Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which $x_n\rightharpoonup x$ for some $x\in X$ while $\limsup \langle f(x_n),x_n-x\rangle\leq 0$, we have $x_n\rightarrow x$.

I am looking for results in this way: $H\subset X$ is an open set with $G\subset H$ and it is possible to extend $F$ to a function $\hat{F}:\overline{H}\rightarrow X^\star$, such that $\hat{F}$ is $S_+$.

Any reference would be appreciated.