On the Stability of the Ritz-Galerkin Method for Hammerstein Equations

For the numerical treatment of Hammerstein equations by variational methods which has been considered by Hertling, we establish the stability in the sense of Mikhlin, Stetter and Tucker. Introduction. If one uses a variational method for the numerical treatment of Hammerstein equations, one obtains a nonlinear algebraic system of equations. In order to investigate the stability of the computing scheme, we will show that one can apply a theorem by Tucker [7]. Tucker's work is based on a paper by Mikhlin [3]. We would also like to refer to a paper by Kasriel and Nashed [2] where the problem of stability has been considered in a very similar way for some classes of nonlinear operator equations. An equivalent general concept of stability and its application to initial-value problems has been given by Stetter [6]. Let B be a bounded measurable set in a finite-dimensional Euclidean space and let the symmetric kernel K(x, y) define an operator A which is selfadjoint in L2 and completely continuous from Iß into V (p > 2, p~1 + q~l = 1): (1.1) Au= jBK(x,y)u(y)dy. Furthermore, we introduce the Nemytsky operator (1.2) h = g(u(y),y) as a continuous operator from Lp into Lq ; we assume that giu, y) is an A/-function and that h is potential. A function g(u, y) is an A^-function if it is continuous with respect to u for almost every y E B and measurable in B with respect to y for every fixed u E (— °°, +°°). An operator h from a Banach space E into the conjugate space E* is called potential on some set // C E, if there exists a functional / such that grad f(x) = h(x) for every x EH. Let G(u, y) be defined by (13) 3G(«, y)/èu = g(u, y) and assume Received August 27, 1973, revised November 29, 1973 and June 4, 1974. AMS (MOS) subject classifications (1970). Primary 45L10, 65R05; Secondary 65N30.