Some Geometric Properities And Duality Invariants In Topological Vector Space

The paper mainly consists of two parts, one is geometric properties related to fixed-point properties of special Banach space, and the other is duality invariants in sequential evaluation convergence of nonlinear series of abstract mapping.In the part one, the following is discussed:(I) Fixed-point property of mean non-expansive mapping. There exist lots of conclusions related to fixed-point properties of non-expansive mapping and many people are interested in this problem. In this paper, mean non-expansive mapping is introduced. By using geometric coefficients of Banach space X , the fixed-point properties of mean non-expansive mapping are extended, and a very general sufficient condition of fixed-point properties of mean non-expansive mapping is obtained, that is, if Garcia-Falset coefficient R ( X ) <2/(1 + 2b + c), then T has a fixed point in K , where K is a weakly compact convex subset of X , and T is a mean non-expansive mapping into itself. Therefore, for a non-expansive mapping, the sufficient condition R ( X ) < 2, which was obtained by Garcia-Falset, is clearly a corollary of this result.(II) The (weakly) locally uniform rotundity and WM properties of Musielak-Orlicz sequence space. It is well known that these properties are related to the fixed-point property. Criteria for locally uniform rotundity of Orlicz space have been obtained, locally uniform rotundity of Musielak-Orlicz function space was discussed and the result and the proof are similar to those of Orlicz. Because of the complicated structure of the Musielak-Orlicz sequence space, although the criteria for rotundity and uniform rotundity were obtained by Anna Kaminska, criterion for locally uniform rotundity has not been found. This paper presents a criteria for (weakly) locally uniform rotundity and WM properties of Musielak-Orlicz sequence space. At the same time, a new method is given to discuss geometric properties of Musielak-Orlicz sequence space without any assumption (such as the condition (*) submitted by Anna Kaminska), a result of WM property of Orlicz space is improved.In the part two, the duality invariants are discussed as follows:...