A note on solutions in \(L^1[0,1]\) to Hammerstein integral equations
The author presents some existence principles for the Hammerstein integral equation \[ y(t)= g(t)+ \int_0^1 k(t,s) f(s,y(s))ds, \] where \(k:[0,1]\times [0,1]\to\mathbb{R}\) and \(f:[0,1]\times \mathbb{R}\to\mathbb{R}\). He considers solutions which lie in \(L^1[0,1]\) (for completeness he also discusses the case \(L^p[0,1]\), \(1<p<\infty\)). His analyses rely on the notion of measure of weak noncompactness and on the Schauder fixed point theorem. It is worth remarking here also that essentially the same reasoning would establish existence principles for Volterra and even Uryson integral equations.
scientific article; zbMATH DE number 1136060
scientific article