= Solution
Put
$$
D=\|\psi_0\|_{H^1(U)}+\|\psi_1\|_{L^2(U)}.
$$
The assumed <energy estimate> for the linear <wave equation> and the first nonlinear estimate give, for $w\in X_{b,\tau}$,
$$
\|Aw\|_{X_\tau}
\leq C_0\left(D+\beta\tau^{1/2}b^2\right).
$$
Choose $b\geq2C_0D$ and then choose $\tau>0$ so small that $C_0\beta\tau^{1/2}b^2\leq b/2$. Then $A$ maps the closed ball $X_{b,\tau}$ into itself.
For $w,\widetilde w\in X_{b,\tau}$, the difference $Aw-A\widetilde w$ solves the linear equation with zero <Cauchy data> and forcing $F(w)-F(\widetilde w)$. The second nonlinear estimate therefore gives
$$
\|Aw-A\widetilde w\|_{X_\tau}
\leq2C_0\gamma\tau^{1/2}(b+b^2)
\|w-\widetilde w\|_{X_\tau}.
$$
Shrinking $\tau$ once more makes the coefficient strictly smaller than one. Since $X_\tau$ is a <Banach space> and its closed ball is complete, $A$ is a contraction on $X_{b,\tau}$.
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