= Monotonicity energy estimate for a semilinear Dirichlet problem
{title2=$\|(u-v)'\|_2^2+\frac12\|u-v\|_2^2\leq\frac12\|f-g\|_2^2$}
Suppose $-u''+u+q(u)=f$ and $-v''+v+q(v)=g$ on a bounded interval, the solutions have the same <Dirichlet boundary conditions>, and $q$ is a <nondecreasing function>. Multiplying the difference equation by $w=u-v$ and using <integration by parts> gives
$$
\|w'\|_2^2+\|w\|_2^2+\int(q(u)-q(v))w=\int(f-g)w.
$$
The nonlinear <integral> is nonnegative. <Young's inequality for products> gives the displayed <energy estimate>. It proves <uniqueness> and <continuous dependence on forcing>. The same argument holds for nodal values when <integration by parts> is replaced by <discrete summation by parts>.
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