Periodic solutions of p-Laplacian differential equations with jumping nonlinearity across half-eigenvalues

This paper aims to investigate the existence of periodic solutions for $p$ -Laplacian differential equations with jumping nonlinearity under the frame of half-eigenvalue. Based on the continuity theorem, some new results are obtained, which enrich and generalize the previous results.

Dynamic Analysis in a Plankton Ecosystem Involving Two Delays

In this paper, a phytoplankton and zooplankton relationship system with two delays is investigated whose coefficients are related to one of the two delays. Firstly, the dynamic behaviors of the system with one delay are given and the stability of positive equilibrium and the existence of periodic solutions are obtained. Using the fact that the system may occur, the stable switching phenomenon is verified. Under certain conditions, the periodic solutions will exist in a wide range as the delay gets away from critical values. Fixing the delay [Formula: see text] in the stable interval, it is revealed that the effect of [Formula: see text] can also cause the vibration of system. This explains that two delays play an important role in the oscillation behavior of the system. Furthermore, using the crossing curve methods, the stable changes of the positive equilibrium in two-delays plane are given, which generalizes the results of systems for which the coefficients do not depend on delay. Some numerical simulations are provided to verify the theoretical results.

On Truly Nonlinear Oscillator Equations of Ermakov-Pinney Type

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

Lumped parameter models for two-ventricle and healthy and failing extracardiac Fontan circulations

Fontan circulations are surgical strategies to treat infants born with single ventricle physiology. Clinical and mathematical definitions of Fontan failure are lacking, and understanding is needed of parameters indicative of declining physiologies. Our objective is to develop lumped parameter models of two-ventricle and single-ventricle circulations. These models, their mathematical formulations and a proof of existence of periodic solutions are presented. Sensitivity analyses are performed to identify key parameters. Systemic venous and systolic left ventricular compliances and systemic capillary and pulmonary venous resistances are identified as key parameters. Our models serve as a framework to study the differences between two-ventricle and single-ventricle physiologies and healthy and failing Fontan circulations.

Periodic solutions of superlinear and sublinear state-dependent discontinuous differential equations

A classical, second-order differential equation is considered with state-dependent impulses at both the position and its derivative. This means that the instants of impulsive effects depend on the solutions and they are not fixed beforehand, making the study of this problem more difficult and interesting from the real applications point of view. The existence of periodic solutions follows from a transformation of the problem into a planar system followed by a study of the Poincaré map and the use of some fixed point theorems in the plane. Some examples are presented to illustrate the main results.

Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces

We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

Periodic Solutions with Prescribed Minimal Period for 2 n th-Order Nonlinear Discrete Systems

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2 n th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2 n th-order discrete systems.