= Solution
Let $f_i=f(ih)$ and $g_i=g(ih)$, and define the scaled <forward difference operator>, <backward difference operator> and <second-order central difference> by
$$
D_x^+W_i=\frac{W_{i+1}-W_i}{h},\qquad D_x^-W_i=\frac{W_i-W_{i-1}}h,\qquad
D_x^+D_x^-W_i=\frac{W_{i+1}-2W_i+W_{i-1}}{h^2}.
$$
The two <finite difference methods> are the nonlinear systems
$$
\boxed{-D_x^+D_x^-U_i+U_i+U_i^3=f_i,\qquad -D_x^+D_x^-V_i+V_i+V_i^3=g_i,\quad 1\leq i\leq N-1,}
$$
with the prescribed <Dirichlet boundary conditions> $U_0=V_0=A$ and $U_N=V_N=B$. These equations determine the interior nodal values.
For clarity distinguish the interior and edge <inner products> and their <norms>:
$$
(P,Q)_h=h\sum_{i=1}^{N-1}P_iQ_i,\quad
(P,Q]_{h}=h\sum_{i=1}^NP_iQ_i,\quad
\|P\|_h^2=(P,P)_h,\quad
\|P]\!\|_h^2=(P,P]_{h}.
$$
The bracketed <norm> includes the last edge $i=N$. Let $W=U-V$, so $W_0=W_N=0$, and put $q_i=D_x^-W_i$ for $1\leq i\leq N$. Direct telescoping gives <discrete summation by parts>:
$$
(-D_x^+D_x^-W,W)_h
=-\sum_{i=1}^{N-1}(q_{i+1}-q_i)W_i
=\sum_{i=1}^Nq_i(W_i-W_{i-1})
=h\sum_{i=1}^Nq_i^2.
$$
Subtract the two nonlinear systems, multiply each equation by $hW_i$ and sum. The resulting discrete <energy identity> is
$$
\|D_x^-W]\!\|_h^2+\|W\|_h^2+h\sum_{i=1}^{N-1}(U_i^3-V_i^3)W_i=(f-g,W)_h.
$$
The nonlinear summands are nonnegative by the factorization in (a). Using <Young's inequality for products> at every node gives
$$
\boxed{\|D_x^-U-D_x^-V]\!\|_h^2+\frac12\|U-V\|_h^2\leq\frac12\|f-g\|_h^2.}
$$
This also proves <uniqueness> of each discrete solution. Existence can be seen by minimizing, with the endpoints fixed, the <strictly convex> <energy functional>
$$
J_f(W)=\frac h2\sum_{i=1}^N(D_x^-W_i)^2+h\sum_{i=1}^{N-1}\left(\frac{W_i^2}2+\frac{W_i^4}4-f_iW_i\right).
$$
It is a <coercive function> of its finitely many interior values, so it attains a minimum. Its first <derivatives> are $h(-D_x^+D_x^-W_i+W_i+W_i^3-f_i)$; thus its <minimizer> solves the scheme. The same argument applies to $g$.
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