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We give a representation of the component of solutions withcharacteristic multiplier 1 in a periodic linear inhomogeneous continuoussystem. It follows from this representation that asymptotic behaviors ofthe component of solutions to the system and to its associated homo-geneous system are quite different, though they are similar in the casewhere the characteristic multiplier is not 1. Moreover, the representationis applicable to linear discrete systems with constant coefficients. 1. Introduction and preliminariesWe consider periodic linear inhomogeneous differential equations of the formddt(1) x(t) = A(t)x(t) +f(t), x(0) = w,where A(t) is a periodic continuous p×p matrix function with period τ > 0 andf : R→ C p a τ-periodic continuous function. Here Cis the set of all complexnumbers and Rthe set of all real numbers.Representations of solutions to the equation (1) have been given in [2, 5, 6].These reformulate the variation of constants formula into the sum of a τ-periodic function and an exponential-like function. In particular, the repre-sentation [6] of the component of solutions with characteristic multiplier µ 6= 1shows that asymptotic behavior of the component of solutions to the equation(1) is the sum of asymptotic behavior of the component of some solution to thehomogeneous equation associated with the equation (1) and the τ-periodic so-lution decided from the equation (1). Roughly speaking, asymptotic behaviorof the component of solutions to the system and to the homogeneous systemassociated with the equation (1) are similar (refer to Remark 3.4). However,it is predicted that in the case where the characteristic multiplier is 1, they |
