Mild formulation of the controlled power NLS

By [F2], the restriction of $W$ to nonnegative times is a strongly continuous semigroup on the Banach space $X$. The two assumed Bochner-integrable functions have the Bochner-integrable linear combination $f=ig(v)-iKh$ by [F5]. The criterion in [F5] then gives ${\int }_0^T\mathrm{\parallel }f(r){\mathrm{\parallel }}_{X}dr<\mathrm{\infty }$. The assumed Axiom of Choice supplies Dependent Choice by [F4]. Therefore the variation-of-constants theorem in [F4] applies with initial vector $v_0$ and this particular forcing. It says that an integral solution is exactly the continuous variation-of-constants function $W(t)v_0+{\int }_0^{t}W(t-r)f(r)dr$.

If $x\in H_D^{\sigma +2}$, [F2] gives the derivative at zero of $W(t)x$ in $X$ as $i{\mathrm{\Delta }}_{D}x$. The definition in [F3] therefore gives $x\in D(B)$ and $Bx=i{\mathrm{\Delta }}_{D}x$. Multiplication of the controlled equation in [F1] by $-i$ gives $v^{\mathrm{\prime }}=i{\mathrm{\Delta }}_{D}v+ig(v)-iKh$, whenever those terms are meaningful in $X$.

Linearity of the Bochner integral splits the last integral into $i{\int }_0^{t}W(t-r)g(v)(r)dr-i{\int }_0^{t}W(t-r)(Kh)(r)dr$. Thus the integral-solution condition is equivalent to the displayed equation. Since $f$ was formed from the given $v$, this equivalence applies to each candidate $v$ satisfying the stated integrability assumptions; it does not use a nonlinear existence theorem.

Every classical solution of $v^{\mathrm{\prime }}=Bv+f$ with the stated initial value has the variation-of-constants formula by [F4], hence the displayed fixed-point equation by step 2.1. Step 1.2 identifies the abstract equation with the controlled equation for a sufficiently regular trajectory taking values in $H_D^{\sigma +2}$ and satisfying the classical-solution conditions. This proves the assertions without claiming that every distributional solution has these regularity properties.