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A geometric property in ℓ p ( · ) and its applications
Author(s) -
Bachar M.,
Khamsi M. A.,
Mendez O.,
Bounkhel M.
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800049
Subject(s) - mathematics , convexity , lacunary function , bounded function , exponent , sequence (biology) , regular polygon , fixed point theorem , fixed point property , pure mathematics , mathematical analysis , geometry , linguistics , philosophy , biology , financial economics , economics , genetics
In this work, we initiate the study of the geometry of the variable exponent sequence space ℓp ( · )wheninf n p ( 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 eitherinf n p ( n ) = 1 orsup n p ( n ) = ∞ remains largely ill‐understood. We state and prove a modular version of the geometric property of ℓp ( · )wheninf 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 ℓp ( · )wheninf n p ( n ) = 1 .