A Short Proof of the Cartwright-Littlewood Fixed Point Theorem

1954 ◽  
Vol 6 ◽  
pp. 522-524 ◽  
Author(s):  
O. H. Hamilton
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Euclidean Plane ◽  
Short Proof ◽  
Bounded Continuum

The purpose of this paper is to give a short proof of the Cartwright-Littlewood fixed point theorem (2, p. 3, Theorem A).Theorem A. If T is a (1-1) continuous and orientation preserving transformation of the Euclidean plane E onto itself which leaves a bounded continuum M invariant and if M does not separate E, then some point of M is left fixed by T.

scholarly journals On a generalization of the Cartwright–Littlewood fixed point theorem for planar homeomorphisms

2016 ◽  
Vol 37 (6) ◽  
pp. 1815-1824 ◽  
Author(s):  
J. P. BOROŃSKI
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Point Theorem ◽  
Short Proof ◽  
The Fixed Point Theorem

We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose that $h:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\cup C$ is acyclic. If there is a $c\in C$ such that $\{h^{-i}(c):i\in \mathbb{N}\}\subseteq C$, or $\{h^{i}(c):i\in \mathbb{N}\}\subseteq C$, then $C$ also contains a fixed point of $h$. Our approach is based on Brown’s short proof of the result of Cartwright and Littlewood. In addition, making use of a linked periodic orbits theorem of Bonino, we also prove a counterpart of the aforementioned result for orientation reversing homeomorphisms, that guarantees a $2$-periodic orbit in $C$ if it contains a $k$-periodic orbit ($k>1$).

Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

2016 ◽  
Vol 2017 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Muhammad Usman Ali ◽  
◽  
Tayyab Kamran ◽  
Mihai Postolache ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Integral Inclusion
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