Existence of Periodic Solutions for a Class of Second Order Ordinary Differential Equations

We provide sufficient conditions for the existence of a periodic solution for a class of second order differential equations of the form x¨+g(x)=εf(t,x,x˙,ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ddot{x}+g(x)=\varepsilon f(t, x,\dot{x},\varepsilon )$\end{document}, where ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document} is a small parameter.


Introduction and Statement of the Results
The second order differential equations of the form x + g(x) = εf (t, x,ẋ, ε), have been studied by many authors because they have many applications, see for instance [2, 3, 5, 8-12, 15, 17, 19]. Two of the main families studied are the Duffing equations see [6,7], and the forced pendulum, see the nice survey [14] and the references quoted therein.
The aim of this work is to study periodic solutions of the second order differential equationẍ where n is a positive integer, μ is a small parameter, and the functions and h(x) are smooth, b = 0, p(t) and q(t, x, y, μ) are smooth and periodic with period 2π in the variable t . Let Γ (x) be the Gamma function, see for more details [1], and let α and β be the first Fourier coefficients of the periodic function p(t), i.e.
Now our main result is the following.
Theorem 1 If αβ = 0 then for μ = 0 sufficiently small the differential equation (1) has a 2π -periodic solution x(t, μ) such that Theorem 1 is proved in Sect. 3, where we use the averaging theory for computing periodic solutions, see Sect. 2 for a summary of the results on this theory that we shall need.

The Averaging Theory
We want to study the T -periodic solutions of the periodic differential systems of the form with ε > 0 sufficiently small, where F 0 , F 1 : R × Ω → R n and F 2 : and Ω is an open subset of R n . Let x(t, z, ε) be the solution of the differential system (2) such that x(0, z, ε) = z. Suppose that the unperturbed system has an open set V with V ⊂ Ω such that for each z ∈ V , x(t, z, 0) is T -periodic. Let y be an n × n matrix, and consider the first order variational equation of the unperturbed system (3) on the periodic solution x(t, z, 0). Let M z (t) be the fundamental matrix of the linear differential system (4) with periodic coefficients such that M z (0) is the n × n identity matrix.
The existence of the periodic solution of Theorem 2 is due to Malkin [13] and Roseau [16], for a shorter and easier proof see [4]. The proof for the stability follows in a similar way to the proof of Theorem 11.6 of [18].

Proof of Theorem 1
The differential equation of second order (1) can be written as the first order differential systemẋ = y, In order to apply the averaging theory described in Sect. 2 to this differential system we do the scaling x → μx and y → μy. Hence the differential system (7) becomeṡ This system is written into the normal form (2) for applying the averaging theory described in Sect. 2, where From Sect. 2 the solution x(t, z, 0) = (x(t, z, 0), y(t, z, 0)) of system (8) with ε = 0 satisfies x(0, z, 0) = z = (x 0 , y 0 ), and consequently x(t, z, 0) = x 0 cos t + y 0 sin t, y(t, z, 0) = −x 0 sin t + y 0 cos t.

The fundamental matrix M z (t) = M(t) of the first order variational equation (4) satisfying (9) is
According with Theorem 2 in order to compute the 2π -periodic solutions of the differential system (8) we must compute the integral Doing induction with respect to n it is not difficult to show that 2π 0 sin t (x 0 cos t + y 0 sin t) 2n+1 Therefore we must solve the system This system has a unique solution The determinant (6)  , and by assumptions it is positive because αβb = 0. In summary all the assumptions of Theorem 2 hold and consequently from Theorem 2 it follows Theorem 1.