral equations by applying Fourier or Laplace transforms. For example, suppose that we are interested in bounded solutions x of the Barbashin equation (16.1) with the property that both x and ax/at belong to the space with mixed norm [L1 +- Xl, where X is some regular ideal space over [a, b) and L1 = L1(Ht). This means that we want x to satisfy
1+00-00 IIx(t,·)lIx dt < 00, Assume that both the multiplier c and the kernel kin (16.1) do not depend on t, the function s ~ c(s) is bounded, and the function (s,D') ~ k(s,D') belongs to [X +- X1, where X' is the associate space to X (see Subsection 4.1). Taking now the Fourier transform of the corresponding equation
Bx(t s) l'th = c(s)x(t,s) + II k(.9,D')x(t,D')dD'+f(t,s) with respect to the variable t leads to the equation
A 1+00 -i(tx(e,s) = -00 z(t,s)e dt, A 1+00 -i(tfee,s) = -00 f(t,s)e dt. This is an integral equation for the unknown function z:IR. x [a, b] ....
~. Similarly, if both u and 8ulBI{) in (16.5) belong to [L1 +- Xl, where X is some regular ideal space over T x S, the coefficient c in (16.5) is bounded, and the kernels I, m and n do not depend on I{) and generate regular integral operators in the space X, one may transform the
