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

AbstractWe discuss iterative methods of the form xn: = μ0Φ(xx − 1) + μ1xn − 1 + … + μkxn − 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 μ0,…, μk 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