Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

Abstract

This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation ( ) = ( , ( ), ( )), ∈ R, where the nonlinearity : R 3 → R is continuous and ( , , ) is 2 -periodic in . Under certain inequality conditions that ( , , ) may be superlinear growth on ( , ), an existence result of odd 2 -periodic solutions is obtained via Leray-Schauder fixed point theorem.

Existence and uniqueness of periodic solution to second‐order impulsive differential equations

“…Remark Without regard for impulsive effects, our result presented in this theorem is incompatible with the results presented in Li and Guo 31 . In fact, in Li and Guo, 31 if the nonlinear term be $$$ Ax+f\left(t,x,y\right) $$$, then condition $$$ (F2) $$$ implies $$$$ {x}^Tf\left(t,x,y\right)\le \left(a-{k}_{\tau}\right){x}^2+b{y}^2+{L}_2; $$$$ here, $$$ {k}_{\tau }=\min \left\{|{k}_1|,\cdots, |{k}_n|\right\},a+b<1,{L}_2>0 $$$. Hence, $$$$ a+b-{k}_{\tau }<1-{k}_{\tau }.$$…”

“…Without regard for impulsive effects, our result presented in this theorem is incompatible with the results presented in Li and Guo. 31 In fact, in Li and Guo, 31 if the nonlinear term be Ax + 𝑓 (t, x, 𝑦), then condition (F2) implies…”

Odd periodic waves and stability results for the defocusing mass‐critical Korteweg‐de Vries equation

“…In fact, let L >0 be fixed. As we have already mentioned in the introduction, Angulo and Natali bring a more general approach to determine the existence of odd periodic waves for nonlinear second‐order differential equations. However, in our case, we would like to construct periodic waves, which arise as a minimizer of the energy with fixed momentum in order to obtain good spectral properties concerning the linearized operator in .…”

“…The existence of odd periodic solutions associated with nonlinear second‐order differential equations is an important qualitative aspect in ordinary differential equations, and it has been studied for a huge class of researchers. For instance, we refer the reader to Angulo and Natali (and references therein) for a general approach, which guarantees the existence of odd periodic solutions for a wide class of nonlinear second‐order differential equations as where f is a smooth function in all variables and satisfying a convenient general set of conditions.…”