Citation: Amri, A.E.; Khamsi, M.A.; Méndez, O.D. A Fixed Point Theorem in the Lebesgue Spaces of Variable Integrability L p(·) . Symmetry 2023, 15, 1999. https://doi.org/10.3390/sym 15111999 Academic Editor: Alexander Zaslavski Received: 30 September 2023 Revised: 24 October 2023 Accepted: 27 October 2023 Published: 30 October 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry S S Article A Fixed Point Theorem in the Lebesgue Spaces of Variable Integrability L p(· ) Amnay El Amri 1 , Mohamed Amine Khamsi 2, * and Osvaldo D. Méndez 3 1 Faculté des Sciences Ben Msik (LAMS), Hassan II University, Casablanca 20023, Morocco; amnay.elamri-etu@etu.univh2c.ma or amnayelamri95@gmail.com 2 Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates 3 Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA; osmendez@utep.edu * Correspondence: mohamed.khamsi@ku.ac.ae Abstract: We establish a fixed point property for the Lebesgue spaces with variable exponents L p(·) , 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 L p(·) , satisfying the “condition of uniform decrease” that we define subsequently. Keywords: electrorheological fluid; fixed point; modular vector space; Nakano; strictly convex; uniformly convex 1. Introduction Sequence spaces l p n of variable exponent ( p n ) were introduced in 1931 by W. Orlicz [1] when addressing a question related to Fourier series. In the same work, Orlicz generalized the idea of Lebesgue spaces of variable integrability by defining the class of measurable functions f such that R 1 0 | f ( x)| p(x) dx < ∞. Lebesgue spaces of variable exponent arise naturally in the study of hydrodynamic equations that describe the behavior of non-Newtonian fluids [2,3]. Electrorheological fluids, characterized by dramatic and sudden changes in viscosity when exposed to an electric or magnetic field, are typical examples. A vigorous mathematical research effort is being devoted to electrorheological fluids and their applications to civil engineering, military science and medicine, among others [4–7]. Though variable exponent Lebesgue spaces appeared for the first time in [1], they were first studied as Banach spaces in [8]. The core observation of our approach is the immediate effect the variability of the exponent p on the topology of L p(·) , namely the mod- ular structure engenders a topology that differs fundamentally from that induced by the Luxemburg norm. This phenomenon is only visible when the exponent p is not constant (in the classical case of constant p, the modular is topologically equivalent to the norm) and is particularly striking in the endpoint cases, namely when the exponent is finite everywhere but is either unbounded or takes up values arbitrarily close to 1. Modular uniform convexity and its applications to fixed point theory and proxim- inality are now well-understood in the case when the exponent p is unbounded, i.e., if ess sup x∈Ω p( x)= p + = ∞,[9], but remains unexplored in the remaining endpoint situation, namely the case when ess inf x∈Ω p( x)= p - = 1. This work aims at remedying this situation. Proposition 2 and Theorems 3 and 4 are our main results and fill the existing gap in the literature for p - = 1. This work of exploiting the modular structure of L p(·) , new convexity properties of the modular, are discussed in the endpoint case p - = 1 and concrete applications of these new properties to modular fixed point theory are presented. The reader interested in the Symmetry 2023, 15, 1999. https://doi.org/10.3390/sym15111999 https://www.mdpi.com/journal/symmetry