A Köthe-Toeplitz dual of a generalized Cesaro difference sequence space, a degenerate Lorentz space, their corresponding function spaces and fpp

Source abstract

In 1970, Cesaro sequence spaces was introduced by Shiue. In 1981, Kızmaz defined difference sequence spaces for ${\ell }^{\infty }$, ${\mathrm{c}}_0$ and $\mathrm{c}$. Then, in 1983, Orhan introduced Cesaro difference sequence spaces. Both works used difference operator and investigated the Köthe-Toeplitz dual 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 and for the corresponding function space of the Köthe-Toeplitz dual. Moreover, we consider another generalized space which is a degenerate Lorentz space because the spaces we consider are somehow related in terms of their construction. In fact, the Köthe-Toeplitz dual is contained in $\ell^1$, the corresponding function space contains Lebesgue space, the Banach space of Lebesgue integrable functions on $[0,1]$, $L_1[0,1]$ but for the degenerate Lorentz space and its corresponding function space, this is reversed completely. Then, as an important result, for both function spaces we show that they fail the weak fixed point property for isometries and even for contractions. As the second main result, by passing to counting measure, we take the corresponding sequence spaces, and for both spaces we get large classes of closed, bounded and convex subsets satisfying the fixed point property for nonexpansive mappings as a Goebel and Kuczumow analogy after noting that in $\ell^1$ Goebel and Kuczumow found a large class with the fixed point property for nonexpansive mappings.