inequality with inferior limit for maximal monotone operators

Question

Let $X$ be a real reflexive Banach space and $A\colon X \to X'$ be a monotone operator. I am interested in the following statement: if $u_n \rightharpoonup u$ in $X$ and $A(u_n)\rightharpoonup f$ in $X'$, then $\langle f,u\rangle \leq \displaystyle \liminf_{n \to \infty} \langle A(u_n),u_n\rangle$.

Answer 1

If $u\in dom(A)$, then by monotonicity $$ \langle Au_n - Au,u_n-u\rangle \ge0 . $$ Hence $$ \liminf \langle Au_n,u_n\rangle \ge \liminf( \langle Au_n ,u\rangle + \langle Au,u_n-u\rangle) = \langle f ,u\rangle . $$