= Solution
Write $w=u-v$ and $r=f-g$. Subtraction of the two <boundary value problems> gives $-w''+w+u^3-v^3=r$, with $w(0)=w(1)=0$. Multiply by $w$ and apply <integration by parts>:
$$
\int_0^1(-w'')w\,dx=-[w'w]_0^1+\int_0^1(w')^2\,dx=\int_0^1(w')^2\,dx.
$$
Consequently the required <energy identity> is
$$
\int_0^1\left((w')^2+w^2+(u^3-v^3)w\right)dx=\int_0^1rw\,dx.
$$
The nonlinear term is nonnegative pointwise, since
$$
(u^3-v^3)(u-v)=(u-v)^2(u^2+uv+v^2),\qquad u^2+uv+v^2=(u+v/2)^2+3v^2/4\geq0.
$$
This is the <monotonicity energy estimate for a semilinear Dirichlet problem> for the <monotone function> $s\mapsto s^3$. Apply <Young's inequality for products>, $rw\leq(r^2+w^2)/2$, inside the last <integral> and discard the nonnegative nonlinear term. Moving $\tfrac12\int_0^1w^2dx$ to the left gives
$$
\int_0^1(u'-v')^2dx+\frac12\int_0^1(u-v)^2dx\leq\frac12\int_0^1(f-g)^2dx.
$$
In particular, equal forcing gives $w=0$, proving <uniqueness> and a quantitative <continuous dependence on forcing> estimate for this <boundary value problem>.
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