= Solution
Work with the <L1 norm> on $\mathbb R^2$. The scalar Lipschitz hypothesis implies
$$
\begin{aligned}
\|\mathbf f(x,\mathbf y)-\mathbf f(x,\mathbf z)\|_1
&=|y_2-z_2|+|F(x,\mathbf y)-F(x,\mathbf z)|\\
&\leq(1+L)\|\mathbf y-\mathbf z\|_1.
\end{aligned}
$$
We may take $L\geq0$. The <compact set> $S$ and continuity give a finite bound $M=\max_S\|\mathbf f\|_1$. Choose
$$
\boxed{0<\eta\leq\min\left\{h,\frac{k}{M+1},\frac1{2(1+L)}\right\}.}
$$
Let $I=[a-\eta,a+\eta]$. The space $C_\eta$ of continuous maps $I\to B_k(\mathbf b)$ is complete in the <supremum norm>: it is a closed subset of the <Banach space> $C(I,\mathbb R^2)$. Define the <Picard operator on a closed ball> by
$$
(\mathcal P\mathbf g)(x)=\mathbf b+\int_a^x\mathbf f(t,\mathbf g(t))\,dt.
$$
For any $\mathbf g\in C_\eta$, the <integral triangle inequality> gives
$$
\|\mathcal P\mathbf g(x)-\mathbf b\|_1\leq |x-a|M\leq\eta M<k.
$$
The integrand is continuous, so the result is continuous and remains in the prescribed <closed ball>. Thus $\mathcal P$ is a self-map of the complete space. Moreover,
$$
\|\mathcal P\mathbf g-\mathcal P\mathbf h\|_\infty
\leq\eta(1+L)\|\mathbf g-\mathbf h\|_\infty
\leq\frac12\|\mathbf g-\mathbf h\|_\infty.
$$
It is a <contraction mapping>. The <contraction mapping theorem> gives a unique <fixed point> $\mathbf y$. By the <fundamental theorem of calculus>, the fixed-point identity makes $\mathbf y$ continuously differentiable and gives $\mathbf y'=\mathbf f(x,\mathbf y)$ and $\mathbf y(a)=\mathbf b$. Its first component is twice continuously differentiable, because its derivative is the second component.
Conversely, integrating any solution that remains in the ball makes it a <fixed point>, so it equals $\mathbf y$. A solution starting at $\mathbf b$ cannot first reach the boundary of this ball during $I$: before such an exit the same estimate gives distance at most $\eta M<k$, contradicting distance $k$ at the first exit. This proves \b[existence and uniqueness on $|x-a|\leq\eta$], not merely formal convergence of an iteration. It is the local <Picard-Lindelöf theorem> construction.
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