In a previous paper, the first author derived an explicit quantitative version of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behaviour of Mann iterations of nonexpansive mappings of convex sets $C$ in normed linear spaces. This quantitative version, which was obtained by a logical analysis of the ineffective proof given by Borwein, Reich, and Shafrir, could be used to obtain strong uniform bounds on the asymptotic regularity of such iterations in the case of bounded $C$ and even weaker conditions. In this paper, we extend these results to hyperbolic spaces and directionally nonexpansive mappings. In particular, we obtain significantly stronger and more general forms of the main results of a recent paper by W. A. Kirk with explicit bounds. As a special feature of our approach, which is based on logical analysis instead of functional analysis, no functional analytic embeddings are needed to obtain our uniformity results which contain all previously known results of this kind as special cases.

This paper contains a review of results concerning “generalized” attractors for a large class of iterated function systems $\left\{w_i:i\in I\right\}$ acting on a complete separable metric space. This generalization, which originates in the Banach contraction principle, allows us to consider a new class of sets, which we call semi-attractors (or semifractals). These sets have many interesting properties. Moreover, we give some fixed-point results for Markov operators acting on the space of Borel measures, and we show some relations between semi-attractors and supports of invariant measures for such Markov operators. Finally, we also show some relations between multifunctions and transition functions appearing in the theory of Markov operators.

A constrained least squares problem in a Hilbert space $H$ is considered. The standard Tikhonov regularization method is used. In the case where the set of the constraints is the nonempty intersection of a finite collection of closed convex subsets of $H$, an iterative algorithm is designed. The resulting sequence is shown to converge strongly to the unique solution of the regularized problem. The net of the solutions to the regularized problems strongly converges to the minimum norm solution of the least squares problem if its solution set is nonempty.

We show a construction of domains in complex reflexive Banach spaces which are locally uniformly convex in linear sense in their Kobayashi distance. We also show connections between norm and Kobayashi distance properties.