Positive T-periodic Solutions Research Articles

Existence of periodic solutions of a Li\xe9nard equation with a singularity of repulsive type

In this paper, the problem of positive periodic solutions is studied for the Liénard equation with a singularity of repulsive type, \t\t\tx″+f(x)x′−α(t)xμ=h(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}$$ x''+f(x)x'-\\frac{\\alpha(t)}{x^{\\mu}}=h(t), $$\\end{document} where f:(0,+infty)rightarrow R is continuous, α, h are continuous with T-periodic and alpha(t)ge0 for all tin R, mu in(0,+infty) is a constant. By means of a Manásevich-Mawhin’s continuation theorem, a sufficient and necessary condition is obtained for the existence of positive T-periodic solutions of the equation. The interesting point is that the weak singularity of restoring force frac{alpha(t)}{x^{mu}} at x=0 is allowed and f may have a singularity at x=0.

Periodic solution and stationary distribution of stochastic SIR epidemic models with higher order perturbation

In this paper, we investigate two stochastic SIR epidemic models with higher order perturbation. For the nonautonomous periodic case of the model, by using Has’minskii’s theory of periodic solution, we show that the system has at least one nontrivial positive T-periodic solution. For the system disturbed by both the white noise and telephone noise, we establish sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the positive solution.

Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay

In this paper, we consider two SEIR epidemic models with distributed delay in random environments. First of all, by constructing a suitable stochastic Lyapunov function, we obtain the existence of stationarity of the positive solution to the stochastic autonomous system. Then we establish sufficient conditions for extinction of the disease. Finally, by using Khasminskii's theory of periodic solutions, we prove that the stochastic nonautonomous epidemic model admits at least one nontrivial positive T-periodic solution under a simple condition.

Periodic solutions for a stochastic non-autonomous Holling–Tanner predator–prey system with impulses

A stochastic non-autonomous Holling–Tanner predator–prey system with impulsive effect is investigated. First, we prove the existence and uniqueness of the global positive solution by constructing the equivalent system without impulse. Second, we give the sufficient condition for the existence of positive T-periodic solution by choosing a suitable Lyapunov function. Third, we prove the existence and global attraction of the boundary periodic solution by using the comparison theorem under a certain condition. Finally, numerical simulations illustrate our theoretical results, which show that the impulse or the white noises can result in the extinction of the predator in a certain condition.

Periodic solutions for a kind of prescribed mean curvature Liénard equation with a singularity and a deviating argument

In this paper, we study the existence of periodic solutions to the following prescribed mean curvature Lienard equation with a singularity and a deviating argument: $$\biggl(\frac{u'(t)}{\sqrt{1+(u'(t))^{2}}}\biggr)'+f\bigl(u(t)\bigr)u'(t)+g \bigl( u(t-\sigma)\bigr)=e(t), $$ where g has a strong singularity at $x=0$ and satisfies a small force condition at $x=\infty$ . By applying Mawhin’s continuation theorem, we prove that the given equation has at least one positive T-periodic solution. We will also give an example to illustrate the application of our main results.

Existence and uniqueness of a positive periodic solution for Rayleigh type ϕ-Laplacian equation

By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some sufficient condition for the existence and uniqueness of positive T-periodic solutions for a generalized Rayleigh type ϕ-Laplacian operator equation. The results of this paper are new and they complement previous known results.MSC:34K13, 34C25.

Periodic solutions of Liénard equation with a singularity and a deviating argument

In this paper, we study the existence of periodic solutions of the Liénard equation with a singularity and a deviating argument x″+f(x)x′+g(t,x(t−σ))=0. When g has a strong singularity at x=0 and satisfies a new small force condition at x=∞, we prove that the given equation has at least one positive T-periodic solution.

Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight

We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODE u ″ + λ a ( t ) g ( u ) = 0 , λ > 0 , where g ( x ) is a positive function on R + , superlinear at zero and sublinear at infinity, and a ( t ) is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities g ( x ) when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.

Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics

We prove the existence of a pair of positive T-periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic-like dynamics for the scalar second order ODE u ″ + a λ , μ ( t ) g ( u ) = 0 , where g ( x ) is a positive function on R + , superlinear at zero and sublinear at infinity, and a λ , μ ( t ) is a T-periodic and sign indefinite weight of the form λ a + ( t ) − μ a − ( t ) , with λ , μ > 0 and large.

Existence results on positive periodic solutions for impulsive functional differential equations

A class of first order nonlinear functional differential equations with impulses is studied. It is shown that there exist one or two positive T-periodic solutions under certain assumptions, and no positive T-periodic solution under some other assumptions. Applications to some impulsive biological models and an example, which can not be covered by known results, are given to illustrate the main results.

Coexistence and Asymptotic Periodicity in a Competition Model of Plankton Allelopathy

In this paper, the two species Lotka-Volterra competition model of plankton allelopathy from aquatic ecology is discussed. The authors study the existence of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions and consider the existence, stability and global attractivity of time-periodic solutions for a coupled parabolic equations in a bounded domain. Their results show that this model possesses at least one coexistence state if cross-diffusions and self-diffusions are weak. The existence of the positive T-periodic solutions and the global stability as well as the global attractivity for the parabolic system are also given.

