A geometric property in ℓp(·) and its applications

In this work, we initiate the study of the geometry of the variable exponent sequence space ℓp(·) when infnp(n)=1. In 1931 Orlicz introduced the variable exponent sequence spaces ℓp(·) 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 ℓp(·) is uniformly convex if and only if the exponent is bounded away from 1 and infinity. The geometry of ℓp(·) when either infnp(n)=1 or supnp(n)=∞ remains largely ill‐understood. We state and prove a modular version of the geometric property of ℓp(·) when infnp(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 ℓp(·) when infnp(n)=1.