In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu ...In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu '(\theta ) + o(1), \rho \to \infty \hfill \\ \end{gathered} \right.$$ , where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ, ∈ [0, 2?], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c < 0 and μ(θ) ≠ 0, θ ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″+f(x)x′+ax +?bx ?+?(x)=p(t) has unbounded solutions provided that a, b satisfy $1/\sqrt a + 1/\sqrt b = 2/n$ and ?(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.展开更多
In this paper,we prove that the forced pendulum-type equation +G<sub>x</sub>（x,t）=p（t）,where G（x,t）∈C<sup>1</sup>（R<sup>2</sup>） with 1-periodicity in x satisfies the condi...In this paper,we prove that the forced pendulum-type equation +G<sub>x</sub>（x,t）=p（t）,where G（x,t）∈C<sup>1</sup>（R<sup>2</sup>） with 1-periodicity in x satisfies the conditions:sup<sub>（x,t）∈R<sup>2</sup></sub>|G<sub>x</sub>（x,t）|【+∞ and lim sup<sub>t→∞</sub>{sup<sub>x∈R</sub>|（G<sub>t</sub>（x,t））/t|}=0,possesses infinitely many unbounded solutions on a cylinder S<sup>1</sup>×R for any almost periodic function p（t） with nonvauishing mean value.展开更多
In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under d...In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.展开更多
By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parame...By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.展开更多
This paper studies the asymptotic behavior of solutions of the difference equation X（n＋1）=max{C1/Xn,C2/X（n-1）},n=0,1,…,where the parameters C1, C2 and the initial conditions x（-1）, xo are nonzero real numbers...This paper studies the asymptotic behavior of solutions of the difference equation X（n＋1）=max{C1/Xn,C2/X（n-1）},n=0,1,…,where the parameters C1, C2 and the initial conditions x（-1）, xo are nonzero real numbers. More precisely, it has been proved that： （1） if Ct 〈 0 and C2 〉 0, then every solution of the equation is eventually periodic; （2） if Ct 〈 0 and C2 〈 0, then every solution of the equation is unbounded when C1≠P C2 or is eventually periodic when C1 = C2.展开更多
In this paper,we consider the periodic solution problems for the systems with unbounded delay,and the existence,uniqueness and stability of the periodic solution are dealt with unitedly.First we establish the suitable...In this paper,we consider the periodic solution problems for the systems with unbounded delay,and the existence,uniqueness and stability of the periodic solution are dealt with unitedly.First we establish the suitable delay-differential inequality,then study seperately the problems of periodic solution for the systems with bounded delay,with unbounded delay and the Volterra integral-dlfferentlal systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré’s periodic operator in the different phase spaces.A series of simple criteria for the existence,uniqueness and stability of these systems are obtained.展开更多
We establish the coexistence of periodic solution and unbounded solution, the infinity of largeamplitude subharmonics for asymmetric weakly nonlinear oscillator x' + a2x+ - b2x- + h(x) = p(t) with h(±∞) - 0 ...We establish the coexistence of periodic solution and unbounded solution, the infinity of largeamplitude subharmonics for asymmetric weakly nonlinear oscillator x' + a2x+ - b2x- + h(x) = p(t) with h(±∞) - 0 and xh(x) → +∞(x →∞), assuming that M(τ ) has zeros which are all simple and M(τ ) 0respectively, where M(τ ) is a function related to the piecewise linear equation x' + a2x+ - b2x- = p(t).展开更多
基金the National Natural Science Foundation of China(Grant No.10471099)the Fund of Beijing Education Committee(Grant No.KM200410028003)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,Ministry of Education of China
文摘In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu '(\theta ) + o(1), \rho \to \infty \hfill \\ \end{gathered} \right.$$ , where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ, ∈ [0, 2?], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c < 0 and μ(θ) ≠ 0, θ ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″+f(x)x′+ax +?bx ?+?(x)=p(t) has unbounded solutions provided that a, b satisfy $1/\sqrt a + 1/\sqrt b = 2/n$ and ?(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.
基金Supported by the National Natural Science Foundation of China(Grant No.19671007)
文摘In this paper,we prove that the forced pendulum-type equation +G<sub>x</sub>（x,t）=p（t）,where G（x,t）∈C<sup>1</sup>（R<sup>2</sup>） with 1-periodicity in x satisfies the conditions:sup<sub>（x,t）∈R<sup>2</sup></sub>|G<sub>x</sub>（x,t）|【+∞ and lim sup<sub>t→∞</sub>{sup<sub>x∈R</sub>|（G<sub>t</sub>（x,t））/t|}=0,possesses infinitely many unbounded solutions on a cylinder S<sup>1</sup>×R for any almost periodic function p（t） with nonvauishing mean value.
基金Supported by the NNSF of China(60464001) Guangxi Science Foundation(0575092).
文摘In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.
基金the National Natural Science Foundation of China(Nos.10671179 and 10772158)
文摘By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.
文摘This paper studies the asymptotic behavior of solutions of the difference equation X（n＋1）=max{C1/Xn,C2/X（n-1）},n=0,1,…,where the parameters C1, C2 and the initial conditions x（-1）, xo are nonzero real numbers. More precisely, it has been proved that： （1） if Ct 〈 0 and C2 〉 0, then every solution of the equation is eventually periodic; （2） if Ct 〈 0 and C2 〈 0, then every solution of the equation is unbounded when C1≠P C2 or is eventually periodic when C1 = C2.
文摘In this paper,we consider the periodic solution problems for the systems with unbounded delay,and the existence,uniqueness and stability of the periodic solution are dealt with unitedly.First we establish the suitable delay-differential inequality,then study seperately the problems of periodic solution for the systems with bounded delay,with unbounded delay and the Volterra integral-dlfferentlal systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré’s periodic operator in the different phase spaces.A series of simple criteria for the existence,uniqueness and stability of these systems are obtained.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10071055).
文摘We establish the coexistence of periodic solution and unbounded solution, the infinity of largeamplitude subharmonics for asymmetric weakly nonlinear oscillator x' + a2x+ - b2x- + h(x) = p(t) with h(±∞) - 0 and xh(x) → +∞(x →∞), assuming that M(τ ) has zeros which are all simple and M(τ ) 0respectively, where M(τ ) is a function related to the piecewise linear equation x' + a2x+ - b2x- = p(t).
