Consider the scalar nonlinear delay differential equationddt[x(t)-f(t,x(t-τ))]+g(t,x(t-δ))=0,tt 0,where τ,δ(0,∞),f,gC([t 0,∞)×R,R)and xg(t,x)0 for tt 0,x∈R.The author obtains sufficient conditions for the ...Consider the scalar nonlinear delay differential equationddt[x(t)-f(t,x(t-τ))]+g(t,x(t-δ))=0,tt 0,where τ,δ(0,∞),f,gC([t 0,∞)×R,R)and xg(t,x)0 for tt 0,x∈R.The author obtains sufficient conditions for the zero solution of this equation to be uniformly stable as well as asymptotically stable.展开更多
In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence an...In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions. Our results not only solve an open problem proposed by Pankov, but also greatly improve some existing ones even for some special cases.展开更多
Consider the retarded difference equation x<sub>n</sub>-x<sub>n-1</sub>=F（-f（x<sub>n</sub>）+g（x<sub>n</sub>-k））, （*） where k is a positive integer, F,f,g:R→R a...Consider the retarded difference equation x<sub>n</sub>-x<sub>n-1</sub>=F（-f（x<sub>n</sub>）+g（x<sub>n</sub>-k））, （*） where k is a positive integer, F,f,g:R→R are continuous, F and f are increasing on R, and uF（u）】0 for all u≠0. We show that when f（y）≥g（y）（resp. f（y）≤g（y）） for y∈R, every solution of （*） tends to either a constant or -∞ （resp. ∞） as n→∞. Furthermore, if f（y）≡g（y） for y∈R, then every solution of （*） tends to a constant as n→∞.展开更多
文摘Consider the scalar nonlinear delay differential equationddt[x(t)-f(t,x(t-τ))]+g(t,x(t-δ))=0,tt 0,where τ,δ(0,∞),f,gC([t 0,∞)×R,R)and xg(t,x)0 for tt 0,x∈R.The author obtains sufficient conditions for the zero solution of this equation to be uniformly stable as well as asymptotically stable.
基金supported partially by the Specialized Fund for the Doctoral Program of Higher Eduction (Grant No.20071078001)Key Project of National Natural Science Foundation of China (Grant No. 11031002)+1 种基金Natural Science and Engineering Research Council of Canada (NSERC)Project of Scientific Research Innovation Academic Group for the Education System of Guangzhou City
文摘In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions. Our results not only solve an open problem proposed by Pankov, but also greatly improve some existing ones even for some special cases.
基金Project supported by NNSF (19601016) of China NSF (97-37-42) of Hunan
文摘Consider the retarded difference equation x<sub>n</sub>-x<sub>n-1</sub>=F（-f（x<sub>n</sub>）+g（x<sub>n</sub>-k））, （*） where k is a positive integer, F,f,g:R→R are continuous, F and f are increasing on R, and uF（u）】0 for all u≠0. We show that when f（y）≥g（y）（resp. f（y）≤g（y）） for y∈R, every solution of （*） tends to either a constant or -∞ （resp. ∞） as n→∞. Furthermore, if f（y）≡g（y） for y∈R, then every solution of （*） tends to a constant as n→∞.
