Rates of Convergence for Asymptotically Weakly Contractive Mappings in Normed Spaces

We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalized convergence proofs along with explicit rates of convergence. More specifically, we define a new notion of being asymptotically psi-weakly contractive with modulus, and present a series of abstract convergence theorems which both generalize and unify known results from the literature. Rates of convergence are formulated in terms of our modulus of contractivity, in conjunction with other moduli and functions which form quantitative analogues of additional assumptions that are required in each case. Our approach makes use of ideas from proof theory, in particular our emphasis on abstraction and on formulating our main results in a quantitative manner. As such, the paper can be seen as a contribution to the proof mining program.

Automated summary

The study introduces a new concept of being asymptotically weakly contractive with modulus, and provides generalized convergence proofs and explicit rates of convergence, further generalizing and unifying known results. The research utilizes ideas from proof theory and formulates main results in a quantitative manner.