Abstract: We consider a system of nth-order ordinarydifferential equations whose right-hand side is the sum of a linear function of the solution with aconstant matrix, an essential nonlinearity of the relay type with hysteresis, and a perturbingcontinuous periodic function. The matrix of the linear function has only real simple nonzeroeigenvalues, of which at least one is positive. We study the question of whether such systems havecontinuous solutions with two switching points in the state space (two-point oscillatory solutions)such that the time in which the solution returns to each of these points coincides with the periodof the perturbing function or is an integer fraction of the latter. A sufficient condition for thenonexistence of such solutions is established, and a theorem is proved that gives sufficientconditions for the existence of a two-point oscillatory solution with return time equal to the periodof the perturbing function. A corroborating example is given.
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