## Abstract

In this work, we initiate the study of the geometry of the variable exponent sequence space (Formula presented.) when (Formula presented.). In 1931 Orlicz introduced the variable exponent sequence spaces (Formula presented.) 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 (Formula presented.) is uniformly convex if and only if the exponent is bounded away from 1 and infinity. The geometry of (Formula presented.) when either (Formula presented.) or (Formula presented.) remains largely ill-understood. We state and prove a modular version of the geometric property of (Formula presented.) when (Formula presented.), 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 (Formula presented.) when (Formula presented.).

Original language | British English |
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Pages (from-to) | 1931-1940 |

Number of pages | 10 |

Journal | Mathematische Nachrichten |

Volume | 292 |

Issue number | 9 |

DOIs | |

State | Published - 1 Sep 2019 |

## Keywords

- 47E10
- 47H10
- electrorheological fluids
- fixed point
- modular vector spaces
- Nakano
- nonexpansive
- Primary: 47H09
- Secondary: 46B20
- uniformly convex in every direction

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