Convergence theorems for integrated semigroups
The paper provides approximation results for the abstract homogeneous \[ {du\over dt}=Au,\quad u(0)= u_ 0,\tag{1} \] and nonhomogeneous \[ {du\over dt}= Au(t)+ f(t),\quad u(0)=x\tag{2} \] Cauchy problems. The novelty consists in the search for integral solutions to (1) and (2). Then \(u(t)\) satisfies the integrated equation \[ u(t)= u_ 0+ A \int^ t_ 0 u(s)ds,\tag{3} \] in the case (1), and \[ u(t)= x+ A\int^ t_ 0 u(s)ds+ \int^ t_ 0 f(s)ds,\tag{4} \] in the second case. This idea was introduced by \textit{G. Da Prato} and \textit{E. Sinestrari} [Ann. Sc. Norm. Sup. Pisa, Cl. Sci., IV. Ser. 14, No. 2, 285-344 (1987; Zbl 0652.34069)]. It is useful when \(u_ 0\) and \(x\) are not sufficiently regular. The main theorem formulated and proved in the present paper concerns the generalization of the famous Trotter-Kato theorem of semigroup theory. In the case under study it states that resolvent consistency, under appropriate conditions, implies convergence of integral solutions. One simple example illustrates the possibility of application to the population model. The paper is clearly written and of interest to specialists in differential equations in Banach spaces.
scientific article; zbMATH DE number 68010