  Techniques are proposed for solving integral equations of the first kind with
an input known not precisely. The requirement that the solution sought for
includes a given number of maxima and minima is imposed. It is shown that when
the deviation of the approximate input from the true one is sufficiently small
and some additional conditions are fulfilled the method leads to an approximate
solution that is necessarily close to the true solution. No regularization is
required in the present approach. Requirements on features of the solution at
integration limits are also imposed. The problem is treated with the help of an
ansatz proposed for the derivative of the solution. The ansatz is the most
general one compatible with the above mentioned requirements. The techniques
are tested with exactly solvable examples. Inversions of the Lorentz, Stieltjes
and Laplace integral transforms are performed, and very satisfactory results
are obtained. The method is useful, in particular, for the calculation of
quantum-mechanical reaction amplitudes and inclusive spectra of
perturbation-induced reactions in the framework of the integral transform
approach.
