= Schauder solution of a small Hammerstein integral equation
{c}
{title2=$u(x)=\mu\int_0^1G(x,y)f(y,u(y))\,dy$}
For <continuous> $G$ and $f$, restrict to the closed <supremum norm>[supremum-norm] ball of radius $R$ in $C([0,1])$. Boundedness on the compact <null space>[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 theorem>[Arzelà-Ascoli] makes that image relatively compact, and <Schauder fixed-point theorem>[Schauder] gives a solution. No global <Lipschitz continuity>[Lipschitz] hypothesis or uniqueness conclusion is needed.
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