Multiple positive solutions for nonlinear periodic problems

We consider a nonlinear periodic equation driven by the scalar p-Laplacian and with a Caratheodory asymptotically (p−1)-linear nonlinearity. Using variational methods coupled with suitable truncation techniques, we show that the problem has at least two positive solutions. For the semilinear case (p=2), using Morse theory we show that the problem has three distinct positive solutions.     

On the periodic boundary-value problem for systems of second-order nonlinear ordinary differential equations

The periodic boundary value problem for systems of secondorder ordinary nonlinear differential equations is considered. Sufficient conditions for the existence and uniqueness of a solution are established.     

The monotone method for periodic differential equations with the non well-ordered upper and lower solutions

In this paper, we consider the following periodic problem:

     

Multiple positive solutions of singularly perturbed differential systems with different orders

In this paper, we consider the multiplicity of positive solutions for a class of singular higher-order perturbed differential systems with different orders. By employing a well-known fixed point theorem, some new existence results are given under the case where nonlinearity can be sign changing.     

New results of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T$T$-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(t,x^′(t))+g(t,x(t))=e(t).$x^{″}+f(t,x^{\prime }\left(t\right))+g(t,x\left(t\right))=e\left(t\right)\text{.}$

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

Multiple periodic solutions of scalar second order differential equations

We prove multiplicity of periodic solutions for a scalar second order differential equation with an asymmetric nonlinearity, thus generalizing previous results by Lazer and McKenna (1987) [1] and Del Pino, Manasevich and Murua (1992) [2]. The main improvement lies in the fact that we do not require any differentiability condition on the nonlinearity. The proof is based on the use of the Poincaré-Birkhoff Fixed Point Theorem.     

Positive periodic solutions of singular systems with a parameter

The existence and multiplicity of positive periodic solutions for second-order non-autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. Our results provide a unified treatment for the problem and significantly improve several results in the literature. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.     

Stability, resonance and Lyapunov inequalities for periodic conservative systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

The existence of solutions of infinite boundary value problems for first-order impulsive differential systems in Banach spaces

In this paper, the existence of solutions of infinite boundary value problems for first-order impulsive differential systems is obtained by means of the Schauder fixed point theorem in a Banach space.     

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

[φ_p([x(t)−c(t)x(t−σ)]⁽ⁿ⁾)]^(m)+f(x(t))x^′(t)+g(t,x(t−τ(t)))=e(t)     

On periodic solutions of second-order differential equations with attractive-repulsive singularities

Sufficient conditions for the existence of a solution to the problem

     

Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems

The purpose of this paper is two-fold. Firstly, we will give some parabolic-like conditions which improve the well-known angle conditions and allow further computations of the critical groups both at degenerate critical points and at infinity. As an application, we then consider the second-order Hamiltonian systems

     

Stable phase-locked periodic solutions in a delay differential system

For a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function, we obtain the existence, stability and limiting profile of a phase-locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory.     

Multiple periodic solutions of a delayed stage-structured predator–prey model with non-monotone functional responses

A delayed stage-structured predator–prey model with non-monotone functional responses is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age, and that immature predators do not have the ability to attack prey. Some new and interesting sufficient conditions are obtained for the global existence of multiple positive periodic solutions of the stage-structured predator–prey model. Our method is based on Mawhin’s coincidence degree and novel estimation techniques for the a priori bounds of unknown solutions to Lx = λNx. An example is given to illustrate the feasibility of our main result.     

Existence and multiplicity results for nonlinear nonautonomous second-order systems

In this paper we consider nonlinear periodic systems driven by the ordinary p$p$-Laplacian and having a nonsmooth potential function. Under minimal and natural hypotheses on the nonsmooth potential and using variational methods based on the nonsmooth critical point theory we prove four existence theorems and a multiplicity theorem. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (i.e. it is unbounded both above and below).