Acceleration of iteration methods for interval fixed point problems

Abstract

We consider the fixed point equation x = F ( x ) with a continuous and inclusion isotone interval function F : R n → R n . The iteration x k+1= F ( x k) converges monotonically ( x k+1⊆ x k ) to a fixed point of F if x 1⊆ x 0 . We prove a theorem on an accelerated monotone iteration and apply it to systems of linear and nonlinear equations. For linear fixed point equations ( x = Ax + b ), we also present a modified single step method.