Periodic Solutions Of Equation Research Articles

The Existence of Periodic Solutions of a Class of <i>n</i>-Degree Polynomial Differential Equations*

This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.

An exact expression of positive periodic solution for a first-order singular equation

The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

Positive periodic solution for p-Laplacian neutral damped Duffing equation with strong singularities of attractive and repulsive type

This paper is devoted to investigating the following p-Laplacian neutral damped Duffing equation with singularity: \t\t\t(ϕp(u(t)−cu(t−τ))′)′+Pu′(t)+g(u(t))=e(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(\\phi _{p}\\bigl(u(t)-cu(t-\\tau )\\bigr)' \\bigr)'+Pu'(t)+g\\bigl(u(t)\\bigr)=e(t), $$\\end{document} where g has a singularity at u=0. Applying the Manásevich–Mawhin theorem on a continuous case of topological degree, we obtain the existence of a positive periodic solution for this equation.

Periodic Solutions of Second-Order Differential Equations with Large Parameters

A second-order differential equation containing a large parameter is considered. Such an equation can be interpreted as an equation of constrained oscillations of a mechanical system with one degree of freedom, provided that the fundamental frequency of the system substantially exceeds the external frequency. We provide a new proof of the existence of a periodic solution of that equation such that it is close to the periodic solution of the corresponding degenerate equation. That proof is obtained by means of the Poincare method.

Almost Periodic Solutions of First-Order Ordinary Differential Equations

Approaches to estimate the number of almost periodic solutions of ordinary differential equations are considered. Conditions that allow determination for both upper and lower bounds for these solutions are found. The existence and stability of almost periodic problems are studied. The novelty of this paper lies in the fact that the use of apparatus derivatives allows for the reduction of restrictions on the degree of smoothness of the right parts. In our work, regarding the number of periodic solutions of equations first order, we don’t require a high degree of smoothness and no restriction on the smoothness of the second derivative of the Schwartz equation. We have all of these restrictions lifted. Our new form presented also emphasizes this novelty.

On the Study Of Asymptotically Almost Periodic Solutions of a Class of Impulsive Population Models

Based on the Mawhin continuous theorem, the existence of strictly positive asymptotically almost periodic solutions of a class of impulsive population models is studied. The conclusion generalizes the conclusion of the existing literatures. Since the Mawhin continuous theorem is only used to prove the existence of periodic solutions or almost periodic solutions of equations (for example:impulsive differential equation, functional differential equation, integral equation, Lienard equation, P-Laplacian equation), the main result is innovative.

New results on positive almost periodic solutions for first-order neutral differential equations

In this paper, a class of first-order neutral differential equations with time-varying delays and coefficients is considered. Some results on the existence of positive almost periodic solutions for the equations are obtained by using the contracting mapping principle and the differential inequality technique. In addition, an example is given to illustrate our results.

Asymptotically Almost Periodic Solutions of Equations with Delays and Nonfixed Times of Pulse Action

By using the properties of asymptotically almost periodic solutions, we prove the theorems on existence of an asymptotically stable piecewise continuous solution of a system of differential equations with delays and nonfixed times of pulse action.

Almost periodic motions in semi-group dynamical systems and Bohr/Levitan almost periodic solutions of linear difference equations without Favard's separation condition

The discrete analogue of the well-known Favard theorem states that the linear difference equation with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of the homogeneous equations where In this paper, we prove that the linear difference equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equation are homoclinic to zero in the positive direction (i.e. for all relatively compact solutions of (3)). If the coefficients of (1) are Bohr almost periodic and all relatively compact solutions of all limiting equations (2) tend to zero as , then equation (1) admits a unique almost automorphic solution. We study the problem of existence of Bohr/Levitan almost periodic solutions of equation (1) in the framework of general non-autonomous dynamical systems (cocycles).

Periodic solutions for a third-order differential equation without asymptotic behavior on the potential

We consider the third-order differential equation $u^{'''} +au'' + g(u')+cu=p(t)$, where $g \colon \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. We prove the existence of $\omega$-periodic solution for this equation, using coincidence degree theories.

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x ″ + a ( t ) x − b ( t ) x 2 + c ( t ) x 3 = 0 , where a ( t ) , b ( t ) , c ( t ) are T -periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T -periodic solutions for this equation, using coincidence degree theories.

Existence and infinity of periodic solutions of some second order differential equations with isochronous potentials

In this paper, we deal with the existence and infinity of periodic solutions of differential equations, $$x^{\prime\prime}+f(x^{\prime})+V^{\prime}(x)+g(x)=p(t),$$ where V is a 2π/n-isochronous potential. When f, g are bounded, we give sufficient conditions to ensure the existence of periodic solutions of this equation. We also prove that the given equation has infinitely many 2π-periodic solutions under resonant conditions by using the topological degree approach.

