In 1970, Cesàro sequence spaces was introduced by Shiue. In 1981, Kızmaz defined difference sequence spaces for ℓ ∞ , c 0 and c. Then, in 1983, Orhan introduced Cesàro difference sequence spaces. Both works used difference operator and investigated Köthe-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, ℓ 1 with fixed point property for nonexpansive mappings. So we consider a Goebel and Kuczumow analogue result for a Köthe-Toeplitz dual of a generalized Cesàro 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.