On Some Applications of Geometry of Banach Spaces and Some New Results Related to the Fixed Point Theory in Orlicz Sequence Spaces (Journal of Mathematical Study, 2021)
We present some applications of the geometry of Banach spaces in the approximationtheory and in the theory of generalized inverses. We also give some newresults, on Orlicz sequence spaces, related to the fixed point theory. After a short introduction, in Section 2 weconsider the best approximation projection from a Banach space $X$ onto its non-emptysubset and proximinality of the subspaces of order continuous elements in variousclasses of Köthe spaces. We present formulas for the distance to these subspaces ofthe elements from the outside of them. In Section 3 we recall some results and definitionsconcerning generalized inverses of operators (metric generalized inverses andMoore-Penrose generalized inverses). We also recall some results on the perturbationanalysis of generalized inverses in Banach spaces. The last part of this section concernsgeneralized inverses of multivalued linear operators (their definitions and representations).The last section starts with a formula for modulus of nearly uniformsmoothness of Orlicz sequence spaces $\ell^Φ$ 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 $R(a,l_Φ)$ is also presented, whencea sufficient condition for the weak fixed point property of the space $\ell^Φ$ is obtained.