Periodic solutions to Mckean–Vlasov SDEs under Lyapunov conditions

The existence of periodic solutions to certain impulsive differential equations

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.

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.

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

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.

As an important tool to study practical problems of biology, engineering and image processing, the cellular neural networks (CNNs) has caused more and more attention. Some interesting results about the existence of solution for cellular neural networks have been obtained. In this paper, by means of iterative analysis, the existence of periodic solution and the uniform stability of the equilibrium point of impulsive cellular neural networks with distributed delays are considered. Some new results are obtained.

Due to resource limitation, nonlinear impulsive control tactics related to integrated pest management have been proposed in a generalized pest-natural enemy model, which allows us to address the effects of nonlinear pulse control on the dynamics and successful pest control. The threshold conditions for the existence and global stability of pest-free periodic solution are provided by Floquet theorem and analytic methods. The existence of a nontrivial periodic solution is confirmed by showing the existence of nontrivial fixed point of the stroboscopic mapping determined by time snapshot, which equals to the common impulsive period. In order to address the applications of generalized results and to reveal how the nonlinear impulses affect the successful pest control, as an example the model with Holling II functional response function is investigated carefully. The main results reveal that the pest free periodic solution and a stable interior positive periodic solution can coexist for a wide range of parameters, which indicates that the local stability does not imply the global stability of the pest free periodic solution when nonlinear impulsive control is considered, and consequently the resource limitation (i.e. nonlinear control) may result in difficulties for successful pest control.

The existence of periodic solutions for the third‐order differential equation is studied. We give some conditions for this equation in order to reduce it to a second‐order nonlinear differential equation. We show that the existence of periodic solutions for the second‐order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second‐order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions if has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).

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

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