The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global exist...The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in C without the linear growth condition. Then, under the local Lipschitz condition in C, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existence-uniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples are provided to show the effectiveness of the theoretical results.展开更多
In this paper, we investigate the dissipativity behavior and give the range estimate for solutions of impulsive functional differential equations. Also, some criteria on asymptotical stability and exponential stabilit...In this paper, we investigate the dissipativity behavior and give the range estimate for solutions of impulsive functional differential equations. Also, some criteria on asymptotical stability and exponential stability of the zero solution are obtained.展开更多
The existence of the multiple positive periodic solutions to a delayed statedependent predator-prey system with non-monotonic functional response is considered. Using the continuation theorem based on Mawhin＇s coinci...The existence of the multiple positive periodic solutions to a delayed statedependent predator-prey system with non-monotonic functional response is considered. Using the continuation theorem based on Mawhin＇s coincidence degree, some more generalized results are obtained.展开更多
基金The Central Level,Scientific Research Institutes for Basic R & D Special Fund Business (No.2060302)National Natural Science Foundation of China(No.40841016,No.40372131 and No.40702056)Old Experts Science Foundation of China Earthquake Administration(No.201039)
基金supported by National Natural Science Foundation of China (Grant Nos.11271270, 11201320 and 11101298)Youth Foundation of Sichuan University (Grant No. 2011SCU11111)
文摘The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in C without the linear growth condition. Then, under the local Lipschitz condition in C, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existence-uniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t. Two examples are provided to show the effectiveness of the theoretical results.
基金the NNSF of China under Grants 10671133,60628302 and 60425307the Scientific Research Fund of Sichuan Provincial Education Department under Grant 2006B071the Scientific Research Fund of Chongqing Normal University under Grant 06XLB025
文摘In this paper, we investigate the dissipativity behavior and give the range estimate for solutions of impulsive functional differential equations. Also, some criteria on asymptotical stability and exponential stability of the zero solution are obtained.
基金the National Natural Science Foundation of China under Grant 10671133
文摘The existence of the multiple positive periodic solutions to a delayed statedependent predator-prey system with non-monotonic functional response is considered. Using the continuation theorem based on Mawhin＇s coincidence degree, some more generalized results are obtained.
