TY - JOUR T1 - On Some Applications of Geometry of Banach Spaces and Some New Results Related to the Fixed Point Theory in Orlicz Sequence Spaces AU - Cui , Yunan AU - Hudzik , Henryk AU - Kaczmarek , Radosław AU - Ma , Haifeng AU - Wang , Yuwen AU - Zhang , Meiling JO - Journal of Mathematical Study VL - 4 SP - 325 EP - 378 PY - 2016 DA - 2016/12 SN - 49 DO - http://doi.org/10.4208/jms.v49n4.16.02 UR - https://global-sci.org/intro/article_detail/jms/10117.html KW - Approximative compactness, proximinality, Kadec-Klee property, uniform rotundity, Orlicz spaces, Banach lattices, quasi-linear projection, generalized inverses. AB -

We present some applications of the geometry of Banach spaces in the approximation theory and in the theory of generalized inverses. We also give some new results, on Orlicz sequence spaces, related to the fixed point theory. After a short introduction, in Section 2 we consider the best approximation projection from a Banach space $X$ onto its non-empty subset and proximinality of the subspaces of order continuous elements in various classes of Köthe spaces. We present formulas for the distance to these subspaces of the elements from the outside of them. In Section 3 we recall some results and definitions concerning generalized inverses of operators (metric generalized inverses and Moore-Penrose generalized inverses). We also recall some results on the perturbation analysis of generalized inverses in Banach spaces. The last part of this section concerns generalized inverses of multivalued linear operators (their definitions and representations). The last section starts with a formula for modulus of nearly uniform smoothness of Orlicz sequence spaces $\ell^\Phi$ equipped with the Amemiya-Orlicz norm. From this result a criterion for nearly uniform smoothness of these spaces is deduced. A formula for the Domínguez-Benavides coefficient $R(a,l_\Phi)$ is also presented, whence a sufficient condition for the weak fixed point property of the space $\ell^\Phi$ is obtained.