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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Fixed points for discontinuous quasi-monotone maps in $\textbf {R}^ n$
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by Shou Chuan Hu PDF
Proc. Amer. Math. Soc. 104 (1988), 1111-1114 Request permission

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 \[ \lim \inf \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne - \infty ,\] or $- f$ is quasi-monotone and \[ \lim \sup \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne + \infty \].
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 104 (1988), 1111-1114
  • MSC: Primary 47H10; Secondary 26B35, 47H05, 65H10
  • DOI: https://doi.org/10.1090/S0002-9939-1988-0937846-1
  • MathSciNet review: 937846