Vol. 29, No. 2, 1969

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ISSN: 0030-8730
A Green’s function approach to perturbations of periodic solutions

Carl Kallina

Vol. 29 (1969), No. 2, 325–334
Abstract

Consider the nonlinear differential system

˙y = A(t) ⋅y+ 𝜖q(t,y,𝜖)
(1.1)

where y, q are column n-vectors, q is continuous in (t,y,𝜖) and has continuous second partial derivatives with respect to y, 𝜖 for all values of t, 0 y R for some R > 0 and 0 𝜖 𝜖0 for some 𝜖0 > 0. Further assume A(t) is an n × n matrix such that A C1, and both A and q are periodic in t with period T. Associated with system (1.1) are the general homogeneous and nonhomogeneous equations

˙y = A (t)⋅y
(1.2)

˙y = A(t) ⋅y+ f(t)
(1.3)

where f(t) is an arbitrary n-vector function periodic in t of period T. In this paper we consider the classical problem of proving the existence of T-periodic solutions y = y(t) of (1.1) when the homogeneous system (1.2) has nontrivial T-periodic solutions.

Mathematical Subject Classification
Primary: 34.45
Milestones
Received: 24 June 1968
Published: 1 May 1969
Authors
Carl Kallina