Periodic Solutions And Quasi-Periodic Solutions Of Newton's Equations

The delay differential equation is the powerful tool to examine the periodic-resonance regularity with time-delay phenomena. In the sixties and seventies of the twentieth century, there has been widely concerned on Newton's equation with time-delay in the promotion of the application. It is focusing on the existence of periodic solutions to the equation in the classic non-resonance conditions from the actual background of physics and mechanics. The Work in this area can be found in Layton at 1980 in [42]. In the non-resonance condition of the promotion Lazer-type, Layton in [42] studied the periodic solutions of the equation(?) (1)After that, many scholars had studied the periodic solutions of the Newton's equations with the second order delay and higher order delay, in [25, 33, 34, 75, 23, 101, 88, 72, 73, 21, 84, 35, 63, 58, 60, 83, 87, 22] and so on.In recent years, the second order delay Newton's equations are researched depthly in the classic non-resonant conditions, but there are few results for the existence of periodic solutions to the high order delay Newton's equation. A natural question is whether there is the similar result for the high order delay Newton's equations? Furtherly, are there some sufficient conditions when the high order delay Newton's equations have the 2Ï€-periodic solutions?Inspired by literatures in [43, 52, 11, 12, 13, 42], this paper has two parts to research these issues depthly.In the first part, we will extend the results of the periodic solutions of the second order delay Newton's Equations. The proof relies on Schauder's fixed point theorem and Fourier analysis techniques. And it is proved that there is the existence and uniqueness of periodic solution to the Newton's equation with the (2κ)th-order delay in non-resonant conditions of the Lazer-type, separately for one-delay and multiple delays circumstances. We firstly discuss Newton's equation with the (2κ)th-order delay(?) (2)(?) (3) and take the assumptions as following:(H1) The function f is continuous and differentiate. For (?)(t, u, 0)∈×R×R, we havewhere M₁ is a positive constant. (H2) There are an integer m and two positive constants p, q, satisfied the following inequality(H5) The function H is continuous on R^4k+1, and is 2Ï€-periodic on the first variable. In addition, there is a positive number M₂ satisfied the following inequalityTheorem 2.2.1 Assumed (H1) and (H2) are satisfied, the equation (2) has at least one 2Ï€-periodic solution.Theorem 2.2.4 Provided (H1), (H2) and (H5) are satisfied, the equation (3) has at least one 2Ï€-periodic solution.Futhermore, we also discussed the (2κ)th-order Newton's equation with n-dimensional delay(?) (4)and the (2κ)th-order Newton's equation with multiple delays(?) (5)We have the similar results to the previous conclusions in the non-resonance conditions of the promotive Lazer.In the second part, we research the existence of the perodic solutions for the (2κ+1)th-order delay Newton's equation. We obtain some sufficient conditions for the existence of periodic solutions for the (2k + 1)th-order delay Newton's equations, separately for one-delay and multiple delays cases. The proof relies on Extension Theorem and uses the technology of the priori estimates and the cut-off function.In this part, we consider the (2k+1)th-order delay Newton's equations(?) (6)and the (2k + 1)th-order Newton's equations with multiple delays(?). (7)In this paper, we always assume that(A1) There is a positive constantσsuch that(A2) There are positive constants b₀ > 2σand c_i,i = 0,1,…,2k, such thatFor |x_i|≥a_i, (x_i+1,…, x_2k)∈R^2k-i,we haveFor |x₀|≥a₀, (x₁,…, x_2k)∈R^2k, we have(A3) There are constants b₀, c_i, i = 0,1,…, 2k, such that (A4) For (x₀,x₁,…,x_2k)∈R^2k+1, there are positive constants d₀ > 2σand e_j,j = 0,1,…,2k, such that(A5) There are positive constants d₀, e_j, j = 0,1,…, 2k, such thatOur main result is the following:Under hypothesis (A1-A3) or (A1,A4,A5), the equation (6) or (7) has at least one 2Ï€-periodic solution.It is widely concerned on the Lagrange stability to the pendulum-type's equations in today's mathematical physics and applied mathematics. This problem was first raised by the Moser [76]. Moser[77], Levi[46] and You[99] studied the periodic situation separately. In [3], Bibikov developed the KAM theorem of almost integrable Hamiltonian systems with a degree of freedom at quasi-periodic perturbations. Bibikov applied this theorem to discuss the balance stability for a class of second-order nonlinear differential equations. In [55], Lin and Wang studied the double quasi-periodic system, and changed the Diophantine conditions which is stronger than the usual conditions, and had the result that was a sufficient condition to ensure the Lagrange stability. In [47],[100], the authors developed the Quasi-periodic Monotony Reverse theorem.In the third part, we study the stability of pendulum-type equations with quasi-periodic conditons. We have some necessary and sufficient conditions to the Lagrange stability of the pendulum-type's equations when the frequencies satisfies the usual Diophantine conditions, so as to solve the Moser's problem in quasi-periodic case. In this part, we research the Pendulum-type equation(?), (8)where p(t,x + 1) = p(t,x), p(t,x) is quasi-periodic function on t, andω= (ω₁,…,ω_n), is the undamental frequency that is,p(t,x) = f(wt,x),where f(θ, x)has the definition on Tⁿ×T¹, and Tⁿ = Rⁿ/Zⁿ is an n-dimensional torus. Assuming f(θ, x) is a real-analytic function on Tⁿ×T¹, and the frequency ? satisfied the following Diophantine condition:(?), (9)whereν> 0 is a given arbitrary constant, <,> is the usual inner product, and we have the following main result: Theorem 4.1 Under hypothesis (9), the equation (8) is Lagrange stable if and only if(?). (10)Furthermore, under the conditions (9) and (10) holded, the equation (8) has an infinite number of quasi-periodic solutions with n+1 fundamental frequencies includingω₁,…,ω_n.