A large class in Köthe-Toeplitz duals of generalized Cesàro difference sequence spaces with fixed point property for nonexpansive mappings

Abstract

In 1970, Ces\`{a}ro sequence spaces was introduced by Shiue.  In 1981, K{\i}zmaz  defined difference sequence spaces for ${\ell }^{\infty }$, ${\mathrm{c}}_0$ and $\mathrm{c}$. Then, in 1983, Orhan introduced Ces\`{a}ro difference sequence spaces. Both works used difference operator and investigated K\"{o}the-Toeplitz duals for the new Banach spaces they introduced. Later, various authors generalized these new spaces, especially the one introduced by Orhan. In this study, first we discuss the fixed point property for these spaces. Then, we recall that Goebel and Kuczumow showed that there exists a very large class of closed, bounded, convex subsets in Banach space of absolutely summable scalar sequences, ${\ell }^1$ with fixed point property for nonexpansive mappings. So we consider a Goebel and Kuczumow analogue result for a K\"{o}the-Toeplitz dual of a generalized Ces\`{a}ro difference sequence space. We show that there exists a large class of closed, bounded and convex subsets of  these spaces with fixed point property for nonexpansive mappings.