Applications of uniform asymptotic regularity to fixed point theorems

Abstract

Abstract. In this paper, we show that there are no nontrivial surjective uniformly asymptotically regular mappings acting on a metric space, and we derive some consequences of this fact. In particular, we prove that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings on a bounded τ -compact subset of a Banach space has a common fixed point, and we give a qualitative complement to the Markov-Kakutani theorem.Mathematics Subject Classification. Primary 47H10; Secondary 46B20, 47H09.

Common fixed point theorems for nonexpansive mappings using the lower semicontinuity property

“…The property from the above lemma was called the subsurjectivity in [BW2]. Now we can state Theorem 2.3.…”

Asymptotic regularity, fixed points and successive approximations

“…There exist some simple results in which asymptotic regularity implies the existence of a fixed point, like the following ( [33], [58], [122], [81], [138],...). For some other related results, see also [26], [80], [82], [142]. Since is α DP -condensing, it follows that O (x) is compact.…”

“…If is asymptotically regular at some point of M, then has a unique fixed point. For other references on Problem 2, see [33], [43], [105], [115], [122], [127], [140], [58], [168], [103], [120], [52], [60], [14], [10], [138], [74]- [79], [26],...…”

Strong Convergence Theorems for a Finite Family of Enriched Strictly Pseudocontractive Mappings and ΦT-Enriched Lipschitizian Mappings Using a New Modified Mixed-Type Ishikawa Iteration Scheme with Error

“…Several generalizations of nonexpansive mappings in different directions have been studied by different researchers in the current literature; see, for instance, Refs. [5][6][7][8][9][10][11][12][13] and the references therein. Note that, in particular that, if Φ T is not necessarily nondecreasing and satisfies Φ T (r) < r for r > 0, then T is known as a nonlinear contraction on C.…”