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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
##### Keywords:
invertibility; functional-differential equations; examples
Zbl 0227.34008
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