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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive Banach spaces
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by Athanassios G. Kartsatos PDF
Trans. Amer. Math. Soc. 350 (1998), 3967-3987 Request permission


A more systematic approach is introduced in the theory of zeros of maximal monotone operators $T:X\supset D(T)\to 2^{X^{*}}$, where $X$ is a real Banach space. A basic pair of necessary and sufficient boundary conditions is given for the existence of a zero of such an operator $T$. These conditions are then shown to be equivalent to a certain asymptotic behavior of the resolvents or the Yosida resolvents of $T$. Furthermore, several interesting corollaries are given, and the extendability of the necessary and sufficient conditions to the existence of zeros of locally defined, demicontinuous, monotone mappings is demonstrated. A result of Guan, about a pathwise connected set lying in the range of a monotone operator, is improved by including non-convex domains. A partial answer to Nirenberg’s problem is also given. Namely, it is shown that a continuous, expansive mapping $T$ on a real Hilbert space $H$ is surjective if there exists a constant $\alpha \in (0,1)$ such that $\langle Tx-Ty,x-y\rangle \ge -\alpha \|x-y\|^{2},~x,~y\in H.$ The methods for these results do not involve explicit use of any degree theory.
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Additional Information
  • Athanassios G. Kartsatos
  • Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
  • Email:
  • Received by editor(s): March 6, 1995
  • Received by editor(s) in revised form: November 7, 1996
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 3967-3987
  • MSC (1991): Primary 47H17; Secondary 47H05, 47H10
  • DOI:
  • MathSciNet review: 1443880