How to solve nonlinear operator equation \(A(v^{2})+Cv=f\)
In the function space \(X=W_2^1[a,b]\), the equation \(A(v^2)+Cv=f\) is studied, where \(A\) and \(C\) are bounded linear operators. Using a reproducing kernel, the nonlinear equation is reduced to a linear equation in \(W=X\otimes X\) whose solutions correspond to the solutions of the nonlinear equation if they have the form \(w(x,y)=v(x)v(y)\). This form is automatic under some uniqueness hypotheses which, however, are hard to verify. In order to construct a minimum norm solution of the linear equation (or, more general, of any linear equation in Hilbert spaces), the Gram-Schmidt orthogonalization process is applied for the adjoint equation. In some numerical examples, the error is small already after a few iterations. Some confusions of the fonts make it sometimes difficult to read the paper.
scientific article; zbMATH DE number 2082059