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Modular Uniform Convexity in Every Direction in L p (·) and Its Applications

Author

Listed:
  • Mostafa Bachar

    (Department of Mathematics, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia
    These authors contributed equally to this work.)

  • Osvaldo Méndez

    (Department of Mathematical Sciences, University of Texas at El Paso, 500W University Ave. 124 Bell Hall, El Paso, TX 79968, USA
    These authors contributed equally to this work.)

Abstract

We prove that the Lebesgue space of variable exponent L p ( · ) ( Ω ) is modularly uniformly convex in every direction provided the exponent p is finite a.e. and different from 1 a.e. The notion of uniform convexity in every direction was first introduced by Garkavi for the case of a norm. The contribution made in this work lies in the discovery of a modular, uniform-convexity-like structure of L p ( · ) ( Ω ) , which holds even when the behavior of the exponent p ( · ) precludes uniform convexity of the Luxembourg norm. Specifically, we show that the modular ρ ( u ) = ∫ Ω | u ( x ) | d x possesses a uniform-convexity-like structure even if the variable exponent is not bounded away from 1 or ∞ . Our result is new and we present an application to fixed point theory.

Suggested Citation

  • Mostafa Bachar & Osvaldo Méndez, 2020. "Modular Uniform Convexity in Every Direction in L p (·) and Its Applications," Mathematics, MDPI, vol. 8(6), pages 1-12, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:870-:d:364347
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    Cited by:

    1. Mohamed A. Khamsi & Osvaldo D. Méndez, 2022. "Remark on a Fixed-Point Theorem in the Lebesgue Spaces of Variable Integrability L p (·)," Mathematics, MDPI, vol. 11(1), pages 1-6, December.

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