A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexi...A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.展开更多
In this paper, we attempt to give a unified approach to the existing several versions of Ekeland＇s variational principle. In the framework of uniiorm spaces, we introduce p-distances and more generally, q-distances....In this paper, we attempt to give a unified approach to the existing several versions of Ekeland＇s variational principle. In the framework of uniiorm spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance （particularly, a p-distance）, which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi＇s nonconvex minimization theorem, a generalized Ekeland＇s variational principle and a generalized Caristi＇s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland＇s variational principle, we deduce a number of particular versions of Ekeland＇s principle, which include many known versions of the principle and their improvements.展开更多
基金Supported by the Scientific Research Foundation of Hubei Province (D200613001)the National Natural Science Foundation of China (10371093)
文摘A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.
基金Supported by National Natural Science Foundation of China(Grant No.10871141)
文摘In this paper, we attempt to give a unified approach to the existing several versions of Ekeland＇s variational principle. In the framework of uniiorm spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance （particularly, a p-distance）, which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi＇s nonconvex minimization theorem, a generalized Ekeland＇s variational principle and a generalized Caristi＇s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland＇s variational principle, we deduce a number of particular versions of Ekeland＇s principle, which include many known versions of the principle and their improvements.
