On a class of Hammerstein integral equations

By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) conditions for the unique existence of positive solutions of the Hammerstein integral equation $$begin{array}{*{20}c} {u(x) = intlimits_D {f(y,u(y)) k(x,y) dy ,} } & {x in D,} end{array} $$ and for the convergence of successive approximations towards the solution. Further we study the corresponding nonlinear eigenvalue problem. Essentially we assume that the integral kernel k satisfies appropriate positivity conditions and that, for the nonlinearity f and any y ∈ D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases.