On a class of ill‐posed non‐linear integral equations arising in the interpretation of indirect measurements

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Abstract
In this paper we consider a class of specific Urysohn integral equations for which the solutions are only determined with the exception of rearrangements of function values and associated arguments. As an alternative to Tikhonov's regularization method approximating minimum‐norm solutions for this ill‐posed class of inverse problems, a constrained least‐squares approach is presented. This approach is aimed at finding decreasing rearrangements serving as appropriate solution representatives. It is shown that the inverses of these decresing solutions solve a Fredholm linear integral equation of the first kind.