Remarks on quadratic equations in Banach space. (English) Zbl 0706.47003
Consider the quadratic equation
\[
(*)\quad x=y+B(x,x),
\]
B a symmetric bilinear form on a Banach space. If for each z the linear operator B(z): \(x\mapsto B(z,x)\) has only 0 as fixed point, then solutions to (*) are unique. Existence theorems for solutions of (*) are formulated in terms of the convergence of sequences \(B(z_ n)B(z_{n-1})...B(z_ 1)x\) for certain vectors x and vector seuences \(z_ 1,...,z_ n,... \).
Reviewer: G.Ph.A.Thijsse
MSC:
47A50 | Equations and inequalities involving linear operators, with vector unknowns |
15A63 | Quadratic and bilinear forms, inner products |
47J05 | Equations involving nonlinear operators (general) |