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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Fixed points for discontinuous quasi-monotone maps in $ {\bf R}\sp n$


Author: Shou Chuan Hu
Journal: Proc. Amer. Math. Soc. 104 (1988), 1111-1114
MSC: Primary 47H10; Secondary 26B35, 47H05, 65H10
MathSciNet review: 937846
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Abstract: Let $ {K_n}$ be the unit cube in $ {R^n}$ and $ f = ({f_1},{f_2}, \ldots ,{f_n}):{K_n} \to {R^n}$. It is known that $ f$ has maximal and minimal fixed points in $ {K^n}$ if $ f:{K_n} \to {K_n}$ and $ f$ is monotone increasing. In this paper, a weaker condition, namely quasi-monotonicity, is considered and it is proved that the above mentioned conclusion is still true if $ f$ is either quasi-monotone and

$\displaystyle \mathop {\lim \inf }\limits_{t \to 0} \frac{{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne - \infty ,$

or $ - f$ is quasi-monotone and

$\displaystyle \mathop {\lim \sup }\limits_{t \to 0} \frac{{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne + \infty $

.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0937846-1
PII: S 0002-9939(1988)0937846-1
Keywords: Dini derivatives, quasi-monotone and discontinuous maps, fixed points
Article copyright: © Copyright 1988 American Mathematical Society