Multilevel methods for the approximation of singular solutions of completely continuous operator equations
Multilevel methods in Banach spaces are considered for approximating singular solutions of the parameter-dependent problem \(u-K(u,\lambda)=0\), where \(K: (u,\lambda)\in U\times\mathbb{R}^ p\to U\) is a completely continuous \(C^ \nu\)-mapping \((\nu\geq 1)\) and \(U\) is a Banach space. By embedding the singular problem into an extended system, the author discusses an iterative method to approximate the solution with incorporation of multilevel methods for the linearized problems in Banach spaces. Convergence of the method is established under appropriate approximations of \(K\) with collectively compact operators and projections.
completely continuous operator equations
scientific article; zbMATH DE number 205198
scientific article