PROJECTION METHODS FOR EQUATIONS OF THE 2ND KIND

A class of projection methods, differing from the classical projection methods, is studied for the equation y = f + Ky, where K is a compact linear operator in a Banach space E, and f ε{lunate} E, In these methods K is approximated by a finite-rank operator Kn, which is constructed with the aid of certain projection operators, and which satisfies Knz = Kz for all z belonging to a chosen subspace Un ⊂ E. Under certain conditions, it is shown that the convergence of the approximate solution is faster than that of any classical projection method based on the subspace Un. In an example, Un is taken to consist of piecewise constant functions, and the projections are so chosen that the method becomes equivalent to a single iteration of a classical method, the collocation method; in this case the error (in the supremum norm) is O( 1 n2), compared with O( 1 n) for the collocation method. © 1979.