Endpoint correction and small-ball contraction estimates

The selector steers $W(\cdot )v_0+D(-i\phi h_v)$ from $v_0$ to $v_T-N(v)(T)$. Adding $N(v)$, which vanishes at time zero, gives $\mathcal{T}(v)(0)=v_0$ and $\mathcal{T}(v)(T)=v_T$. Linearity of $G$ gives $h_v-h_u=-G(0,N(v)(T)-N(u)(T))$. These are the same solution and selector maps as in the endpoint feedback lemma.

Since multiplication by $i$ preserves the norm of $H$, the forcing and nonlinear estimates give $\mathrm{\parallel }N(v){\mathrm{\parallel }}_{X_T}\le C_0\mathrm{\parallel }ig(v){\mathrm{\parallel }}_{L^1(0,T;H)}\le C_0^2\mathrm{\parallel }v{\mathrm{\parallel }}_{X_T}^m$. Similarly, $\mathrm{\parallel }N(v)-N(u){\mathrm{\parallel }}_{X_T}\le C_0^2(\mathrm{\parallel }v{\mathrm{\parallel }}_{X_T}^{m-1}+\mathrm{\parallel }u{\mathrm{\parallel }}_{X_T}^{m-1})\mathrm{\parallel }v-u{\mathrm{\parallel }}_{X_T}$. These estimates apply to the forcing before $D$ and do not require multiplication by $i$ to preserve a real $X_T$.

The selector and multiplier bounds imply $\mathrm{\parallel }\phi h_v{\mathrm{\parallel }}_{L^1(0,T;H)}\le T{C}_{\phi }{C}_G(\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_H+\mathrm{\parallel }{v}_T{\mathrm{\parallel }}_H+\mathrm{\parallel }N(v)(T){\mathrm{\parallel }}_H)$. Terminal evaluation is bounded by one. Combining this with the homogeneous and forcing bounds and step 1.2 yields the asserted size estimate with $C_1,C_2$; $C_1$ also bounds the coefficient of $\mathrm{\parallel }{v}_T{\mathrm{\parallel }}_H$.

The selector difference in step 1.1, terminal evaluation, and the same forcing and multiplier bounds give $\mathrm{\parallel }\mathcal{T}(v)-\mathcal{T}(u){\mathrm{\parallel }}_{X_T}\le (1+C_{0}T{C}_{\phi }{C}_G)\mathrm{\parallel }N(v)-N(u){\mathrm{\parallel }}_{X_T}$. The difference estimate follows from step 1.2 and the definition of $C_2$.

If both endpoint norms are at most $\delta$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_{X_T}\le M$, then step 2.1 gives $\mathrm{\parallel }\mathcal{T}(v){\mathrm{\parallel }}_{X_T}\le 2C_1\delta +C_2{M}^m\le M\mathrm{/}2+M\mathrm{/}4\le M$. For two points in the ball, step 2.2 gives a Lipschitz constant at most $2C_2{M}^{m-1}\le 1\mathrm{/}2$. Because $m>1$ and $C_1,C_2$ are fixed, a positive $\delta$ satisfying the displayed smallness condition exists.

The ball is nonempty and complete: it contains zero, and a Cauchy sequence in it converges in the Banach space $X_T$ to a point still in the ball by the triangle inequality. The contraction theorem gives exactly one fixed point in this ball. Its endpoint values are those in step 1.1. This proves the stated conclusion without asserting uniqueness outside the ball.