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.
In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.
In this paper, a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered. By establishing a nonautonomous -operator impulsive delay inequality and using the properties of ρ-cone and stochastic analysis technique, we obtain the p-attracting set and p-invariant set of the impulsive stochastic functional differential equation. An example is also discussed to illustrate the efficiency of the obtained results.
