A geometric property in ℓp(·) and its applications. Issue 9 (21st May 2019)

• Record Type: Journal Article
• Authors: Bachar, M.; Khamsi, M. A.; Mendez, O.; Bounkhel, M.
• Is Part Of: Mathematische Nachrichten. Volume 292:Issue 9(2019)
• Page Start: 1931
• Page End: 1940
• Publication Date: 2019-05-21
• Subjects: electrorheological fluids -- fixed point -- modular vector spaces -- Nakano -- nonexpansive -- uniformly convex in every direction -- Primary: 47H09 -- Secondary: 46B20 -- 47H10 -- 47E10; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201800049
• Languages: English
• ISSNs: 0025-584X

Abstract

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 ) = ∞ 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 .