Approximate solutions of eigenvalue problems with reproducing nonlinearities (Q762458)

Let \(E=\{u_ i\}\), \(i=1,2,..\). be complete orthonormal system of the separable Hilbert space H and let N: domain(N)\(\subset H\to H\) be a nonlinear operator. N is said to be ''finitely reproducing relative to E'' if \[ (1)\quad N(\sum^{n}_{i=1}c_ iu_ i)=\sum^{m(n)}_{i=1}b_ iu_ i \] where \(b_ i\) are given functions of \((c_ 1,c_ 2,...,c_ n)\). Let \(H=L^ 2(0,\pi)\) and \(E=\{u_ i;u_ i(x)=\sqrt{2/\pi}\sin (ix)\}\). The author gives a complete characterization of (1) for \(N(u)=([u^{(g)}]^ p)^{(l)}\) and uses it in the numerical approximation of the equation \(- u''+N(u)=\lambda u\).