Three periodic solutions for a nonlinear first order functional differential equation

This paper is concerned with the existence of three positive T-periodic solutions of the first order functional differential equations of the form x ′ ( t ) = a ( t ) x ( t ) - λ b ( t ) f ( t , x ( h ( t ) ) ) , where a ( t ) , b ( t ) and τ ( t ) are positive T-periodic functions, and λ is a positive parameter. Leggett–Williams multiple fixed point theorem has been used to prove our results. Our results extend and improve some results in the literature.

Existence of Positive Periodic Solutions for the Liénard Differential Equations with Weakly Repulsive Singularity

In this paper we discuss the existence of positive T-periodic solutions for the following second order differential equation $$\ddot{x}+f(x)\dot{x}+g(x)=c(t),$$ where f, g:ℝ+=(0,+∞)→ℝ are continuous functions, g has a repulsive singularity at the origin and c(t) is a continuous T-periodic function. The novelty of the present article is that for the first time we show that a weakly repulsive singularity enables the achievement of a new existence criterion of positive T-periodic solutions through a basic application of the coincidence degree theory. Recent results in the literature are generalized and significantly improved.

COEXISTENCE AND ASYMPTOTIC PERIODICITY IN A COOPERATING MODEL

In this paper, the cooperating two-species Lotka–Volterra model is discussed. The authors study the existence of solutions to a strongly coupled elliptic system with homogeneous Dirichlet boundary conditions and consider the existence, stability, and global attractivity of time-periodic solutions for a coupled parabolic equations in a bounded domain. Their results show that this model possesses at least one coexistence state if cross-diffusions are weak. The existence of the positive T-periodic solutions, the local stability, and the global attractivity for the parabolic system are also given.

FURTHER RESULTS ON POSITIVE PERIODIC SOLUTIONS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS AND APPLICATIONS

AbstractA class of first-order impulsive functional differential equations with forcing terms is considered. It is shown that, under certain assumptions, there exist positive T-periodic solutions, and under some other assumptions, there exists no positive T-periodic solution. Applications and examples are given to illustrate the main results.

Multiple positive periodic solutions for nonlinear first order functional difference equations

Sufficient conditions have been obtained for the existence of at least three positive T-periodic solutions for the first order functional difference equations of the forms Δx(n) = −a(n)x(n) + λb(n)f(n, x(h(n))) and Δx(n) = a(n)x(n) − λb(n)f(n, x(h(n))). Leggett-Williams multiple fixed point theorem have been used to prove our results. We have applied our results to some mathematical models in population dynamics and obtained some interesting results. The results are new in the literature.

Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation

This paper discusses a randomized non-autonomous logistic equation d N ( t ) = N ( t ) [ ( a ( t ) − b ( t ) N ( t ) ) d t + α ( t ) d B ( t ) ] , where B ( t ) is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164–172], the authors show that E [ 1 / N ( t ) ] has a unique positive T-periodic solution E [ 1 / N p ( t ) ] provided a ( t ) , b ( t ) and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and ∫ 0 T [ a ( s ) − α 2 ( s ) ] d s > 0 . We show that this equation is stochastically permanent and the solution N p ( t ) is globally attractive provided a ( t ) , b ( t ) and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and min t ∈ [ 0 , T ] a ( t ) > max t ∈ [ 0 , T ] α 2 ( t ) . By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded.

Positive periodic solutions of state-dependent functional differential equations

Consider the existence of positive periodic solutions of the non-autonomous state-dependent delay differential equation with the impulses as follows It is shown that under reasonable conditions, the equation has positive periodic solutions and under certain conditions the equation has no positive T-periodic solution.

Positive periodic solutions of nonlinear functional difference equations depending on a parameter

In this paper, we use the upper and lower solutions method to show that there existsa λ *, such that the nonlinear functional difference equation of the form χ(n + 1) = a(n)χ(n) + λh(n)ƒ(χ(n − τ(n))), n ε Z, has at least one positive T-periodic solutions for λ ε (0, λ * and does not have any positive T-periodic solutions for λ > λ *, where a(n), h(n), and τ(n) are T-periodic functions.

A note on nonautonomous logistic equation with random perturbation

This paper discusses a randomized nonautonomous logistic equation d N ( t ) = N ( t ) [ ( a ( t ) − b ( t ) N ( t ) ) d t + α ( t ) d B ( t ) ] , where B ( t ) is 1-dimensional standard Brownian motion. We show that E [ 1 N ( t ) ] has a unique positive T-periodic solution E [ 1 N p ( t ) ] provided a ( t ) , b ( t ) , and α ( t ) are continuous T-periodic functions, a ( t ) > 0 , b ( t ) > 0 and ∫ 0 T [ a ( s ) − α 2 ( s ) ] d s > 0 .