Monotonicity energy estimate for a semilinear Dirichlet problem ($\Vert (u-v{)}^{\prime }{\Vert }_2^2+\frac{1}{2}\Vert u-v{\Vert }_2^2\le \frac{1}{2}\Vert f-g{\Vert }_2^2$)

Suppose $-u^{\prime \prime }+u+q(u)=f$ and $-v^{\prime \prime }+v+q(v)=g$ 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 givesThe 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.