Can you please give a reference where I can get the exact proof of the Minty-Browder theorem stated as follows: Observe that I below the operator $T$ is demicontinuous and need not be continuous. For continuous $T$, I know some references but I need it for demicontinuos $T$. Thank you very much.
Let $X$ be any real, separable reflexive Banach space and $T:X\to X^{*}$ satisfies the following properties: (i.) bounded, i.e., $T$ maps bounded sets to a bounded set. (ii.) demicontinuous, i.e., for every sequence $u_n\to u$ in $X$, we have $T(u_n)\rightharpoonup T(u)$ in $X^*$. (iii.) coercive, i.e., $$\lim_{||u||\to\infty}\frac{\langle T(u),u\rangle}{||u||}=\infty$$ and (iv.) $T$ is monotone, i.e., $$\langle T(u)-T(v),u-v\rangle\ge 0\ \forall u,v\in X.$$ Then for every $v\in X^*$, the equation $$ T(u)=v $$ has a unique solution $u\in X$. Moreover, $T$ is strict monotone, i.e., $$ \langle T(u)-T(v),u-v \rangle >0\ \forall u \ne v\in X, $$ then for every $v\in X^*$, the equation $$T(u)=v$$ has unique solution $u\in X$.
You can find this exact result in the book "Linear and Nonlinear Functional Analysis with Applications" by Ciarlet, see Theorem 9.14. Also in the german book "Nichtlineare Funktionalanalysis" by Ruzicka and in the following lecture notes, Theorem 9.13.
Note that for a monotone operator demicontinuity and hemicontinuity are equivalent, see Proposition 9.11 in the lecture notes. So you find a lot of formulations of the Browder-Minty with hemicontinuity as assumption.