On the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive Banach spaces

Abstract

A more systematic approach is introduced in the theory of zeros of maximal monotone operators T: X D D(T) -2X, 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, a]nd 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 morlotone operator, is improved by including non-convex domains. A partial answer to Nirenberg's problem is also given. Namely, it; is shown that a continzuous, expansive mapping T on a real Hilbert space 1t is surjective if there exists a constant ae G (0, 1) such that KTx Ty, x y) > -caX y 112, x, y E H. The methods for these results do not involve explicit use of any degree theory.