My question comes from Chapter 4 of "Partial Differential Equations with Numerical Methods" by Stig Larsson and Vidar Thomee.
Consider the two-point boundary value problem
\begin{equation} \mathcal{A}u := -au'' + bu' +cu = f \ \ \text{in} \ \mathcal{\Omega} = (0,1), \\ u(0) = u_0, \ u(1) = u_1 \end{equation}
where $a = a(x), b = b(x)$, and $c = c(x)$ are smooth functions satisfying $a(x) > 0$ and $c(x) > 0$ in $\bar{\mathcal{\Omega}}$, and where the function $f = f(x)$ is given.
Now, we introduce mesh points $0 = x_0 < x_1 < \cdots < x_M = 1$ with $x_j = jh$, $j = 0, \ldots, M$, $h = 1/M$ and approximations $U_j = u(x_j)$, $b_j = b(x_j)$, $c_j = c(x_j)$, $f_j = f(x_j)$. By defining $\hat{\partial}U_j = (U_j - U_{j-1}) /h$, and $\partial \bar{\partial}U_j = (U_{j+1} - 2U_j + U_{j-1})/h^2$ we can introduce a finite difference approximation of our two-point boundary value problem as
\begin{equation} \mathcal{A}_hU_j := -a_j \partial \bar{\partial}U_j + b_j \hat{\partial}U_j + c_j U_j = f_j, \ \ j = 1, \ldots, M - 1, \\ U_0 = u_0, U_m = u_1. \end{equation}
And so, at interior point $x_j$ we have
$$ (2a_j + h^2c_j)U_j - (a_j - \frac{1}{2}hb_j)U_{j+1} - (a_j + \frac{1}{2}hb_j)U_{j-1} = h^2f_j. $$
An important result given is
which is the maximum principle applied to our finite difference method.
Now, defining $|U|_S = \underset{x_j \in S}{\text{max}} |U_j|$ we arrive at the following result
As the result of Lemma 4.2 assumes $b = 0$, I am trying to see how the same conclusion can be reached when $b \neq 0$ if $h$ is sufficiently small and we have a mesh function $W$ such that $\mathcal{A}_h W_j \geq 1$ for $x_j \in \mathcal{\Omega}$ and $W_j \geq 0$ for $x_j \in \bar{\mathcal{\Omega}}$.
The textbook does not provide a proof of this fact, but says one should consider a similar proof of Lemma 4.2 with $w(x) = e^\lambda - e^{\lambda x}$ for a suitable $\lambda$.
If someone sees how to properly construct the mesh function $W$ such that the result can follow with a similar proof I would enjoy to see how it works!