Abstract
In this work, we initiate the study of the geometry of the variable exponent sequence space l(p().()) when inf(n) p(n) = 1. In 1931 Orlicz introduced the variable exponent sequence spaces l(p().()) while studying lacunary Fourier series. Since then, much progress has been made in the understanding of these spaces and of their continuous counterpart. In particular, it is well known that l(p().()) is uniformly convex if and only if the exponent is bounded away from 1 and infinity. The geometry of l(p().()) when either inf(n) p(n) = 1 or sup(n) p(n) = infinity remains largely ill-understood. We state and prove a modular version of the geometric property of l(p().()) when inf(n) p(n) = 1, known as uniform convexity in every direction. We present specific applications to fixed point theory. In particular we obtain an analogue to the classical Kirk's fixed point theorem in l(p().()) when inf(n) p(n) = 1.