Numerical treatment of hammerstein-equations by variational methods

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

“…Let G(u, y) be defined by one of the authors [1] has considered the numerical solution by means of a Ritz-Galerkin scheme and by using subspaces of spline functions and finite elements. We shall establish here the stability of this approximating scheme.…”

“…If we represent a function in L2m by EjjSL,, u¡w¡iy) and if we define xpÇEjL. u^xvAy)) = Giul,-• •, um) = G(u), then it has been shown in [1] that there exists a positive constant C. such that We will denote the system (2.9) by (2-11) Tmum=0. Definition 1 [7].…”

On the stability of the Ritz-Galerkin method for Hammerstein equations

“…For the Hammerstein equation (1.5) u = Ahu, one of the authors [1] has considered the numerical solution by means of a Ritz-Galerkin scheme and by using subspaces of spline functions and finite elements. We shall establish here the stability of this approximating scheme.…”

“…If we represent a function in L2m by EjjSL,, u¡w¡iy) and if we define xpÇEjL. u^xvAy)) = Giul,-• •, um) = G(u), then it has been shown in [1] that there exists a positive constant C. such that where u0 is the solution of (2.5), wm the unique function which minimizes the func- Let us remark that we did not use the existence of the second derivative of the functional (2.3) as has been done by Mikhlin [3] and Schiop [5]. On the other hand, we have to assume (2.14).…”

“…Let us finally say that, with the machinery of [1], an analogous proof for the stability of the computing scheme for Hammerstein equations with quasi-definite kernels can be given.…”