Unboundedness of solutions of a class of planar Hamiltonian systems

In this paper, the unboundedness of solutions for the following planar Hamilton system Ju ′ = ?H (u) + h (t) is discussed, where the function H (u) ∈ C²(R², R) is positive for u ≠ 0 and is positively (q, p)‐quasihomogeneous of quasi‐degree pq, where p > 1 and + = 1, h: S¹ → R² with h ∈ L^∞(0, 2π) is 2π ‐periodic and J is the standard symplectic matrix. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics analysis of a class of planar systems with time-varying delays

This paper formulates and studies a model of planar systems. The model can well describe many practical architectures of delayed neural networks, which is generalization of some existing neural networks under a time-varying environment. Without assuming the smoothness, monotonicity and boundedness of the activation functions, the existence and global exponential stability of its periodic solutions are investigated. Some explicit and conclusive results are established. Our approach is based on the continuation theorem of the coincidence degree, a priori estimates, and differential inequalities.     

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.     

Sign-Changing Solutions for a Class of Nonlinear Elliptic Problems

In the present paper,the author studies the existence of sign-changing solutions for nonlinear elliptic equations,which have jumping nonlinearities,and may or may not be resonant with respect to Fu(?)ik spectrum,via linking methods under Cerami condition.     

Existence of nontrivial solutions for Kirchhoff-type problems with jumping nonlinearities

In this study, the Fučik spectrum is used to investigate Kirchhoff-type problems with jumping nonlinearities at infinity, and a new result on the existence of nontrivial solutions is obtained.     

Periodic and subharmonic solutions of a class of superquadratic Hamiltonian systems

Periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic Hamiltonian systems by using the minimax methods in critical point theory.     

Periodic solutions for a class of nonautonomous second order Hamiltonian systems

In this paper, some existence theorems of periodic solutions of a class of the nonautonomous second order Hamiltonian systems

     

Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems

Based on new deformation theorems concerning strongly indefinite functionals, we give some new min-max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy Cerami condition instead of Palais-Smale condition. As one application of abstract results, we study existence of multiple periodic solutions for a class of non-autonomous first order Hamiltonian system

     

具有四个时滞的平面系统的周期解的存在性

The sufficient condition for the existence of non-constant periodic solutions of the following planar system with four delays are obtained:     

Periodic solutions for a class of new superquadratic second order Hamiltonian systems

A new superquadratic growth condition is introduced, which is an extension of the well-known superquadratic growth condition due to P.H. Rabinowitz and the nonquadratic growth condition due to Gui-Hua Fei. An existence theorem is obtained for periodic solutions of a class of new superquadratic second order Hamiltonian systems by the minimax methods in critical point theory, specially, a local linking theorem.     

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.     

Uniqueness of positive solutions for a class of elliptic systems

In this article, we consider uniqueness of positive radial solutions to the elliptic system Δu+a(|x|)f(u,v)=0, Δv+b(|x|)g(u,v)=0, subject to the Dirichlet boundary condition on the open unit ball in RN (N?2). Our uniqueness results applies to, for instance, f(u,v)=uqvp, g(u,v)=upvq, p,q>0, p+q<1 or more general cases.     

Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems

Some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second-order Hamiltonian systems by the minimax methods in critical point theory.     

Periodic solutions of a class of degenerate parabolic system with delays

This paper is concerned with a class of periodic degenerate parabolic system with time delays in a bounded domain under mixed boundary condition. Under locally Lipschitz condition on reaction functions, we apply Schauder fixed point theorem to obtain the existence of periodic solutions of the periodic problem. With quasi-monotonicity in addition, we also show that the periodic problem has a maximal and a minimal periodic solutions. Applications of the obtained results are also given to some nonlinear diffusion models arising from ecology.     

Existence and uniqueness of periodic solution for a class of differential systems

In this paper, sufficient conditions are obtained for the existence of a unique periodic solution for a class of differential systems.