Renormings of ℓ1 and C 0 and Fixed Point Properties

Given two (real) normed (linear) spaces $X$ and $Y$, let $X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$, where $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$. It is known that $X\otimes _{1}Y$ is $2$-UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$-dimensional and $Y$ is $k$-UR, then $X\otimes _{1}Y$ is $(m+k)$-UR. In the other direction, we observe that if $X\otimes _{1}Y$ is $k$-UR, then both $X$ and $Y$ are $(k-1)$-UR. Given a monotone norm $\Vert \cdot \Vert _{E}$ on $\mathbb{R}^{2}$, we let $X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where $\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$. It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\Vert \cdot \Vert _{E}$ is strictly monotone, then $X\otimes _{E}Y$ has WFPP. Using the notion of $k$-uniform rotundity relative to every $k$-dimensional subspace we show that this result holds with a weaker condition on $X$.

Download Full-text

Fixed point properties for a degenerate Lorentz–Marcinkiewicz space

TURKISH JOURNAL OF MATHEMATICS ◽

10.3906/mat-1802-79 ◽

2019 ◽

Vol 43 (4) ◽

pp. 1919-1939

Author(s):

Veysel NEZİR

Keyword(s):

Fixed Point ◽

Marcinkiewicz Space ◽

Fixed Point Properties

Download Full-text

Three counterexamples concerning ω-chain completeness and fixed point properties

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500006453 ◽

1981 ◽

Vol 24 (2) ◽

pp. 141-146 ◽

Cited By ~ 1

Author(s):

J. D. Mashburn

Keyword(s):

Fixed Point ◽

Upper Bound ◽

Partially Ordered Set ◽

Ordered Set ◽

Partially Ordered ◽

Fixed Point Properties ◽

Countable Chain ◽

Chain Complete

A partially ordered set, is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.

Download Full-text

On Approximate Coincidence Point Properties and Their Applications to Fixed Point Theory

Journal of Applied Mathematics ◽

10.1155/2012/302830 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 7

Author(s):

Wei-Shih Du

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Metric Spaces ◽

Coincidence Point ◽

Fixed Point Theory ◽

Point Theory ◽

Cone Metric Spaces ◽

Fixed Point Properties ◽

Contractive Maps

We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and some well-known results in the literature.

Download Full-text