Abstract
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 ψ-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.
Highlights
Let X be a real normed space and T : E → X a mapping defined on some closed convex set E ⊆ X
This paper will be based around one such class, the so-called weakly contractive mapping introduced by Alber and Guerre-Delabriere in [3]
In our final and most complex case study, we take as inspiration a paper of Alber, Reich and Yao [7], where approximation sequences {zn} to weakly contractive mappings T : E → X are studied for which zn is projected onto some En ⊆ E
Summary
Let X be a real normed space and T : E → X a mapping defined on some closed convex set E ⊆ X. We use ideas from proof theory to establish a series of general convergence theorems for classes of weakly contractive mappings These strengthen the aforementioned results by weakening parameters and introducing suitably abstract formulations of key properties, but bring them together as part of a unifying scheme. Our rates of convergence can be reformulated and directly compared to those which have been given in the literature, but we are able to give explicit rates of convergence for general theorems which, to the best of our knowledge, are new Another feature of our approach in that our main results were obtained through the careful analysis of existing proofs, where we sought to dispose of superfluous assumptions (which, for instance, were only required to establish existence of a fixed point) and provide abstract versions of others, resulting in generalised theorems in which certain premises have been weakened and others eliminated altogether. It is certainly the case that there are plenty of other classes of mappings and associated convergence theorems which could be abstracted and generalised in a similar fashion, and we leave an exploration of more recent results in this area to future work (cf. Section 7)
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