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A more systematic approach is introduced in the theory of zeros of maximal monotone operators T : X D(T ) ! 2X ,w here Xis a real Banach space. A basic pair of necessary and sucient 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 sucient conditions to the existence of zeros of locally dened, 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 2 (0; 1) such that h Tx Ty;x y i k x y k 2 ;x ; y 2H: The methods for these results do not involve explicit use of any degree theory.

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