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Transactions of the American Mathematical Society

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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

Author: Athanassios G. Kartsatos
Journal: Trans. Amer. Math. Soc. 350 (1998), 3967-3987
MSC (1991): Primary 47H17; Secondary 47H05, 47H10
MathSciNet review: 1443880
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Abstract: 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

Keywords: Maximal monotone operator, resolvent, Yosida resolvent, demicontinuous monotone operator, existence of zeros, Nirenberg’s problem
Received by editor(s): March 6, 1995
Received by editor(s) in revised form: November 7, 1996
Article copyright: © Copyright 1998 American Mathematical Society