Iterative k-step methods for computing possibly repulsive fixed points in Banach spaces

Abstract

We discuss iterative methods of the form x n : = μ 0 Φ( x x − 1 ) + μ 1 x n − 1 + … + μ k x n − k ( n = k, k + 1,…) for computing a fixed point x ∗ of a Fréchet-differentiable self-mapping Φ of a sub-set in a Banach space. By suitably choosing the coefficients μ 0,…, μ k the local convergence is established under certain assumptions on the spectrum of Φ′ x ∗ . This spectrum need not be contained in the unit disk however; if it is, convergence can often be speeded up considerably compared to Picard iteration. The methods are generalizations of the methods of V. N. Kublanovskaya and W. Niethammer for linear systems of equations. A more general type of iteration with nonstationary coefficients is considered also. For the proof of their local convergence a generalization of the convergence theorems of Perron, Ostrowski and Kitchen is presented. The connection with other methods, in particular Mann's iterative processes, is also discussed.