Periodic solutions of a resonant third-order equation (original) (raw)

Periodic solutions of a resonant higher order equation

Portugaliae Mathematica, 2005

We study the existence of periodic solutions for an ordinary differential equation of resonant type. Under suitable conditions we prove the existence of at least one periodic solution of the problem applying Mawhin Coincidence degree theory.

Multiple Nontrivial Solutions for Doubly Resonant Periodic Problems

Canadian Mathematical Bulletin, 2010

We consider semilinear periodic problems with the right-hand side nonlinearity satisfying a double resonance condition between two successive eigenvalues. Using a combination of variational and degree theoretic methods, we prove the existence of at least two nontrivial solutions.

Periodic solutions for completely resonant nonlinear wave equations

2004

We consider the nonlinear string equation with Dirichlet boundary conditions uxx−utt=phi(u)u_{xx}-u_{tt}=\phi(u)uxx−utt=phi(u), with phi(u)=Phiu3+O(u5)\phi(u)=\Phi u^{3} + O(u^{5})phi(u)=Phiu3+O(u5) odd and analytic, Phineq0\Phi\neq0Phineq0, and we construct small amplitude periodic solutions with frequency o\oo for a large Lebesgue measure set of o\oo close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no

Multiple solutions for strongly resonant periodic systems

Nonlinear Analysis: Theory, Methods & Applications, 2007

We considered a semilinear, second order periodic system. We assumed that the differential operator x → −x − Ax has zero as an eigenvalue and has no negative eigenvalues. Also we imposed a strong resonance condition (with respect to the zero eigenvalue) on the potential function F(t, x). Using the second deformation theorem, we established the existence of at least two nontrivial solutions. To do this we needed to conduct a detailed analysis of the Cerami compactness condition, which is actually of independent interest.

On the existence of a three point boundary value problem at resonance in

Journal of Mathematical Analysis and Applications, 2014

We apply a coincidence degree theorem due to Mawhin to obtain a sufficient condition for the existence of at least one solution for a class of nonlinear secondorder differential equations in R n associated with three point boundary conditions.

A Note on the Existence of Periodic Solutions of Second Order Non-Linear Differential Equations

American Journal of Mathematics and Statistics, 2012

Bahman Mehri, using Leray-Schauder fixed point continuation method, established the existence of periodic solutions to equations of the form, x // +(k+h(x)) x / + f(t, x) = p(t),where h is a continuous function, f, p are continuous functions in their respective arguments and periodic with respect to t of period ω and k is a constant.The aim of this research is to extend this result to a wider class of equations of the form, x // +(k+h(x)) x / + F(t, x, x /) = p(t),where F is continuous in t,x and x / ; and periodic with respect to t of period ω and k is a constant, using Leray-Schauder fixed point continuation method.

Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

Mathematical Problems in Engineering

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus Z=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check th...