On The Fixed Point Problem And The Extension Theorems In（2，p）-Normed Space

Fixed Point Theory has an important role in the solution of algebraic equations、functions equations、differential equations etc.．Hahn-Banach extension theorem in thelinear (2，p ) normed space and the strong and weak convergence of points range areimportant parts in applied mathematics，play an active role of other branches ofmathematics．The first chapter put together the concepts and the nature of the cone metric spaces andgeneralized cone metric spaces，as well as the fixed point theorem of cone metric spaces andgeneralized cone metric spaces，then put forward the concept of mixed-monotonic nature ofthe mapping and couple fixed points of generalized cone metric space，then the generalizedcone metric spaces satisfy the following contractive conditions fixed point theorem:Then the mappings F with mixed-monotonic nature have the unique fixed points.The second chapter generalized metrization theorem of generalized cone metric space，firstly introduced the generalized metric space definition and related properties，then thefollowing generalized metric defined in the generalized cone metric space:Generalized cone metric equivalent generalized metric in the sense of sequenceconverges .we obtain the relations between generalized metric and generalized cone metric，that is generalized cone metric can be generalized metrization.The third chapter，after the concepts of strong and weak convergence of(2，p ) sequence in linear (2，p ) normed space and uniformly convex (2，p ) normedspace were defined，the Hahn-Banach extension theorem in linear (2，p ) normed space and the properties of (2，p ) strong convergence and (2，p ) weak convergence werestudied．Eventually the theorem of equivalent of strong convergence and weak convergencewas obtained，that is to say，if { xn }is weak convergence and satisfiesnli→m∞x n，y p =x，ypin uniformly convex linear (2，p ) normed space，{ x n}is strongconvergence．...