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This paper furnishes two classes of methods for calculating the best approximate solution of an operator equation in Banach spaces, where the operator is bounded, linear and has closed range. The best approximate solution can be calculated by an iterative method in Banach spaces stated in terms of an operator parameter. Specifying the parameter yields some new and some old iterative techniques. Another approach is to extend the classical approximation theory of Kantorovich for equations with invertible operators to the singular case. The best approximate solution is now obtained as the limit of the best approximate solutions of simpler equations, usually systems of linear algebraic equations. In particular, a Galerkin-type method is formulated and its convergence to the best approximate solution is established. The methods of this paper can also be used for calculating the best least squares solution in Hilbert spaces or the true solution in the case of an invertible operator.

Calculating the best approximate solution of an operator equation