Strongly monotone operator

In functional analysis, a set-valued mapping ${\displaystyle A:X\to 2^{X}}$ where X is a real Hilbert space is said to be strongly monotone if

${\displaystyle \exists c>0{\mbox{ s.t. }}\langle u-v,x-y\rangle \geq c\|x-y\|^{2}\quad \forall x,y\in X,u\in Ax,v\in Ay}$.

This is analogous to the notion of strictly increasing for scalar-valued functions of one scalar argument.

Equivalently, a binary relation ${\displaystyle R\subseteq X^{2}}$ is strongly monotone if

${\displaystyle \exists c>0{\mbox{ s.t. }}\langle u-v,x-y\rangle \geq c\|x-y\|^{2}\quad \forall \langle x,u\rangle ,\langle y,v\rangle \in R}$.

A function ${\displaystyle f:X\to X}$ is strongly monotone if

${\displaystyle \exists c>0{\mbox{ s.t. }}\langle f(x)-f(y),x-y\rangle \geq c\|x-y\|^{2}\quad \forall x,y\in X}$.

See also

• Monotonic function

References

• Zeidler. Applied Functional Analysis (AMS 108) p. 173
• Bauschke, Heinz H.; Combettes, Patrick L. (28 February 2017). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer Science & Business Media. ISBN 978-3-319-48311-5. OCLC 1037059594.

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