Abstract
Let D be the infinitesimal generator of a strongly continuous periodic one-parameter group of linear operators in a Banach space. Main results: An analog of the resolvent operator (= quasi-resolvent operator of D) is defined for points of the spectrum of D and its evident form is given. The theorem on integral for the operator D, theorems on the existence of periodic solutions of a linear differential equation of the n th order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are obtained. In the case of the existence of periodic solutions, evident forms of all periodic solutions of a linear differential equation of the n th order with constant coefficients and systems of linear differential equations with constant coefficients in Banach spaces are given in terms of resolvent and quasi-resolvent operators of D.MSC:42A, 43, 47D.
Highlights
1 Introduction One-parameter groups of linear operators and periodic one-parameter groups of linear operators in topological vector spaces were investigated by Stone [ ], Dunford [ ], Gelfand [ ] and others
Let α(t) (t ∈ T) be a strongly continuous one-parameter group of bounded linear operators in a Banach space H, and let D be an infinitesimal generator of the group α(t)
In Theorem (iv) below, we prove that Rλ is equal to the resolvent operator of D for every point λ of the resolvent set of D
Summary
One-parameter groups of linear operators and periodic one-parameter groups of linear operators in topological vector spaces were investigated by Stone [ ], Dunford [ ], Gelfand [ ] and others (see [ – ]).Let T be the one-dimensional torus {eit : –π ≤ t < π}. –π exists and Fn(x) ∈ Hn. Proposition Let α be a strongly continuous isometric linear representation of T on a Banach space H.
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