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

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.