Global positive periodic solutions of periodic n-species competition systems

In this paper, an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of generalized n-species Lotka-Volterra type and Gilpin-Ayala type competition systems is obtained. It improves a series of the well-known sufficiency theorems in the literature about the problems mentioned above. The method is based on a well-known fixed point theorem in a cone of Banach space. This approach can be applied to more general competition systems.     

Existence of nonzero positive solutions of systems of second order

Existence of nonzero positive solutions  of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the well-known theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in  ordered Banach spaces.     

Global boundedness of the solutions to a Gurtin-MacCamy system

This paper is concerned with the analysis of a generalized Gurtin-MacCamy model describing the evolution of an age-structured population. The problem of global boundedness is studied. Namely we ask whether there are simple general assumptions that one can make on the vital rates in order to have boundedness of the solution. Next a fully implicit finite difference scheme along the characteristic is considered to approximate the solution of the system. Global boundedness of the numerical solutions is investigated. The optimal rate of convergence of the scheme is obtained in the maximum norm. Numerical examples are presented.     

Banach空间四阶周期边值问题正解的存在性

讨论了Banach空间E中的四阶周期边值问题:( u(4)(t)??u00(t)+′u(t)= f(t; u(t));06 t 61; u(i)(0)= u(i)(1); i =0;1;2;3正解的存在性,其中f :[0;1]￡ P ! P连续, P为E的正元锥,?;′2 R且满足0<′<(?2+2?2)2;?>?2?2;′?4+??2+1>0:通过非紧性测度的估计技巧与凝聚映射的不动点指数理论获得了该问题正解的存在性结果.     

Existence of multiple positive periodic solutions for functional differential equations

In our paper, by employing Krasnoselskii fixed point theorem, we investigate the existence of multiple positive periodic solutions for functional differential equations

     

Global attractivity of positive periodic solutions for nonlinear impulsive systems

In this paper, the existence, uniqueness and global attractivity of positive periodic solutions for nonlinear impulsive systems are studied. Firstly, existence conditions are established by the method of lower and upper solutions. Then uniqueness and global attractivity are obtained by developing the theories of monotone and concave operators. And lastly, the method and the results are applied to the impulsive n$n$-species cooperative Lotka–Volterra system and a model of a single-species dispersal among n$n$-patches.     

Existence of positive solutions for nth-order periodic difference equations

This paper is devoted to the study of difference equations coupled with periodic boundary value conditions. We deduce the existence of at least one positive solution provided that the nonlinear part of the equation satisfies some monotonicity assumptions and the existence of a positive upper solution. The result is obtained from a new fixed point theorem based on the classical Krasnoselskii's cone expansion/contraction theorem and the constant sign properties of the related Green's function.     

非自治时滞微分方程正周期解的存在性

应用Krasnoselskii锥映射不动点定理，研究了具一般时滞非线性非自治Logistic方程的ω-周期解的存在性，获得了存在正周期解的充分条件．     

碰撞Hamiltonian系统的无穷小周期解

本文通过适当的坐标变换将碰撞振子的相平面转变为全平面,应用Poincaré-Birkhoff扭转定理,证明了在原点附近超线性碰撞振子的无穷多弹性周期解的存在性,从而推广了已有的结果.     

Stability of periodic solutions for Lipschitz systems obtained via the averaging method

Existence and asymptotic stability of the periodic solutions of the Lipschitz system is hereby studied via the averaging method. The traditional dependence of on is relaxed to the mere strict differentiability of at for , giving room to potential applications for structured nonsmooth systems.

     

Triple positive periodic solutions of nonlinear singular second-order boundary value problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″(t) + λ(t)u(t) = f(t,u(t)),a.e.t ∈ [0,2π],u(0) = u(2π),u′(0) = u′(2π),where f(t,u) is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett–Williams and Krasnosel'skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive periodic solutions of singular systems of first order ordinary differential equations

The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.     

Periodic segment implies infinitely many periodic solutions

In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.

     

Existence of positive solutions for fractional boundary value problems

In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some $p > 1$, we discuss the existence and multiplicity of positive solutions to the four point boundary value problems of nonlinear fractional differential equations. Our results extend some recent works in the literature.     

Positive periodic solutions and eigenvalue intervals for systems of second order differential equations

In this paper, we employ a well‐known fixed point theorem for cones to study the existence of positive periodic solutions to the n ‐dimensional system x ″ + A (t)x = H (t)G (x). Moreover, the eigenvalue intervals for x ″ + A (t)x = λH (t)G (x) are easily characterized. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multiplicity result of positive solutions for a class of multi-parameter ordinary differential systems

In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As applications, we prove the global existence of positive solutions for a multi-parameter system of second-order ordinary differential equations with respect to parameters. The discussion is based on the result of Leray-Schauder degree on the order interval and the fixed point index theory in cones.     

Three positive solutions for boundary value problem for differential equation with Riemann-Liouville fractional derivative

In this paper, by using the Avery-Peterson fixed point theorem, we establish the existence result of at least three positive solutions of boundary value problem of nonlinear differential equation with Riemann-Liouville''s fractional order derivative. An example illustrating our main result is given. Our results complements and extends previous work in the area of boundary value problems of nonlinear fractional differential equations.