Let $G:X \rightarrow Y$ be a continuously differentiable nonlinear operator, with $X$ and $Y$ being real Hilbert spaces. I want to know what is the weakest hypothesis necessary to guarantee that, for every weakly convergent sequences $\{x^{k}\}_{k \in \mathbb{N}}$ and $\{\lambda^{k}\}_{k \in \mathbb{N}}$ converging weakly to $x \in X$ ($x^{k} \rightharpoonup x$) and $\lambda \in Y$ ($\lambda^{k} \rightharpoonup \lambda$), respectively, we have that, for every, $h \in X$, $$\left\langle G'(x^{k}) h, \lambda^{k} \right\rangle \rightarrow \left\langle G'(x) h, \lambda \right\rangle.$$ This is satisfied for bounded linear operators $G$.
However, some common assumptions for this is assuming that $G'$ is completely continuous, but such a hypothesis seems too strong, and even commonly not satisfied. There are weaker hypotheses for such type of convergence?
This assumption type convergence is required in optimization while studying duality theories in Hilbert spaces. For example, see Lemma 4.9. or Theorem 5.3.