The Existence and Uniqueness of Periodic Solutions for Some Nonlinear nth-Order Differential Equations

and Applied Analysis 3 Now, let f̃ : R 1 → R be a continuous function, T -periodic with respect to the first variable, and consider the nth-order differential equation u n f̃ ( t, u, u′, u′′, . . . , u n−1 ) . 2.5 Lemma 2.1 see 14 . Assume that the following conditions hold. i There exists ρ > 0 such that, for each λ ∈ 0, 1 , one has that any possible T -periodic solution u of the problem u n λf̃ ( t, u, u′, u′′, . . . , u n−1 ) 2.6 satisfies the priori estimation ‖u‖ n−1 < ρ. ii The continuous function F : R → R defined by F x ∫T 0 f̃ t, x, 0, 0, . . . , 0 dt, x ∈ R 2.7 satisfies F −ρ F ρ < 0. Then, 2.5 has at least one T -periodic solution u such that ‖u‖ n−1 < ρ. From Lemma 2.2 in 15 and the proof of inequality 10 in 7, pp 3402 , one obtains the following. Lemma 2.2. Let x t ∈ C1 T . Suppose that there exists a constant D ≥ 0 such that |x τ0 | ≤ D, τ0 ∈ 0, T , 2.8