Nonlinear perturbations of positive semigroups
The main problem of this paper is to study the perturbation of a positive \(M\)-Pettis integrable linear semigroup. Let \(B\) be a Banach space and a linear lattice with a positive cone. Let \(B^+= \{x\in B;\) \(x\geq 0\}\). Let \((T_i)\) be a linear, positive, \(M\)-Pettis integrable semigroup. Let \(L: B^+\to B\) be an \(M\)-measurable (not necessarily linear) operator such that \(L[x(t)]\) is bounded on bounded subintervals whenever \(x(t)\in B^+\) is bounded on bounded subintervals. The author establishes the existence and uniqueness of the following integral equation. \[ V_t x= T_tx+ (MP) \int_{(0, t)} T_s [L(V_{t- s} x)] ds. \] With the help of this and other preliminaries developed in the earlier sections of the paper, the author explains how the supersemigroups and superprocesses due to \textit{F. Fitzsimmons} [Isr. J. Math. 64, No. 3, 337-361 (1988; Zbl 0673.60089)] and \textit{S. Watanabe} [J. Math. Kyoto Univ. 8, 141-167 (1968; Zbl 0159.462)] are particular cases of his results.
perturbation of a positive \(M\)-Pettis integrable linear semigroup
scientific article; zbMATH DE number 563282
scientific article