FIXED POINT PROPERTY FOR NONEXPANSIVE MAPPINGS ON LARGE CLASSES IN KOTHE-TOEPLITZ DUALS OF CERTAIN DIFFERENCE SEQUENCE SPACES

In 1970, Cesaro sequence spaces were introduced by Shiue. In 1981, K & imath;zmaz defined difference sequence spaces for & ell;(infinity), c(0), and c. In 1989, Colak obtained new types of sequence spaces by generalizing K & imath;zmaz's idea. Using Colak's structure, Et and Esi, in 2000, obtained generalized difference sequences. In fact, they found the corresponding Kothe-Toeplitz duals and examined geometric properties for those spaces. We will be interested in their generalizations and we study the Goebel and Kuczumow analogy for the spaces they introduced. We recall that in 1979, Goebel and Kuczumow found that there exists a large class of closed, bounded, convex subsets in & ell;(1) with the fixed point property for nonexpansive mappings. Their study has become a pioneer for researchers investigating if nonreflexive spaces can be renormed to have the fixed point property and even the first example was given by Lin, in 2008. Lin's study was on & ell;(1) and there are, indeed, traces of impression from Goebel and Kuczumow. In the present study, we aim to discuss Goebel and Kuczumow analogous results for Kothe-Toeplitz duals of certain generalized difference sequence spaces studied by Et and Esi. We show that there exists a very large class of closed, bounded, convex subsets in those spaces with the fixed point property for nonexpansive mappings.