If $H$ is a separable Hilbert space, an operator $A \subset H \times H$ is said to be $\lambda$-monotone ($\lambda \in \mathbb{R}$) if $$ \langle v-w, x-y \rangle \ge \lambda |x-y|^2 \quad \text{ for every } (x,v),\, (y,w) \in A.$$ Such condition is also known as hypom-onotonicity if $\lambda <0$.

The study of such operator is extensively treated in the book by Brezis "Operateurs Maximaux Monotones Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert" only in case $\lambda=0$. Many of his results can be easily generalized from the $0$-monotone case to the $\lambda$-monotone case via the transformation $A \mapsto A-\lambda I$. Some other theorems can be proven along the same lines with only few modifications obtaining basically the same results.

However, instead of systematically citing the results by Brezis adding "with obvious adaptations if $\lambda \ne 0$", I would like to find some resource where the results are clearly stated in the general case with $\lambda$ possibly different from $0$. In particular I am interested to the properties of the resolvent operator, characterisations of maximality, the evolution equation $\dot{x}_t \in A(x_t)$.... (the theory corresponding to chapters 2 and 3 of Brezis' book should be enough). Does anyone have any suggestions?



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