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Zbl 0954.06007
Stein, J.D.jun.
Fixed points of inequality-preserving maps in complete lattices.
(English)
[J] Czech. Math. J. 49, No.1, 35-43 (1999). ISSN 0011-4642; ISSN 1572-9141/e

Let $L$ be a complete lattice and $T$ a map of $L$ into $L$. An element $x\in L$ is said to be a fixed point of $T$ if $x=Tx$ holds. The paper presents some conditions sufficient for the existence of a fixed point of $T$. \par Let $p,k$ be positive integers, $m$ their greatest common divisor. If $x,y$ are arbitrary elements in $L$ and $x\le y$ implies $T^p x \le T^k y$, then $T^m$ has a fixed point. Similarly, if $x\le Ty$ implies $T^px\le T^p(Ty)$ and $Tx \le y$ implies $T^k (Tx) \le T^ky$, then $T^m$ has a fixed point. \par If $F$ is a commuting family of maps on $L$ such that $x\le Sy$ implies $Tx \le T(Sy)$ and $Sx \le y$ implies $T(Sx) \le Ty$ for any $S,T \in F$, then all maps in $F$ have a common fixed point. \par Let LIM be a map assigning an element in $L$ to any sequence of elements in $L$. If this map has some natural properties, it is called a lower Banach limit. The paper contains some conditions sufficient for the existence of a fixed point of LIM $T^n$ where $T$ is a map of $L$ into $L$; furthermore, conditions containing the operator LIM and sufficient for the existence of a fixed point of $T$ are formulated. A map $T$ is said to satisfy Fatou's Condition if $T(\text {LIM} \ x_n)\le \text {LIM} \ Tx_n$ holds for any sequence $\{x_n\}$. \par The author proves the following theorem. \par Let $T$ be a map of $L$ into $L$. Suppose that the following conditions are satisfied. \par (i) $T$ satisfies Fatou's Condition. \par (ii) If $x, y \in L$, $x\le y$, there exists $N =N(x,y)$ such that $n\ge N$ implies $T^n x\le T^ny$. \par Then $T$ has a fixed point. \par If $L$ is a complete chain and (ii) holds, then $T$ has a fixed point.
[M.NovotnÃ½ (Brno)]
MSC 2000:
*06B23 Complete lattices

Keywords: complete lattice; complete chain; fixed point; lower Banach limit; Fatou's condition

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