Averaging and fixed points in Banach spaces.

We use various averaging techniques to obtain results in different aspects of functional analysis and Banach space theory, particularly in fixed point theory. Specifically, in the second chapter, we discuss the class of so-called mean nonexpansive maps, introduced in 2007 by Goebel and Japon Pineda, and we prove that mean isometries must be isometries in the usual sense. We further generalize this class of mappings to what we call the affine combination maps, give many examples, and study some preliminary properties of this class. In the third chapter, we extend Browder's and Opial's famous Demiclosedness Principles to the class of mean nonexpansive mappings in the setting of uniformly convex spaces and spaces satisfying Opial's property. Using this new demiclosedness principle, we prove that the iterates of a mean nonexpansive map converge weakly to a fixed point in the presence of asymptotic regularity at a point. In the fourth chapter, we investigate the geometry and fixed point properties of some equivalent renormings of the classical Banach space c0. In doing so, we prove that all norms on `1 which have a certain form must fail to contain asymptotically isometric copies of c0

Strict pseudocontractions and demicontractions, their properties and applications

We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary and sufficient conditions under which a convex combination and composition of strict pseudocontractions as well as demicontractions that share a common fixed point is again a strict pseudocontraction or a demicontraction, respectively. Moreover, we introduce a generalized relaxation of composition of demicontraction and give its properties. We apply these properties to prove the weak convergence of a class of algorithms that is wider than the Douglas-Rachford algorithm and projected Landweber algorithms. We have also presented two numerical examples, where we compare the behavior of the presented methods with the Douglas-Rachford method.Comment: 27 pages, 3 figure

On the structure of fixed-point sets of asymptotically regular semigroups

We extend a few recent results of G\'{o}rnicki (2011) asserting that the set of fixed points of an asymptotically regular mapping is a retract of its domain. In particular, we prove that in some cases the resulting retraction is H\"{o}lder continuous. We also characterise Bynum's coefficients and the Opial modulus in terms of nets.Comment: 11 page

Fixed Point Theorems for Nonexpansive Type Mappings in Banach Spaces

In this paper, we present some fixed point results for a class of nonexpansive type and α-Krasnosel’skiĭ mappings. Moreover, we present some convergence results for one parameter nonexpansive type semigroups. Some non-trivial examples have been presented to illustrate facts.The authors thanks the Basque Government for its support through Grant IT1207-19

Some fixed point results for enriched nonexpansive type mappings in Banach spaces

[EN] In this paper, we introduce two new classes of nonlinear mappings and present some new existence and convergence theorems for these mappings in Banach spaces. More precisely, we employ the Krasnosel'skii iterative method to obtain fixed points of Suzuki-enriched nonexpansive mappings under different conditions. Moreover, we approximate the fixed point of enriched-quasinonexpansive mappings via Ishikawa iterative method. The first author acknowledges the support from the GES 4.0 fellowship, University of Johannesburg, South Africa.Shukla, R.; Pant, R. (2022). Some fixed point results for enriched nonexpansive type mappings in Banach spaces. Applied General Topology. 23(1):31-43. https://doi.org/10.4995/agt.2022.16165314323

Demiclosedness Principle for Total Asymptotically Nonexpansive Mappings in C

We prove the demiclosedness principle for a class of mappings which is a generalization of all the forms of nonexpansive, asymptotically nonexpansive, and nearly asymptotically nonexpansive mappings. Moreover, we establish the existence theorem and convergence theorems for modified Ishikawa iterative process in the framework of CAT(0) spaces. Our results generalize, extend, and unify the corresponding results on the topic in the literature