Invertibility of nonautonomous functional-differential operators.

*(English. Russian original)*Zbl 0646.34016
Math. USSR, Sb. 58, 83-100 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 1, 86-104 (1986).

Let \(C^{(0)}\) denote the space of continuous and bounded functions acting from (-\(\infty,\infty)\) into a finite dimensional Banach space E, with the standard sup norm. Further, let \(C^{(p)}\) be its subspace consisting of all functions x such that \(d^ kx/dt^ k\in C^{(0)}\) \((k=1,2,...,p)\) and equipped with the classical norm. The problem of the invertibility of the operator \(A+d^ m/dt^ m\) is investigated, where an operator A belongs to \(L(C^{(p)},C^{(q)})\) and is assumed to be c- continuous or c-completely continuous [cf. E. Muhamadiev, Dokl. Akad. Nauk SSSR, 196, 47-49 (1971; Zbl 0227.34008)]. Assuming additionally that \(\ker (A+d^ m/dt^ m)=\{\Theta \}\) and A can be approximated in some way by a completely continuous and almost periodic operator it is shown that \((A+d^ m/dt^ m)^{-1}\) exists and is c- continuous. Several particular results of this theorem are considered and many interesting generalizations are also presented. Moreover, the applicability to the theory of functional-differential equations is illustrated by a few nontrivial examples.

Reviewer: J.Banas

##### MSC:

34A55 | Inverse problems involving ordinary differential equations |