Periodic solutions of arbitrary period, variational methods

2006 ◽  
pp. 353-375 ◽  
Author(s):  
Robert H. G. Helleman ◽  
Tassos Bountis
Keyword(s):  
Variational Methods ◽  
Periodic Solutions ◽  
Arbitrary Period

Periodic Solutions of Non-autonomous Allen–Cahn Equations Involving Fractional Laplacian

2020 ◽  
Vol 20 (3) ◽  
pp. 725-737 ◽  
Author(s):  
Zhenping Feng ◽  
Zhuoran Du
Keyword(s):  
Variational Methods ◽  
Periodic Solutions ◽  
Smooth Function ◽  
Fractional Laplacian ◽  
Type Inequality ◽  
Energy Functional ◽  
Large Integer ◽  
Existence Of Periodic Solutions ◽  
Double Well Potential

AbstractWe consider periodic solutions of the following problem associated with the fractional Laplacian: {(-\partial_{xx})^{s}u(x)+\partial_{u}F(x,u(x))=0} in {\mathbb{R}}. The smooth function {F(x,u)} is periodic about x and is a double-well potential with respect to u with wells at {+1} and -1 for any {x\in\mathbb{R}}. We prove the existence of periodic solutions whose periods are large integer multiples of the period of F about the variable x by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.

scholarly journals Periodic Solutions for a Class of Singular Hamiltonian Systems on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xiaofang Meng ◽  
Yongkun Li
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Time Scales ◽  
Critical Point Theory ◽  
Point Theory ◽  
Singular Hamiltonian Systems ◽  
Existence Of Periodic Solutions

We are concerned with a class of singular Hamiltonian systems on time scales. Some results on the existence of periodic solutions are obtained for the system under consideration by means of the variational methods and the critical point theory.

Study on the Periodic Solutions to a Superquadratic Discrete Hamiltonian System Based on Mechanics

2013 ◽  
Vol 394 ◽  
pp. 92-95
Author(s):  
Da Wei Sun ◽  
Jia Rui Liu
Keyword(s):  
Hamiltonian System ◽  
Variational Methods ◽  
Periodic Solutions ◽  
Existence Of Solution ◽  
Discrete Hamiltonian System ◽  
New Type ◽  
Discrete Type

This paper studies the periodic solutions to a superquadratic second-oder discrete type Hamiltonian system in the n dimensional Euclide space. By the variational methods and some discrete computional techniques, this paper proves the existence of solution to a new type discrete Hamiltonian system.

scholarly journals Variational Approach to Quasi-Periodic Solution of Nonautonomous Second-Order Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Juhong Kuang
Keyword(s):  
Periodic Solution ◽  
Variational Methods ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Variational Approach ◽  
Second Order ◽  
New Approach ◽  
Second Order Hamiltonian Systems

We deal with the quasi-periodic solutions of the following second-order Hamiltonian systemsx¨(t)=∇F(t,x(t)), wherex(t)=(x1(t),…,xN(t)), and we present a new approach via variational methods and Minmax method to obtain the existence of quasi-periodic solutions to the above equation.

scholarly journals Variational Methods for Almost Periodic Solutions of a Class of Neutral Delay Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
M. Ayachi ◽  
J. Blot
Keyword(s):  
Variational Methods ◽  
Periodic Solutions ◽  
Delay Equations ◽  
Delay Equation ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Neutral Delay ◽  
Density Result

We provide new variational settings to study the a.p. (almost periodic) solutions of a class of nonlinear neutral delay equations. We extend Shu and Xu (2006) variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings. We obtain results on the structure of the set of the a.p. solutions, results of existence of a.p. solutions, results of existence of a.p. solutions, and also a density result for the forced equations.

scholarly journals Multiple Periodic Solutions for a Class of Second-Order Neutral Impulsive Functional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Jingli Xie ◽  
Zhiguo Luo ◽  
Yuhua Zeng
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Critical Point ◽  
Periodic Solutions ◽  
Critical Point Theory ◽  
Functional Differential Equations ◽  
Point Theory ◽  
Second Order ◽  
Multiple Periodic Solutions ◽  
Functional Differential

In this paper, we study a class of second-order neutral impulsive functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of critical point theory and variational methods. We propose an example to illustrate the applicability of our result.

scholarly journals Variational Approach to Impulsive Problems: A Survey of Recent Results

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Fang-fang Liao ◽  
Juntao Sun
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Periodic Solutions ◽  
Variational Approach ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Homoclinic Solutions ◽  
Nontrivial Solutions ◽  
Impulsive Problems

We present a survey on the existence of nontrivial solutions to impulsive differential equations by using variational methods, including solutions to boundary value problems, periodic solutions, and homoclinic solutions.

A note on periodic solutions of some nonautonomous differential equations

1986 ◽  
Vol 34 (2) ◽  
pp. 253-265 ◽  
Author(s):  
M. R. Grossinho ◽  
L. Sanchez
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Time Dependent ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonautonomous Differential Equations ◽  
Nontrivial Periodic Solutions

We prove the existence of nontrivial periodic solutions of some nonlinear ordinary differential equations with time-dependent coefficients using variational methods.

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