Periodic solutions of Liénard equation with a singularity and a deviating argument

In this paper, we study the existence of periodic solutions of the Liénard equation with a singularity and a deviating argument $x^{″}+f\left(x\right)x^{\prime }+g(t,x(t-\sigma ))=0\text{.}$ When $g$ has a strong singularity at $x=0$ and satisfies a new small force condition at $x=\infty$, we prove that the given equation has at least one positive $T$-periodic solution.     

Existence and uniqueness of periodic solutions for?a?kind of?Liénard equation with multiple deviating arguments

In this paper, a kind of Liénard equation with multiple deviating arguments of the form $$x''(t)+f(x(t))x'(t)+\sum_{k=1}^{n}g_{k}(t,x(t-\tau_{k}(t)))=p(t)$$ is considered. By using coincidence degree theory, some criteria are obtained to guarantee the existence and uniqueness of periodic solutions of this equation. The obtained results are also valid and new for the problem discussed in the previous literature.     

Periodic solutions of the retarded Liénard equation

Consider the equation

New stability and boundedness results of Liénard type equations with multiple deviating arguments

The paper considers a Liénard type equation withmultiple variable deviating arguments. Some sufficient conditions, under which the solution of this equation is asymptotically stable and bounded by means of the Lyapunov functional approach, are found. An example showing the effectiveness of the result is given.     

Quasi-periodic solutions of a semilinear Liénard equation at resonance

We are concerned with the existence of quasi-periodic solutions for the following equation

Periodic solution to p-Laplacian neutral Liénard type equation with variable parameter

Using Mawhin’s continuation theorem we obtain some existence results of periodic solutions for a type of p-Laplacian neutral Liénard equation $$\left( {\phi _p \left( {\left( {x\left( t \right) - c\left( t \right)x\left( {t - \tau } \right)} \right)^\prime } \right)} \right)^\prime + f\left( {x\left( t \right)} \right)x'\left( t \right) + g\left( {x\left( {t - \gamma \left( t \right)} \right)} \right) = e\left( t \right).$$ It is worth noting that c(t) is no longer a constant which is different from the corresponding ones of past work.     

Quasi-periodic solutions of a semilinear Li(?)nard equation at resonance

We are concerned with the existence of quasi-periodic solutions for the follow- ing equation x″ F_x(x,t)x′ ω~2x φ(x,t)=0, where F and φare smooth functions and 2π-periodic in t,ω>0 is a constant.Under some assumptions on the parities of F and φ,we show that the Dancer's function,which is used to study the existence of periodic solutions,also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e.all solutions are bounded).     

On the criteria for integrability of the Liénard equation

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Liénard equation using an approach based on nonlocal transformations. We also obtain some of the previously known criteria for integrability of the Liénard equation as a straightforward consequence of our approach’s application. We illustrate our results by several new examples of integrable Liénard equations.     

Attractive singularity problems for superlinear Liénard equation

Positivity - In this paper, we consider the following quasilinear Liénard equation with a singularity $$\begin{aligned} (\phi _p(x'(t)))'+f(x(t))x'(t)+g(t,x(t))=e(t),...     

Existence of Positive Periodic Solutions for the Liénard Differential Equations with Weakly Repulsive Singularity

In this paper we discuss the existence of positive T-periodic solutions for the following second order differential equation

The localization of attractors of the Liénard equation

A method for localizing the attractors of the Liénard equation is proposed, based on the construction of special piecewise-linear discontinuous comparison systems.