How do I apply Brouwer fixed-point theorem in this claim? 
*

*Let $\zeta:\mathbb{R}\to [0,+\infty)$ be a continuous non-negative function such that $\zeta(0)=0$ and $\tau\mapsto \zeta(\tau)\tau$ is a non-decreasing differentiable function whose derivative is bounded on every compact subset of $\mathbb{R}$.


*Let $\{\phi_{k}, \lambda_{k}\}_{k \in \Bbb N}$ be the the Dirichlet eigenpairs of the n Laplace operator on an open bounded set $\Omega\subset \Bbb R^N$, i.e.,  $\phi_k\in H_0^1(\Omega)$  and $-\Delta \phi_k= \lambda_k\phi_k$. Recall $\{\phi_k\}_{k}$ is an orthonormal basis in $L^2(\Omega)$.


*Question: Let $\mathcal{V}_{k}= \operatorname{span}\{\phi_1,\dotsc, \phi_{k}\}$. According to page 5 Eq (3.3) of Starovoitov - Boundary value problem for a global in time parabolic equation, the Brouwer fixed-point theorem implies the existence of a vector $v_k\in \mathcal{V}_k$ such that
\begin{equation}\label{Star-3.3}
\int_\Omega \nabla v_{k}\cdot \nabla \phi dx + \int_\Omega\zeta(v_k)v_k\phi dx=\int_\Omega f\phi dx\quad\text{for all}\quad\phi\in \mathcal{V}_{k}.
\end{equation}
How can one justify this claim?
My Taught and ideas
In fact, that $\phi_k\in L^\infty (\Omega)$ is the only important property needed from $\phi_k$. So that by assumption the function $\zeta(v_k)v_k$ is  bounded.
Since we are in finite-dimensional space and $\int_\Omega \nabla \phi_{i}\cdot \nabla \phi_j dx=\lambda_i\delta_{ij}$, the above equation reduces into finding $v_k=(v_{k,1}\phi_1+ \dotsb+v_{k,k}\phi_k)$ satisfying
\begin{equation}\label{Star-3.v}
  \sum_{i=1}^k\lambda_iv_{k,i}\phi_i  + \zeta(v_k)v_k = f_k\quad\text{in} \quad \mathcal{V}_{k},
\end{equation}
where $f_k=(f_{k,1}\phi_1+ \dotsb+f_{k,k}\phi_k)$ is the projection of $f$ on $\mathcal V_k$. Note that by abuse of notation we again write $\zeta(v_k)v_k$ to denote its own projection on $\mathcal{V}_k$.

Recall Brouwer fixed-point theorem: Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.

