Computing positive fixed-points of decreasing Hammerstein operators by relaxed iterations

We prove global convergence of (under)relaxed Picard-like methods for fixed-point equations u = A(u),A : C+(Ω) → C+(Ω), Ω being a compact Hausdorff space. The operator A is decreasing and completely continuous, and possesses no pairs of distinct and comparable coupled-fixed points. Infinite- as well as finite-dimensional Hammerstein equations of this type arise in transport theory. As a numerical application, we test Picard, updated Picard, Jacobi, and Gauss-Seidel (under)relaxed iterations on the discrete “decreasing” version of Chandrasekhar H-equation. A comparison with popular Newton-like solvers is also presented.