Solutions with exponential character to a linear functional equation and roots of its characteristic equation
Continuing his work [Exponential behaviour of solutions of a linear functional equation. Aequationes Math., to appear] the author offers, among others, the following result: Let \(\lambda\) be a real number, \((S,\Sigma,\sigma)\) a measure space, \(\sigma\) being a finite measure, and \(M: S\to\mathbb{R}\) be a \(\Sigma\)-measurable bounded function, for which the \(\sigma\)-measure of the set where \(M\neq 0\) is positive. If, for some solution \(\phi\) of (1)\(\;\phi(x)=\int_S\phi(x+M(s))\sigma(ds),\;\) there exists a finite limit (2)\(\;\lim_{x\to\infty}\;\phi(x)e^{-\lambda x}\;/x\neq 0,\) then (3)\(\;\int_S e^{\lambda M(s)}\sigma(ds)=1.\;\) If, on the other hand, for some regular solution \(\phi\) of (1), a finite limit (2) exists, then (3) is satisfied, \(\int_S M(s)e^{\lambda M(s)}\sigma(ds)\leq 0,\;\sigma(M<0)>0\) and \(\sigma(M>0)>0\) hold.
linear integral functional equations and inequalities
scientific article; zbMATH DE number 5083282