On the existence of periodic solutions of a certain third-order differential equation

Abstract

In this paper we shall be concerned with the differential equation in which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t . The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases: under which every solution x(t) of (1) satisfies where t 0 depends on the particular x chosen, and D is a constant depending only on a , b , h and p . These hypotheses are, in the case (2), or, in the case (3), In what follows here we shall refer to (2) and ( H 1 ) collectively as the (boundedness) hypotheses ( BH 1 ), and to (3) and ( H 2 ) as the hypotheses ( BH 2 ). Our object is to examine whether periodic solutions of (1) exist under the hypotheses ( BH 1 ), ( BH 2 ).