A note on periodic solutions for semi-ratio-dependent predator-prey systems

This note investigates the existence of periodic solutions for semi-ratio-dependent predator-prey models with functional response. New sharp criteria without any other nontrivial assumptions are presented by the invariance property of homotopy and analysis technique, which improve and extend many previous work. Some interesting numerical examples are given to illustrate our results.     

A note on periodic solutions of nonautonomous second-order systems

A multiplicity theorem is obtained for periodic solutions of nonautonomous second-order systems with partially periodic potentials by the minimax methods.

     

A note on solutions for asymptotically linear elliptic systems

In this paper, we are concerned with the elliptic system of
{ -△u＋V（x）u=g（x,v）, x∈R^N,
-△v＋V（x）v=f（x,u）, x∈R^N,
where V（x） is a continuous potential well, f, g are continuous and asymptotically linear as t→∞. The existence of a positive solution and ground state solution are established via variational methods.     

A note on regular points for solutions of elliptic systems

A vector valued function u(x), solution of a quasilinear elliptic system cannot be too close to a straight line without being regular.     

A note on the Lyapunov stability of periodic discrete-time systems

We characterize the stability of discrete-time Lyapunov equations with periodic coefficients. The characterization can be seen as the analog of the classical stability theorem of Lyapunov equations with constant coefficients. It involves quantities readily computable with good accuracy.     

A note on the existence of periodic travelling wave solutions with large periods in generalized reaction-diffusion systems

We show that the existence of wave trains with high velocity of generalized reaction-diffusion equations can be easily established by using a theorem of D. V. Anosov on the existence of periodic solutions of singularly perturbed differential systems.     

A remark on periodic solutions of singular Hamiltonian systems

In this note we study the existence of non-collision periodic solutions for singular Hamiltonian systems with weak force. In particular for potential where D is a compact C³-surface in we prove the existence of a non-collision periodic solution.     

A note on some piecewise-linear difference equations with Mersenne-type periodic solutions

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 ^k  ? 1) or 6(2 ^k  ? 1) for some k ≥ 1. The existence of solutions with such periods is also considered.     

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f _x (u, y), whereu y ^M u(y) ^N ,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT ^M +1. In the autonomous case,f=f(x), the above result holds for any .     

A constructive method for finding periodic solutions of differential systems

Coefficient conditions for the existence of periodic solutions of degenerate systems of differential equations are obtained. Iterative algorithms for finding these solutions are developed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 468–476, April, 1995.     

A note on periodic solutions of a differential equation with two parameters

In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε ₁ andε ₂ are real constants andε ₁ andε ₂ are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).     

Non-collision periodic solutions for singular Hamiltonian systems

We study the existence of classical (non-collision) T-periodic solutions of the Hamiltonian system where and is a T-periodic function in t which has a singularity at like Under suitable conditions on H, we prove that if then (HS) possesses at least one non-collision solution and if then the generalized solution of (HS) obtained in [5] has at most one time of collision in its period.     

Multiple periodic solutions for nonautonomous Hamiltonian systems

In this paper, by the use of minimax method, we obtain some existence and multiplicity theorems for periodic solutions of nonautonomous Hamiltonian systems with bounded nonlinearity of the type:¶ J [(x)\dot] + ?H(t, x) + e(t) = 0. J \dot x + \nabla H(t, x) + e(t) = 0.     

Multiple solutions for strongly resonant periodic systems

We considered a semilinear, second order periodic system. We assumed that the differential operator x→−x^″−Ax$x\to -x^{″}-Ax$ has zero as an eigenvalue and has no negative eigenvalues. Also we imposed a strong resonance condition (with respect to the zero eigenvalue) on the potential function F(t,x)$F(t,x)$. Using the second deformation theorem, we established the existence of at least two nontrivial solutions. To do this we needed to conduct a detailed analysis of the Cerami compactness condition, which is actually of independent interest.     

概周期系统概周期解的存在唯一性

运用Ｌｉａｐｕｎｏｖ函数，研究了概周期系统概周期解的存在唯一性，得到了一个方便应用的判定定理．     

A stability criterion for periodic solutions of two-dimensional discontinuous dynamical systems

We establish stability conditions for periodic solutions of two-dimensional systems of ordinary differential equations with pulse influence. We study the properties of the jump operator for such systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1262–1266, September, 1999.