ON A SPECIAL-CLASS OF NONLINEAR FREDHOLM-INTEGRAL-EQUATIONS OF THE FIRST KIND

In this paper a special class of nonlinear Fredholm integral equations of the first kind, the so-called Urysohn equation, is considered, whe re the kernel depends on t only via the unknown function x(t). To over come the ambiguity, a decreasing rearrangement approach is used. Moreo ver, a constrained least squares method helps regularizing the problem . As a specific property, the equation can be decomposed into a well-p osed nonlinear part, the inversion of a function, and an ill-posed lin ear part, a linear Fredholm integral equation of the first kind. The l inear part of this two-stage procedure was already discussed in [8]. I n the present paper the two-stage procedure is compared with a one-sta ge nonlinear least squares approximation which is directly applied to the nonlinear original integral equation. The comparison is explained by means of a computational case study for a specific example arising in optics.