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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. (English) Zbl 0902.47052

Summary: 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 \(\langle Tx-Ty,x-y\rangle \geq -\alpha \| x-y\| ^{2}\), \(x,y\in H\). The methods for these results do not involve explicit use of any degree theory.

MSC:

47J05 Equations involving nonlinear operators (general)
47H04 Set-valued operators
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
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