On a sufficient condition for equality of two maximal monotone operators

Abstract

Abstract: We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators TeX Embedding failed! which share the same convex-like domain TeX Embedding failed! coincide whenever TeX Embedding failed! and TeX Embedding failed! have a nonempty intersection for every TeX Embedding failed!. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their TeX Embedding failed!-subdifferentials intersect at every point of that domain.

Joint work with Juan Enrique Mart

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