Schauder solution of a small Hammerstein integral equation ($u(x)=\mu {\int }_0^{1}G(x,y)f(y,u(y))dy$)

For continuous $G$ and $f$, restrict to the closed supremum-norm ball of radius $R$ in $C([0,1])$. Boundedness on the compact kernel and nonlinearity boxes makes the integral map preserve this ball for sufficiently small positive $\mu$. Uniform continuity of $f$ proves continuity of the map; uniform continuity of $G$ gives an equicontinuous bounded image. Arzelà-Ascoli makes that image relatively compact, and Schauder gives a solution. No global Lipschitz hypothesis or uniqueness conclusion is needed.