Existence Of Periodic Solutions For A Class Nonlinear Second Order Vector Differential Equations

As we all know, among the study of differential equations, the investigation of periodic solutions is very crucial, especially the existence of them, which always play an important role. As a result, lots of mathematicians have showed interests in this field, and obtained many classical conclutions. Meanwhile, some new methods were discovered and developed, which promoted the development of differential equations. some related ineresting results can be found in [7, 9, 10, 13, 18].X. Lu, Y. Li, Y. Su [2] have studied the following equationx" + f(t, x, x') = 0by the homotopy method to obtain the existence of periodic solutions for the above equation, under some conditions were satisfied.K. Schmitt [15] concluded the existence of periodic solutions for the following equation where the function in the rightside is locally Lipschitz, and there exist two constantsα,β, such thatf(tα,0)≤0≤f(t,β,0).S. Sedziwy discussed the equationx" = f(t,x,x'),for which there existsed upper solutions and lower solutions, and f is a Caratheodary function.In this paper, we consider the exsitence of T-periodic solutions of the equationx" + f(t, x, x') = 0,where f : is a continuous function, T-periodic in t for some T > 0. such results are inspired by C. Fabry and P. Habets recording to the existence of solutions for Picard BVP, which can be found in [1].Let G(t, s) be a Green function of operator x"—k²x under periodic boundary value conditionsx(0)=x(T),x'(0) = x'(T).The problem of periodic solution(l) can be transformed into the integral equation Consequently, the problem consists of finding the fixed point of the mappingdefined bywhere C₁ⁿ[0,T] is the Banach space of C¹ functions x : [0,T]→(?)ⁿ, with the following periodic boundary value conditionsThe existence of fixed points for the mapping A will be based on the following theorem, which is a simple and classical application of Leray—Schauder degree theory.Definition 1 The mapping A from Banach space X into itself is called compact if A is continuous and maps bounded sets into relatively compact sets.Lemma 1 Let X be a Banach space, B : X→X be a compact mapping such that I—A is one to one, andΩ, an open boundary set such that 0€(I—B)Ω. Then the compact mapping A : (Ω|-)→X has a fixed point in (Ω|-) if for anyλ∈(0,1), the equationx =λAx + (1 -λ)Bx (4)has no solution x on the boundary (?)ΩofΩ Lemma 2 Let x : [a, b]→(?)ⁿ be an absolutely continuous function with an absolutely continuous derivative. Assume that for almost every t∈[a, b], we haswhere h :(?)⁺→(?)⁺\{0}is continuous and satisfiesthenwhere g is defined byIt is easy to show that g is continuous and increasing on (?)⁺.Our main results state as follows :Theorem 1 Assume that there exists a twice differentiate functionwhere 0, and a continuous functionwhere F(t,x,y) is T-periodic in t for some T > 0. They satisfy the following conditions: for any (t,x,y), t∈[0, T], |x| =φ(t),〈x,y〉= |x|φ′(t) , the scalar product is denoted by (x,y), the norm by \x\. we have:Assume moreover that there exist numbersα∈[0,1),β≥0,such that, for any (t,x,y), t∈[0,T], |x|≤φ(t), y∈(?)ⁿ, then we havewhereis continuous and satisfiesThen, the problem (1) has a T-periodic solution x_*(t), such that |x_*(t)|≤φ(t).Finally, we give some numerical experiments. By means of ho-motopy method, following the path of homotopy of numerical value until the hyperplane ofλ= 1. Concretely, we begin at y⁰ = (x⁰,λ₀), and compute the string of points y¹,y²..., on the curve, such that, each point y(i+1) is obtained throw guessing yⁱ = (xⁱ,λ_i), to get z₀, and correcting it by Newton's method.