Large classes with the fixed point property in a degenerate Lorentz-Marcinkiewicz space

Abstract

Recently, Nezir has renormed l¹ and observed that the resulting space turns out be a degenerate Lorentz-Marcinkiewicz space. Then, fixed point properties have been investigated for the space, its dual and its predual. Also, inspiring from the study of Goebel and Kuczumow, as they showed for the Banach space of absolutely summable sequences l¹, Nezir showed that a class of non-weak* compact, closed, convex and bounded sets in one of these spaces has the fixed point property for affine nonexpansive mappings. In fact, very recently, generalizing the equivalent norm on l¹, Nezir and Mustafa obtained new type of degenerate Lorentz-Marcinkiewicz spaces with their fixed point properties and got the analogy of Goebel and Kuczumow's for the resulting space. In this paper, we show that there exists large classes of non-weak* compact, closed, convex and bounded sets with the fixed point property for affine nonexpansive mappings in the generalized degenerate Lorentz-Marcinkiewicz space.