 Solution

For $f,g\in X$, the Lipschitz condition givesChoose $c$ so that $c+1>Mb$. Then $\varphi$ is a contraction mapping. The space $X$ is complete in the sup norm, and therefore also in the equivalent weighted norm.

For initial value $y_0$, defineIt has the same contraction constant. The Banach fixed-point theorem gives a unique fixed point $f$, which satisfies the integral equation. The fundamental theorem of calculus then givesConversely every solution satisfies that integral equation, proving uniqueness on $[1,b]$. This weighted-norm argument is the Bielecki norm method.