THE EXISTENCE OF PERIODIC SOLUTIONS FOR THREE-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

where |c| < 1, τ is a constant, a2(t), a1(t), a0(t), τi(t), βi(t) (i = 1, 2, ..., n) and p(t) are real continuous functions defined on R with positive period T and gi(x) (i = 1, 2, ..., n) are real continuous functions defined on R. Recently, the existence of periodic solutions for differential equations have arouse extensive attention. Most of the results obtained in literature are about the periodic solutions on delay differential equations. Only a small portion of the results [1, 2, 5, 8, 18, 19, 21] concern the periodic solutions on neutral differential equations. For the detailed basic theory, we would like to recommend interested readers to refer to [3, 6, 10–15, 17, 20, 22–24]. This note is inspired by [7] and [9] which discuss the existence of multiple periodic solutions for neutral differential equations with one and two order, and now we study the existence of periodic solutions for neutral differential equations (1.1) by applying two diverse methods. The following note will be described in these aspects. In Section 2, using Kranoselskii fixed point theorem to reveal that (1.1) exists periodic solutions. In Section 3, we state that Mawhin’s continuation theorem is used for proving the existence of periodic solutions for (1.1).