A new metastable convergence criterion and an application in the theory of uniformly convex Banach spaces

Abstract

We study a convergence criterion which generalises the notion of being monotonically decreasing, and introduce a quantitative version of this criterion, a so called metastable rate of asymptotic decreasingness. We then present a concrete application in the fixed point theory of uniformly convex Banach spaces, in which we carry out a quantitative analysis of a convergence proof of Kirk and Sims. More precisely, we produce a rate of metastability (in the sense of Tao) for the Picard iterates of mappings T which satisfy a variant of the convergence criterion, and whose fixed point set has nonempty interior.