Pseudo almost periodic solutions for equation with piecewise constant argument

By using the roughness theory of exponential dichotomies and the contraction mapping, some sufficient conditions are obtained for the existence and uniqueness of pseudo almost periodic solution of the above differential equation with piecewise constant argument d x d t = A ( t ) x ( t ) + ∑ j = 0 r A j ( t ) x ( [ t − j ] ) + g ( t , x ( t ) , x ( [ t ] ) , … , x ( [ t − r ] ) ) .

Almost periodic solutions of neutral functional differential equations

We study a nonlinear neutral functional differential equation. Applying the properties of almost periodic function and exponential dichotomy of linear system as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of almost periodic solution of the equations.

Existence of asymptotically stable periodic solutions of a Rayleigh type equation

In this work, we investigate the following Rayleigh type equation: x ″ ( t ) + f ( x ′ ( t ) ) + g ( t , x ( t ) ) = e ( t ) . Some criteria for guaranteeing the existence of asymptotically stable (in the sense of Lyapunov) periodic solutions of this equation is presented by using Mawhin’s continuation theorem, Floquet theory, Lyapunov stability theory and some analysis techniques. Moreover, an example is provided to illustrate the validity of our results.

A study on periodic solutions of higher order nonlinear functional difference equations with p-Laplacian†

The study on the existence of periodic solutions of higher order nonlinear functional difference equations with p-Laplacian is made. Sufficient conditions for the existence of at least one periodic solution of these equations are established, respectively. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on Mawhin's continuation theorem. The methods we used to estimate a priori bound on periodic solutions are also new.

On the existence of periodic solution for equation of motion of thick beams having arbitrary cross section with tip mass under harmonic support motion

A cantilever beam having arbitrary cross section with a lumped mass attached to its free end while being excited harmonically at the base is fully investigated. The derived equation of vibrating motion is found to be a non-linear parametric ordinary differential equation, having no closed form solution for it. We have, therefore, established the sufficient conditions for the existence of periodic oscillatory behavior of the beam using Green’s function and employing Schauder’s fixed point theorem.

Periodic solutions of Liénard equations at resonance

In this paper we study the existence of periodic solutions of the second order differential equation $$ x''+f(x)x'+n^2x+\varphi(x)=p(t), \quad n\in {{\bf N}} . $$ We assume that the limits $$ \lim_{x\to\pm\infty}\varphi(x)=\varphi(\pm\infty),\quad \lim_{x\to\pm\infty}F(x)=F(\pm\infty)\quad \Big( F(x) =\int_0^xf(u)du \Big) $$ exist and are finite. We prove that the given equation has at least one $2\pi$-periodic solution provided that (for $A=\int_0^{2\pi}p(t)\sin nt dt, B=\int_0^{2\pi}p(t)\cos nt dt$) one of the following conditions is satisfied: $$ 2(\varphi(+\infty)-\varphi(-\infty))>\sqrt{A^2+B^2} $$ $$ 2n(F(+\infty)-F(-\infty))>\sqrt{A^2+B^2} $$ $$ 2(\varphi(+\infty)-\varphi(-\infty))=\sqrt{A^2+B^2}, \quad F(+\infty)\not=F(- \infty) $$ $$ 2n(F(+\infty)-F(-\infty))=\sqrt{A^2+B^2}, \quad \varphi(+\infty)\not=\varphi(- \infty). $$ On the other hand, we prove the non-existence of $2\pi$-periodic solutions provided that the inequality $$ 2(\varphi(+\infty)-\varphi(-\infty))+2n(F(+\infty)-F(-\infty))\leq\sqrt {A^2+B^2} $$ and other conditions hold. We also deal with the existence of $2\pi$-periodic solutions of the equation when $\varphi$ satisfies a one-sided sublinear condition and $F$ is bounded.

Periodic solutions of equations with oscillating nonlinearities

In this paper, we consider the 2π-periodic problem for the equation x″ + n 2 x = f( x) − b( t), where n is a positive integer, b( t) is continuous and 2π-periodic, and f( x) is bounded and continuous. We give a new formulation for the Lazer-Leach conditions for the existence of 2π-periodic solutions, and new sufficient conditions for the existence of unbounded sequences of such solutions. © 2000 Elsevier Science Ltd. All rights reserved.

On the existence of periodic solutions for the third-order nonlinear ordinary differential equations

In this paper, the existence of periodic solution for the third-order nonlinear ordinary differential equation of the form {\(\dddot x + f\left( {\ddot x} \right) + g\left( {\dot x} \right) + h\left( x \right) = p\left( t \right)\)} is considered, where f, g, h and p are the continuous functions, and p(t+T)=p(t). By using the Leray-Schauder degree method, some sufficient conditions to guarantee the existence of T periodic solution for the e-quation are obtained. Some previous results are extended and improved.