Small-data contraction around a nonzero trajectory

Regard $E$ and $X_T$ as real spaces. In the energy case $X_T$ is Banach by [F3] and continuously embeds in $C([0,T];E)$ by the norm equivalence in [F2]. In the order-two case $E$ is complete by [F3]; a uniformly Cauchy sequence of continuous $E$-valued paths has a uniform limit, and the uniform limit is continuous, so $X_T=C([0,T];E)$ is Banach. Evaluation is bounded in both cases.

Define $Ax=S(\cdot ,0)x$ and $D_{c}h={\int }_0^{\cdot }S(\cdot ,r)(-i\phi h(r))dr$. The homogeneous estimate in [F2] bounds $A:E\to X_T$. By [F3], $\mathrm{\parallel }-i\phi h{\mathrm{\parallel }}_{L_t^{1}E}\le T\mathrm{\parallel }\phi {\mathrm{\parallel }}_{E\to E}\mathrm{\parallel }h{\mathrm{\parallel }}_{C_{t}E}$; the retarded estimate bounds $D_c:C([0,T];E)\to X_T$. These maps and $G_w$ are real-linear. Their controlled solution $Ax+D_c{G}_w(x,y)$ is exactly the one whose endpoints are $x,y$ in [F1].

By [F2], $N_w:X_T\to X_T$ is well defined and obeys $\mathrm{\parallel }{N}_w(z)\mathrm{\parallel }\le C(\mathrm{\parallel }z{\mathrm{\parallel }}^2+\mathrm{\parallel }z{\mathrm{\parallel }}^3)$ and $\mathrm{\parallel }{N}_w(z)-N_w(\tilde{z})\mathrm{\parallel }\le C(\mathrm{\parallel }z\mathrm{\parallel }+\mathrm{\parallel }\tilde{z}\mathrm{\parallel }+\mathrm{\parallel }z{\mathrm{\parallel }}^2+\mathrm{\parallel }\tilde{z}{\mathrm{\parallel }}^2)\mathrm{\parallel }z-\tilde{z}\mathrm{\parallel }$. Here multiplication by $i$ preserves the $E$ norm before application of the retarded estimate. The polynomial gives $N_w(0)=0$, and the zero-length integral gives $N_w(z)(0)=0$.

The maps in steps 1.2 and 1.3 realize the endpoint-feedback framework of [F4] with $m=3$. The selector is bounded, so combining its norm bound with steps 1.2 and 1.3 yields the two displayed estimates for suitable positive $C_1,C_2$. Real-linearity gives $h_z-h_{\tilde{z}}=-G_w(0,N_w(z)(T)-N_w(\tilde{z})(T))$. The controlled linear part has endpoints $z_0$ and $z_T-N_w(z)(T)$; adding $N_w(z)$, whose initial value is zero, gives the stated endpoint identities for every $z$.

Set $M=4C_1\delta$. Choose $\delta >0$ sufficiently small that $C_2(M^2+M^3)\le M\mathrm{/}2$ and $C_2(2M+2M^2)<1$. If ${\delta }^{\mathrm{\prime }}\le 2\delta$, step 2.1 gives $\mathrm{\parallel }{\mathcal{T}}_w(z)\mathrm{\parallel }\le M$ whenever $\mathrm{\parallel }z\mathrm{\parallel }\le M$, and gives a Lipschitz constant strictly below one for pairs in that ball.

Put $B=\{z\in X_T:\mathrm{\parallel }z\mathrm{\parallel }\le M\}$ with metric $d(z,u)=\mathrm{\parallel }z-u\mathrm{\parallel }$. It is nonempty because $0\in B$. A Cauchy sequence $(z_j)$ in $B$ converges in $X_T$ by step 1.1, and its limit $z$ satisfies $\mathrm{\parallel }z\mathrm{\parallel }\le \mathrm{\parallel }z-z_j\mathrm{\parallel }+M$ for every $j$, hence $\mathrm{\parallel }z\mathrm{\parallel }\le M$. Thus $B$ is complete. Step 3.1 shows that ${\mathcal{T}}_w{\mathrm{\mid }}_B:B\to B$ is a contraction with constant $q=C_2(2M+2M^2)<1$. Applying [F5] to this metric space and this restriction gives exactly one fixed point in the stated radius-$4C_1\delta$ ball. The endpoint identities of step 2.1 apply to that fixed point; its selector value belongs to $C([0,T];E)$. Substitution of the fixed point into the definition in [F1] gives the controlled perturbation integral equation, with the factors $i$ retained inside the real-linear propagator. This proves the claim.