Existence Of Positive Periodic Solutions Research Articles

Weak and strong singularities for second-order nonlinear differential equations with a linear difference operator

A class of second-order singular differential equations with a linear difference operator is investigated in this paper. The novelty of the present article is that for the first time we show that weak and strong singularities enable the achievement of a new existence criterion of positive periodic solutions through an applications of a fixed point theorem of Krasnoselskii’s, i.e., our results of the existence of positive periodic solutions reveal a delicate relation between the value of external force e(t) and the velocity of nonlinear term $$f(t,x(t-\tau (t)))$$ approaching towards infinity when $$x(t-\tau (t))$$ tending to zero.

Existence of positive periodic solutions for a neutral Liénard equation with a singularity of repulsive type

The periodic problem is studied in this paper for the neutral Lienard equation with a singularity of repulsive type $$\begin{aligned} (x(t)-cx(t-\sigma ))''+f(x(t))x'(t)+\varphi (t)x(t-\tau )-\frac{r(t)}{x^{\mu }(t)}=h(t), \end{aligned}$$ where $$f:[0,+\infty )\rightarrow R$$ is continuous, $$r: R\rightarrow (0,+\infty )$$ and $$\varphi :R \rightarrow R$$ are continuous with T-periodicity in the t variable, $$c,\mu ,\sigma ,\tau $$ are constants with $$|c|>1,\mu >1,0<\sigma ,\tau <T$$ . Many authors obtained the existence of periodic solutions under the condition $$|c|<1$$ , and we extend their results to the case of $$|c|>1$$ . The proof of the main result relies on a continuation theorem of coincidence degree theory established by Mawhin.

Study on variable coefficients singular differential equation via constant coefficients differential equation

In this paper, we consider the following third-order singular differential equation with variable coefficients: \t\t\tx‴+a(t)x=f(t,x)+e(t).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x'''+a(t)x=f(t,x)+e(t). $$\\end{document} By using the Green’s function of the linear differential equation with constant coefficients and some fixed point theorems, i.e., Leray–Schauder alternative principle and Schauder’s fixed point theorem, we prove the existence of positive periodic solutions of the above equation.

Multiperiodicity to a Certain Delayed Predator–Prey Model

The delayed predator–prey system with generalized non-monotonic functional responses and stage structure was investigated in the present paper. By virtue of Mawhin’s coincidence degree and the application of inequalities technique, we are successful to generate some novel conditions to guarantee that the system has at least two positive periodic solutions. It is shown that all parameters of the system have effects on the existence of positive periodic solutions and the period of the coefficients can also affect the existence of positive periodic solutions. In the end, an illustrative example is presented to the feasibility of the main results.

On the existence and stability of positive periodic solution of a nonautonomous commensal symbiosis model with Michaelis-Menten type harvesting

A non-autonomous commensal symbiosis model of two populations with Michaelis-Menten type harvesting is proposed and studied in this paper. By using a continuation theorem based on Gaines and Mawhin's coincidence degree, we study the global existence of positive periodic solutions of the system. By constructing a suitable Lyapuonov function, sufficient conditions which ensure the global attractivity of the positive periodic solution are obtained. Numeric simulations are carried out to show the feasibility of the main results.

ANALYSIS OF A STOCHASTIC RECOVERY-RELAPSE EPIDEMIC MODEL WITH PERIODIC PARAMETERS AND MEDIA COVERAGE

This paper addresses, motivated by mathematical work on infectious disease models, the impacts of environmental noise and media coverage on the dynamics of recovery-relapse infectious diseases. A susceptibleinfectious-recovered-infectious model is formulated with both vertical transmission and horizontal transmission. The existence and uniqueness of the positive global solution is studied by constructing suitable Lyapunov-type function. Then, the existence of positive periodic solutions is verified by applying Khasminskii's theory. The existence of positive periodic solutions indicates the continued survival of the diseases. Besides, sufficient conditions for the extinction of the diseases are obtained. Numerical simulations then demonstrate the dynamics of the solutions. The paper extends the results of the corresponding deterministic system.

Existence of positive periodic solutions for third-order nonlinear delay differential equations with variable coefficients

In this paper, the following third-order nonlinear delay differential equation with periodic coefficients x'''(t) + p(t)x''(t) + q(t)x'(t) + r(t)x(t) = f(t,x(t), x(t - t (t))) + d/dt g (t, x(t - t (t))), is considered. By employing Green's function and Krasnoselskii's fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order nonlinear delay differential equation.

非自治时滞捕食–食饵模型正周期解的存在性

非自治时滞捕食–食饵模型正周期解的存在性

Dynamics Analysis of an Avian Influenza A (H7N9) Epidemic Model with Vaccination and Seasonality

H7N9 virus in the environment plays a role in the dynamics of avian influenza A (H7N9). A nationwide poultry vaccination with H7N9 vaccine program was implemented in China in October of 2017. To analyze the effect of vaccination and environmental virus on the development of avian influenza A (H7N9), we establish an avian influenza A (H7N9) transmission model with vaccination and seasonality among human, birds, and poultry. The basic reproduction number for the prevalence of avian influenza is obtained. The global stability of the disease‐free equilibrium and the existence of positive periodic solution are proved by the comparison theorem and the asymptotic autonomous system theorem. Finally, we use numerical simulations to demonstrate the theoretical results. Simulation results indicate that the risk of H7N9 infection is higher in colder environment. Vaccinating poultry can significantly reduce human infection.

The supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations

In this paper, we provide supplementation of the theory of periodic solutions for a class of nonlinear diffusion equations ut−Δum=a(x,t)up with m > 0, p > 0. By using a new method to deal with the “blow up sequence”, we obtain the L∞ estimate with optimal exponent 1<pm<ps, where ps is the Sobolev critical exponent, then we apply topological degree method to prove the existence of positive periodic solutions. If pm≥ps,N≥3 and Ω is star–shaped, we establish Rellich–Pohozaev type identity to prove the nonexistence of positive periodic solutions. We improve the results of Mizoguchi [10, Therem 1] by filling the gap m>1,pm≥1+2mN. In addition, we also give quite complete results about the existence of periodic solutions for the fast diffusion case 0 < m < 1. As far as we know, there are no results about the fast diffusion case before present work.

Periodic Solution for a Stochastic Non-autonomous Predator-Prey Model with Holling II Functional Response

A biological population may be subjected to stochastic disturbance and exhibit periodicity. In this paper, a stochastic non-autonomous predator-prey system with Holling II functional response is proposed, and the existence of a unique positive solution is derived. We give sufficient conditions for extinction and strong persistence in the mean by analyzing a corresponding one-dimensional stochastic system. Also we establish the existence of positive periodic solutions for this stochastic non-autonomous predator-prey system. Finally, we use numerical simulations to illustrate our results and we present some conclusions and future directions. The results of this paper provide methods for other stochastic population models, which we hope to analyze in the future.

Existence of positive periodic solutions for the impulsive Lotka\u2013Volterra cooperative population model with time-delay and harvesting control on time scales

In this paper, we consider a class of impulsive Lotka–Volterra cooperative population models with time delay and harvesting control on time scales. Using the fixed point theorem of strict-set-contraction, we analyze the existence conditions of positive periodic solutions for this model. As applications, we analyze the existence conditions of positive periodic solutions for some common Lotka–Volterra systems on time scales.

Existence of positive periodic solutions for Li\xe9nard equations with an indefinite singularity of attractive type

In this paper, we study the periodic problem for the Liénard equation with an indefinite singularity of attractive type \t\t\tx″+f(x)x′+φ(t)x+r(t)xμ=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x''+f(x)x'+\\varphi (t)x+\\frac{r(t)}{x^{\\mu }}=0, $$\\end{document} where f:(0,+infty )rightarrow R is continuous and may have singularities at zero, r, varphi : Rrightarrow R are T-periodic functions, and μ is a positive constant. Using the method of upper and lower functions, we obtain some new results on the existence of positive periodic solutions to the equation.

Systems of delay differential equations: Analysis of a model with feedback

Using topological degree theory, we prove the existence of positive periodic solutions of a system of delay differential equations for models with feedback arising on regulatory mechanisms in which self-regulation is relevant, e.g. in cell physiology. We study different models based on the cycle of testosterone and generalizations. The method in the present work allows to analyse and extend known results from a different perspective, shortening proofs and giving an alternative approach for the study of complex models.

Positive periodic solutions for singular fourth-order differential equations with a deviating argument

In this paper, we study the singular fourth-order differential equation with a deviating argument:By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.

Existence of positive periodic solution of second-order neutral differential equations

In this work, we consider two types of second-order neutral differential equations and we obtain sufficient conditions for the existence of positive $\om$-periodic solutions for these equations. We employ Krasnoselskii's fixed point theorem for the sum of a completely continuous and a contraction mapping. An example is included to illustrate our results.

A delayed impulsive food chain system with prey refuge and mutual inference of predator

In this paper, a three-species delayed food chain system with mutual interference and impulses is established. By utilizing the continuation theorem, the comparison theorem, some analysis techniques, and constructing a suitable Lyapunov functional, some sufficient conditions for the existence of positive periodic solutions, permanence, and global attractivity of the system are obtained. The conditions obtained are related to the mutual interference constant m, prey refuge constant θ, delays, and impulses. An example is given to show the feasibility of the results.

A note on the existence of time periodic solution of a superlinear heat equation

In this paper, we consider the existence of positive periodic solutions for a superlinear heat equation with and being the Sobolev critical exponent, subject to homogeneous Dirichlet boundary condition. We relax the technical assumption on a(t) and convex assumption on the domain , which generalizes the previous results.

Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale

Let T be a periodic time scale. The purpose of this paper is to use Krasnoselskii's fixed point theorem to prove the existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale. We invert these equations to construct a sum of a contraction and a compact map which is suitable for applying the Krasnoselskii's theorem. The results obtained here extend the work of Candan &lt;cite&gt;c1&lt;/cite&gt;.

Positive periodic solutions for singular high-order neutral functional differential equations

Abstract In this paper, we establish new results on the existence of positive periodic solutions for the following high-order neutral functional differential equation (NFDE) ( x ( t ) − c x ( t − σ ) ) ( 2 m ) + f ( x ( t ) ) x ′ ( t ) + g ( t , x ( t − δ ) ) = e ( t ) . $$\begin{array}{} (x(t)-cx(t-\sigma)) ^{(2m)}+f(x(t)) x'(t)+g(t,x(t-\delta))=e(t). \end{array}$$ The interesting thing is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the corresponding ones known in the literature. Two examples are given to illustrate the effectiveness of our results.