 A: What is needed is an a-priori $L_\infty$ bound for the solution $v_k$. If you know such an a-priori bound, you can modify $\zeta$ outside of this bound, and you can assume without of generality that $\zeta(u)=0$ for large $|u|$. (More precisely, you need the same a-priori bounded for the modified equation, that is, you have to know that any solution of the modified equation is also a solution of the original equation.)
Then the finite-dimensional equation is of the form
$$Av+F(v)=0$$
where $A$ is linear and positive definite, and $F$ is continuous and bounded. In particular, $A^{-1}$ exists, and the equation thus is equivalent to
$$v=-A^{-1}F(v)\text.$$
The range of the map $G=-A^{-1}F$ is contained in some ball. In particular, $G$ maps this ball into itself, and so Brouwer's fixed point theorem implies that $G$ has a fixed point which thus is a solution of the finite-dimensional equation.
A: Only now I realize the condition that $\zeta$ is nonnegative. (Was it really there in the first formulation of the question?)
With this condition, it is possible to get the required a-priori bound required for my other reply by a simple sign argument:
Choose the test function $\varphi=v_k$ in the equation.
Then the first summmand in that equation is bounded from below by $c\lVert v_k\rVert_{L_2}^2$ where $c>0$ is a constant coming from Poincaré's inequality, the second summand is nonnegative, and the absolute value of the last summand is bounded from above by $\lVert v_k\rVert_{L_2}$ by Cauchy-Schwarz. Hence, the equation cannot hold if $\lVert v_k\rVert_{L_2}\ge R$, where $R>0$ is independent of the particular form of $\zeta$.
Hence, you can replace $\zeta$ by $\widetilde\zeta(v)=\lambda(\lVert v\rVert_{L_2})\zeta(v)$ where $\lambda\colon[0,\infty)\to[0,1]$ is continuous with $\lambda|_{[0,R]}=1$ and $\lambda_{[R+1,\infty)}=0$, and for both equations the solutions have $L_2$-norm at most $R$, where the equations coincide. In other words: The original equation with $\zeta$ and the modified equation with $\widetilde\zeta$ have the same solutions.
For the modified equation $v\mapsto\widetilde\zeta(v)v$ is globally bounded, and the argument from the other comment applies.
A: First using Brouwer
Let $w\in \mathcal{V}_k$, necessarily $\zeta(w)$ is a bounded function since $\phi_k $'s are also bounded. The Lax-Milgram lemma implies there is a unique function $\widehat{w}\in  \mathcal{V}_k$
such that
\begin{align}
\int_\Omega\nabla\widehat{w}\nabla\phi +\zeta(w)\widehat{w}\phi -f\psi dx=0\quad\text{for all}\quad\phi\in \mathcal{V}_{k}.
\end{align}
The Poincar'{e}--Friedrichs inequality yields
\begin{equation}
\int_\Omega|\nabla\widehat{w}|^2dx  + \int_\Omega \zeta(w )\widehat{w}^2\, d x\leq \|f\|_{L^{2}(\Omega)}\,\|\widehat{w}\|_{L^{2}(\Omega)}\leq  C\|f\|_{L^{2}(\Omega)}\, \|\widehat{w}\|_{H_0^1(\Omega )} 
\end{equation}
Thus, letting $R=C\,\|f\|_{L^{2}(\Omega)}$, since $\varphi\geq0 $ we obtain the following estimates
\begin{align}\label{eq:boundedmapT}
&\|\widehat{w}\|_{H_0^1(\Omega )}\leq R
\quad\text{ and }\quad
 \int_\Omega \zeta(w )\widehat{w}^2\,dx\leq R^2.
\end{align}
We let $ \mathcal{B}_R=\big\{ w\in \mathcal{V}_k: \|w\|_{H_0^1(\Omega)} \leq R\big\}$, be the closed ball in $\mathcal{V}_k$ of radius $R$ centered at the origin.  Clearly, the mapping $T:\mathcal{V}_k\to \mathcal{B}_R$ with $Tw=\widehat{w}$  is well defined.
It remains to prove that $T$ is a continuous mapping. Indeed, let
$\{w_n\}$ be a sequence in $\mathcal{V}_k$ with $w_n= \lambda_{1,n}\phi_1+\cdots+  \lambda_{k,n}\phi_k$ converging in $\mathcal{V}_k$ to a function $w= \lambda_1\phi_1+\cdots+  \lambda_k\phi_k$; i.e., $\lambda_{\ell,n}\xrightarrow{n\to\infty } \lambda_\ell$, $\ell=1,2,\cdots,k$.
By continuity we have $ \varphi(w_n)\xrightarrow{n\to \infty}\varphi(w)$ almost everywhere. In addition, the convergence in $L^2(\Omega)$ also holds, since the continuity gives $\sup_{n\geq 0} \|\varphi(w_n) \|_{L^\infty(\Omega)} <\infty$ because $\sup_{n\geq 0}\|w_n\|_{L^\infty(\Omega) } <\infty$.
On the other side, in virtue of  the first estimate above the sequence
$\{Tw_n\}$ is bounded in finite dimensional space $\mathcal{V}_k$ and thus converges $\mathcal{V}_k$ up to a subsequence  to some $w_*\in \mathcal{V}_k$. Altogether, it follows that, for all  $\phi\in \mathcal{V}_k\subset L^\infty(\Omega)$
%
\begin{align*}
(f,\phi)= \lim_{n\to \infty} \int_\Omega\nabla\widehat{w}_n\nabla\phi  
+\zeta(w_n)\widehat{w}_n\phi dx=
\int_\Omega\nabla\widehat{w}\nabla\phi  +\zeta(w)w_* \phi. 
\end{align*}
The uniqueness of $\widehat{w}$ entails  $w_*=\widehat{w}=Tw$ and hence  the whole sequence $\{Tw_n\}$ converges in $Tw$ in $\mathcal V_k$, which gives the continuity of $T$.
Therefore, by the Brouwer fixed-point theorem,  $T$ has a fixed point $v_k\in \mathcal V_k$, i.e., $v_k=Tv_k$ which clearly satisfies  the announced relation.
An alternative.
The given problem is equivalent to the minimization problem
\begin{align}
 \mathcal{J}(v_k)=  \min_{v\in \mathcal{V}_k}  \mathcal{J}(v)\quad \text{with}\quad  \mathcal{J}(v):=  \frac12 \int_\Omega|\nabla v|^2 dx + \int_\Omega G(v)d x + \int_\Omega fvd x
 \end{align}
and we define the function $G(v)=  \int_0^v\zeta(\tau)\tau d \tau $.  Note that $G$ is non-negative since $\zeta(\tau)\geq 0$ and that $\mathcal J$ is continuous on $\mathcal V_k$. Using the Poincaré-Friedrichs inequality we find that $\mathcal{J}(v)\to \infty$, as $\|v\|_{L^2(\Omega)} \to \infty$ and $v\in \mathcal{V}_k$. Which implies the existence of a minimizer $v_k\in \mathcal{V}_k$ of \mathcal{J}, since we are in finite dimension space.
