# Uniform convexity in $\ell_{p(\cdot)}$

Volume 10, Issue 10, pp 5292--5299 Publication Date: October 20, 2017       Article History
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### Authors

Mostafa Bachar - Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia. Messaoud Bounkhel - Department of Mathematics, College of Sciences, King Saud University, Riyadh, Saudi Arabia. Mohamed A. Khamsi - Department of Mathematics & Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. - Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, USA.

### Abstract

In this work, we investigate the variable exponent sequence space $\ell_{p(\cdot)}$. In particular, we prove a geometric property similar to uniform convexity without the assumption $\limsup_{n \to \infty} p(n) < \infty$. This property allows us to prove the analogue to Kirk's fixed point theorem in the modular vector space $\ell_{p(\cdot)}$ under Nakano's formulation.

### Keywords

• Fixed point
• modular vector spaces
• nonexpansive mapping
• uniformly convex
• variable exponent spaces

•  47H09
•  46B20
•  47H10
•  47E10

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