Approximate Solution of Bisingular Integro-Differential Equations

When solving bisingular integrodifferential equations by collocation and Galerkin methods one naturally asks whether the approximate solutions exist, are uniquely determined and converge to the exact solution. These problems were studied in [2],[6) for Toeplitz and singular integral operators by means of Banach algebra techniques. The integro-differential operator treated here acts from one Banach space E1 into another Banach space E2 , where E2. Thus, there is no multiplication operation in the set £(E1 , E2 ) of all bounded linear operators. This necessitates the consideration of special paraalgebras which allows us to reduce the original problem of the applicability of collocation and Galerkin methods to the investigation of the invertibility of certain elements in a quotient paraalgebra A/J. This problem can be solved using a local principle for paraalgebras (cf. [3]) generalizing the well-known local principle of Gohberg-Krupnik [5]. We note that some results on the approximate solution of pseudodifferential equations are already contained in [9].