A Fixed Point Theorem in the Lebesgue Spaces of Variable Integrability <i>L</i>^<i>p</i>(·)

Abstract

We establish a fixed point property for the Lebesgue spaces with variable exponents Lp(·), focusing on the scenario where the exponent closely approaches 1. The proof does not impose any additional conditions. In particular, our investigation centers on ρ-non-expansive mappings defined on convex subsets of Lp(·), satisfying the “condition of uniform decrease” that we define subsequently.

Keywords

• electrorheological fluid
• fixed point
• modular vector space
• Nakano
• strictly convex
• uniformly convex