In functional analysis, a set-valued mapping A : X → 2 X {\displaystyle A:X\to 2^{X}} where X is a real Hilbert space is said to be strongly monotone if
∃ c > 0 s.t. ⟨ u − v , x − y ⟩ ≥ c ‖ x − y ‖ 2 ∀ x , y ∈ X , u ∈ A x , v ∈ A y {\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 R ⊆ X 2 {\displaystyle R\subseteq X^{2}} is strongly monotone if
∃ c > 0 s.t. ⟨ u − v , x − y ⟩ ≥ c ‖ x − y ‖ 2 ∀ ⟨ x , u ⟩ , ⟨ y , v ⟩ ∈ R {\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 f : X → X {\displaystyle f:X\to X} is strongly monotone if
∃ c > 0 s.t. ⟨ f ( x ) − f ( y ) , x − y ⟩ ≥ c ‖ x − y ‖ 2 ∀ x , y ∈ X {\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.