Numerical solutions of nonlinear integral equations by approximations of fixed-points

This paper considers a numerical method for approximating solutions of Hammerstein’s equation on the functional space consisting of Riemann-integrable functions. The fundamental feature of the method is to solve numerically a fixed-point problem of an operator defined on the functional space, having the same solution as the Hammerstein’s equation. Two kinds of approximation solutions are constructed by employing Lagrange’s interpolation and a natural projection from the view-point of collocation methods. For the approximation solutions, convergence in L2-norm is shown and when the functional space is restricted to the subspace consisting of continuous functions, especially convergence in the uniform norm is shown. Also analysis of errors is done for the approximation solutions and a scheme of numerical computation for obtaining approximation solutions is proposed in the case where Hammerstein’s equation has sufficiently smooth solutions. Particularly the scheme is applicable to problems having one or more than one solution. Several examples are included to verify utility of the scheme and to illustrate convergence of approximation solutions in problems possessing discontinuity. [In Japanese.]