Let $H$ be a real Hilbert space and $F:H\rightarrow H$ be a mapping such that $$ (A)\qquad\qquad(u_n\rightharpoonup u_*, F(u_n)\rightarrow F(u_*))\; \Longrightarrow\;(u_n\rightarrow u_*) $$ We are easy to verify that if $F$ is strongly monotone, i.e., there exists $\alpha>0$ such that $$ \langle F(u)-F(v), u-v\rangle\geq\alpha\|u-v\|^2\quad \forall u,v\in H, $$ then the property $(A)$ is satisfied.
I have some difficulties in the following questions:
If $F$ satisfies the property $(A)$ what do we call $F$?
Could we find a class of mappings $F$ such that the property $(A)$ is satisfied but $F$ is not strongly monotone.
I do not know any references about the mapping $F$ satisfying $(A)$.
Thank you for your kind help.