Existence of positive periodic solution for variable–coefficient third‐order differential equation with singularity

In this paper, we investigate a class of singular third‐order differential equations with variable coefficients. By application of Green's functions and Schauder's fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established. Copyright © 2013 John Wiley & Sons, Ltd.     

Study on a kind of $p$-Laplacian neutral differential equation with multiple variable coefficients

In this paper, we first discuss some properties of the neutral operator with multiple variable coefficients $(Ax)(t):=x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)$. Afterwards, by using an extension of Mawhin''s continuation theorem, a kind of second order $p$-Laplacian neutral differential equation with multiple variable coefficients as follows $$\left(\phi_p\left(x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)\right)''\right)''=\tilde{f}(t,x(t),x''(t))$$ is studied. Finally, we consider the existence of periodic solutions for two kinds of second-order $p$-Laplacian neutral Rayleigh equations with singularity and without singularity. Some new results on the existence of periodic solutions are obtained. It is worth noting that $c_i$ ($i=1,\cdots,n$) are no longer constants which are different from the corresponding ones of past work.     

带阻尼项的二阶奇异耦合系统的正周期解

We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′+ p_1(t)x′+ q_1(t)x = f_1(t, y(t)) + c_1(t),y′′+ p_2(t)y′+ q_2(t)y = f_2(t, x(t)) + c_2(t),where pi, qi, ci ∈ C(R/T Z; R), i = 1, 2; f_1, f_2 ∈ C(R/T Z ×(0, ∞), R) and may be singular near the zero. The proof relies on Schauder's fixed point theorem and anti-maximum principle.Our main results generalize and improve those available in the literature.     

Existence of time periodic solutions for the Nicholson's blowflies model with Newtonian diffusion

This paper is concerned with the existence of periodic solutions of the Nicholson's blowflies model with Newtonian diffusion. By constructing some suitable Lyapunov functionals and combining with Leray–Schauder fixed point theorem, we establish the existence of nonnegative time periodic solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiplicity results of positive solutions for fourth‐order nonlinear differential equation with singularity

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed‐point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation, which extend and improve significantly existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

具有脉冲的分数阶Bagley-Torvik 模型边值问题

将具有脉冲的分数阶Bagley-Torvik微分方程边值问题巧妙地转化为积分方程,定义加权Banach空间及全连续算子,运用不动点定理获得该边值问题解的存在性定理.举例说明了定理的应用.最后提出有趣的研究问题.     

Positive solutions for fourth‐order singular nonlinear differential equation with variable‐coefficient

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation with variable‐coefficient, which extend and improve significantly existing results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Global stability of periodic solutions of predator-prey system with Holling type III functional response

This paper studied some properties of a predator-prey system with Holling type III functional response. Based on Mawhin''s Continuation Theorem, some sufficient conditions for the existence of periodic solutions are obtained. Moreover, the global stability of the periodic solution is built with the help of a suitable Lyapunov function.     

A Positive Solution of Mixed Non-Linear Fractional Delay Differential Equations with Integral Boundary Conditions

In this paper, we study mixed non-linear fractional delay differential equations with integral boundary conditions. We obtain an equivalence result between the proposed problem and non-linear Fredholm integral equation of the second kind. Further, we establish existence and uniqueness of positive solutions for the problem using Guo-Krasnoseleskii''s fixed point theorem and Banach contraction principle.     

Multiple periodic solutions for non-linear difference systems involving the p-Laplacian

Using the three critical points theorem, Clark's theorem and the Morse theory, multiple periodic solutions for non-linear difference systems involving the p-Laplacian are obtained by variational methods.     

锥度量空间中基于Ekeland变分原理的向量均衡问题的解的存在性

本文研究了向量均衡问题.利用在锥度量空间中给出的Ekeland变分原理,我们推导了向量均衡问题解的存在性定理.本文的结论是新的并推广了相关文献中的结论.     

Cohen-Fischman-Westreich's Double Centralizer Theorem for Almost-triangular Hopf Algebras

In this paper, the authors study the Cohen-Fischman-Westreich''s double centralizer theorem for triangular Hopf algebras in the setting of almost-triangular Hopf algebras.     

Existence results of periodic solutions for third‐order nonlinear singular differential equation

Using Green's function for third‐order differential equation and some fixed‐point theorems, i.e., Leray‐Schauder alternative principle and Schauder's fixed point theorem, we establish three new existence results of periodic solutions for nonlinear third‐order singular differential equation, which extend and improve significantly existing results in the literature.     

Periodic solutions of an iterative functional differential equation with variable coefficients

In this paper, we use Krasnoselskii's fixed point theorem to study the existence and uniqueness of periodic solutions of an iterative functional differential equation Copyright © 2016 John Wiley & Sons, Ltd.     

Positive almost periodic solution for impulsive Nicholson's blowflies model with multiple linear harvesting terms

In this paper, by means of the Cauchy matrix and fixed pointed theorem, we give sufficient conditions for the existence and exponential stability of almost periodic solutions for the impulsive Nicholson's blowflies model with multiple linear harvesting terms and infinite delay. An example is provided to illustrate the effectiveness of the new results. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiplicity results of fourth-order singular nonlinear differential equation with a parameter

In this paper, we investigate a class of fourth-order singular nonlinear differential equation with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. By applications of Green''s function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established.     

广义阿贝尔微分方程的周期解

本文讨论了广义阿贝尔微分方程.利用不动点定理,得到了方程存在两个非零周期解的充分条件.同时,我们还讨论了不存在非零周期解和存在唯一非零周期解的情况.     

The existence of periodic solutions for three-order neutral differential equations

In this paper, by using Kranoselskii fixed point point theorem and Mawhin''s continuation theorem, we establish two existence theorem on the periodic solutions for a class of three-order neutral differential equations.     

Traveling Waves of a Reaction-Diffusion SIRQ Epidemic Model with Relapse

This paper is concerned with the traveling waves of a reaction-diffusion SIRQ epidemic model with relapse. We find that the existence and nonexistence of traveling waves are determined by the basic reproduction number of the system and the minimal wave speed. This threshold dynamics is proved by Schauder''s fixed-point theorem combining Lyapunov functional with the theory of asymptotic spreading. Moreover, the numerical simulations are provided to illustrate our analytical results and the effect of the relapse is also discussed.     

A study on periodic solutions of higher order nonlinear functional difference equations with p-Laplacian†

The study on the existence of periodic solutions of higher order nonlinear functional difference equations with p-Laplacian is made. Sufficient conditions for the existence of at least one periodic solution of these equations are established, respectively. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on Mawhin's continuation theorem. The methods we used to estimate a priori bound on periodic solutions are also new.