The existence of periodic solutions to certain impulsive differential equations

Novel Existence and Stability Criteria of Periodic Solutions for Impulsive Delayed Neural Networks Via Coefficient Integral Averages

In this article, we first investigate the linear difference operator (Ax)(t):=x(t)-sum_{i=1}^{n}c_{i}(t)x(t- delta _{i}(t)) in a continuous periodic function space. The existence condition and some properties of the inverse of the operator A are explicitly pointed out. Afterwards, as applications of properties of the operator A, we study the existence of periodic solutions for two kinds of second-order functional differential equations with this operator. One is a kind of second-order functional differential equation, by applications of Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive periodic solutions are established. Another one is a kind of second-order quasi-linear differential equation, we establish the existence of periodic solutions of this equation by an extension of Mawhin’s continuous theorem.

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

A mathematical model on schistosomiasis governed by periodic differential equations with a time delay was studied. By discussing boundedness of the solutions of this model and construction of a monotonic sequence, the existence of positive periodic solution was shown. The conditions under which the model admits a periodic solution and the conditions under which the zero solution is globally stable are given, respectively. Some numerical analyses show the conditional coexistence of locally stable zero solution and periodic solutions and that it is an effective treatment by simply reducing the population of snails and enlarging the death ratio of snails for the control of schistosomiasis.

We study the existence of positive periodic solutions of second-order singular differential equations. The proof relies on Schauder’s fixed point theorem. Our results generalized and extended those results contained in the studies by Chu and Torres (2007) and Torres (2007) . In some suitable weak singularities, the existence of periodic solutions may help.

Stability analysis of a Komarova type model for the interactions of osteoblast and osteoclast cells during bone remodeling

This paper studies the existence and uniqueness of periodic solutions to a class of nonlinear neutral matrix difference systems with multiple delays. The analysis is based on the construction of a suitable Green operator combined with fixed-point methods under exponential dichotomy assumptions. The existence of periodic solutions is established using Krasnoselskii’s fixed-point theorem, while uniqueness is demonstrated under a natural contraction condition via Banach’s principle. The results extend previous contributions on neutral difference systems and provide discrete analogues of related differential models. Examples are included to illustrate the applicability of the theory.

Existence of Periodic Solutions in Impulsive Differential Equations

Unequal partitioning of the molecular content at cell division has been shown to be a source of heterogeneity in a cell population. We propose to model this phenomenon with the help of a scalar, nonlinear impulsive differential equation (IDE). To study the effect of molecular partitioning at cell division on the effector/memory cell-fate decision in a CD8 T-cell lineage, we study an IDE describing the concentration of the protein Tbet in a CD8 T-cell, where impulses are associated to cell division. We discuss how the degree of asymmetry of molecular partitioning can affect the process of cell differentiation and the phenotypical heterogeneity of a cell population. We show that a moderate degree of asymmetry is necessary and sufficient to observe irreversible differentiation. We consider, in a second part, a general autonomous IDE with fixed times of impulse and a specific form of impulse function. We establish properties of the solutions of that equation, most of them obtained under the hypothesis that impulses occur periodically. In particular, we show how to investigate the existence of periodic solutions and their stability by studying the flow of an autonomous differential equation. Then we apply those properties to prove the results presented in the first part.

Existence and global attractivity of periodic solution for an impulsive delay differential equation with Allee effect

The existence of multiple periodic solutions of the following differential delay equation x′(t) = −f(x(t − r)) is established by applying variational approaches directly, where x ∈ ℝ, f ∈ C(ℝ, ℝ) and r > 0 is a given constant. This means that we do not need to use Kaplan and Yorke′s reduction technique to reduce the existence problem of the above equation to an existence problem for a related coupled system. Such a reduction method introduced first by Kaplan and Yorke in (1974) is often employed in previous papers to study the existence of periodic solutions for the above equation and its similar ones by variational approaches.

Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations

As the impulsive differential equations are useful in modelling many real processes observed in physics, chemistry, biology, engineering, etc., see [1, 11, 13, 20, 21, 22, 25, 26, 27], there has been an increasing interest in studying such equations from the point of view of stability, asymptotic behavior, existence of periodic solutions, and oscillation of solutions. The classical theory can be found in the monographs [9, 18]. Recently, the oscillation theory of impulsive differential equations has also received considerable attention, see [2, 14] for the Sturmian theory of impulsive differential equations, and [15] for a Picone type formula and its applications. Due to difficulties caused by the impulsive perturbations the solutions are usually assumed to be continuous in most works in the literature. In this paper, we consider second order non-selfadjoint linear impulsive differential equations with discontinuous solutions. Our aim is to derive a Picone type identity for such impulsive differential equations, and hence extend and generalize several results in the literature. We consider second order linear impulsive differential equations of the form

A system of functional differential equations is used to modelthe single microorganism in the chemostat environment with aperiodic nutrient and antibiotic input. Based on the techniqueof Razumikhin, we obtain the sufficient condition for uniformpersistence of the microbial population. For general periodicfunctional differential equations, we obtain a sufficientcondition for the existence of periodic solution, therefore,the existence of positive periodic solution to the chemostat-typemodel is verified.

Periodic solutions for a class of neutral functional differential equations with infinite delay