= Solution
Use $u_i=u(x_i)$ for the sampled exact solution, and let $e_i=U_i-u_i$. The <Dirichlet boundary conditions> imply $e_0=e_N=0$. The definition of the <truncation error> $\varphi$ gives the exact discrete equation
$$
-D_x^+D_x^-u_i+u_i+u_i^3=f_i-\varphi_i.
$$
Subtracting it from the numerical equation therefore gives
$$
-D_x^+D_x^-e_i+e_i+(U_i^3-u_i^3)=\varphi_i.
$$
The sign on the right is positive because $\varphi_i$ was defined as the difference approximation minus the exact <second derivative>. Multiply by $he_i$, sum over the interior nodes and apply <discrete summation by parts>:
$$
\|D_x^-e]\!\|_h^2+\|e\|_h^2+h\sum_{i=1}^{N-1}(U_i^3-u_i^3)e_i=(\varphi,e)_h.
$$
Again $(U_i^3-u_i^3)e_i=e_i^2(U_i^2+U_iu_i+u_i^2)\geq0$. By <Young's inequality for products>, $(\varphi,e)_h\leq\tfrac12\|\varphi\|_h^2+\tfrac12\|e\|_h^2$. Hence
$$
\boxed{\|D_x^-U-D_x^-u]\!\|_h^2+\frac12\|U-u\|_h^2\leq\frac12\|\varphi\|_h^2.}
$$
No fourth <derivative> is needed for this step: the assumed twice continuous differentiability suffices to define $\varphi$ and obtain the discrete <energy estimate>.
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