Maximum principle for functional equations in the space of discontinuous functions of three variables (Q868779)

The author considers the functional equation \[ { u(t, x, y) = \left(T_{u} \right)(t, x, y) + f(t, x, y),\quad (t, x, y) \in [0, + \infty)^{3},} \tag{1} \] where \(T\) is a linear continuous operator from the space \(L^{\infty}_{[0, +\infty)^{3}}\) of all measurable essentially bounded real functions defined on \([0, + \infty)^{3}\) into itself, \(u \in L^{\infty}_{[0, +\infty)^{3}}\) is an unknown function and \(f \in L^{\infty}_{[0, +\infty)^{3}}\) is a given one. We say that \textit{the maximum principle} for equation (1) holds if for each \(f \in L^{\infty}_{[0, +\infty)^{3}}\) there exists a unique solution \(u \in L^{\infty}_{[0, +\infty)^{3}}\) of (1) and this solution is nonnegative (strongly positive) for nonnegative (strongly positive) \(f\). Among others the following two results are given. 1. If \(T\) is a positive operator, then the maximum principle for (1) holds if and only if \ the spectral radius of \ \(T\) is less than one. 2. If \ \(T\) may be represented as the difference of \ two positive operators \(T^{+}\) and \(T^{-}\) and the spectral radius of \ \(| T| = T^{+} + T^{-}\) is less than one, then a suitable maximum principle for (1) holds.