Solvability and asymptotic behavior of solutions of ordinary differential equations with variable operator coefficients. (English) Zbl 0772.34046

Kozlov, V.; Maz’ya, V.

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1992, No.V, 12 p. (1992).

Suppose \((H_ q,\|\cdot\|_ q)\) are Hilbert spaces \((q=0,1,\dots,n)\) such that \(H_ n\subset H_{n-1}\subset\cdots\subset H_ 0\), \(\| u\|_ q\leq\| u\|_{q+1}\) \((q=0,1,\dots,n-1)\) and \(H_ n\) is dense in \(H_ 0\). Introduce the operator pencil \({\mathcal A}(\lambda)=\sum^ n_{q=0}A_{n-q}\lambda^ q:H_ n\to H_ 0\), where \(A_ q\) is a linear bounded operator from \(H_ q\) into \(H_ 0\). Next, let \(D_ t=i^{-1}\partial/\partial t\) and let \(L(t,D_ t)\) denote a perturbation of the operator \({\mathcal A}(D_ t)\). The paper under review is devoted to the study of the differential equation of the form (1) \(L(t,D_ t)=f\), for \(t\in R\). Under some hypotheses (guaranteeing that \({\mathcal A}(\lambda)\) is the Fredholm pencil, among others) a few results on the existence and uniqueness of solutions of (1) are derived. Moreover, some estimates of solutions of (1) are obtained and asymptotic properties of solutions are also studied.

Reviewer: J.Banaś (Rzeszów)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34C11	Growth and boundedness of solutions to ordinary differential equations
34E05	Asymptotic expansions of solutions to ordinary differential equations

Keywords:

Hilbert spaces; operator pencil; perturbation; differential equation; Fredholm pencil; existence; uniqueness; estimates; asymptotic properties