One-parameter family of solutions for one class of Hammerstein nonlinear equations on a half-line
The existence of continuously many positive solutions with at most linear growth of the Hammerstein equation \[ \varphi(x)=\int_0^\infty K(x,t)Y(\varphi(t))dt \] on the half-line \((0,\infty)\) is proved where \(K(x,t)\geq K_0(x-t)\) is in a certain specified sense close to a convolution kernel \(K_0\), \(K_0\) being the Laplace transform of a measure with certain properties, and the nonlinearity \(Y\) has the rather particular form \(Y(u)=u-\omega(u)\) with \(\omega\geq0\) being decreasing for large arguments. Also results about factorizing the corresponding linear integral operator are obtained. The proof is in a sense constructive, using a variant of successive approximation with starting value depending on a scalar parameter; for different parameters, different solutions are obtained.
scientific article; zbMATH DE number 5664307