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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.

Write $w=u-v$ and $r=f-g$. Subtraction of the two boundary value problems gives $-w^{\prime \prime }+w+u^3-v^3=r$, with $w(0)=w(1)=0$. Multiply by $w$ and apply integration by parts:Consequently the required energy identity isThe nonlinear term is nonnegative pointwise, sinceThis is the monotonicity energy estimate for a semilinear Dirichlet problem for the monotone function $s\mapsto s^3$. Apply Young's inequality for products, $rw\le (r^2+w^2)/2$, inside the last integral and discard the nonnegative nonlinear term. Moving $\frac{1}{2}{\int }_0^1{w}^{2}dx$ to the left givesIn particular, equal forcing gives $w=0$, proving uniqueness and a quantitative continuous dependence on forcing estimate for this boundary value problem.