Remarks on asymptotic regularity and fixed points

Abstract

Asymptotic regularity allows to provide simple proofs of Banach's theorem and Kannan's theorem. Using asymptotic regularity and Kannan's type conditions we generalize these results, in particular, the Banach contraction principle (see Theorem 2.6 and Corollary 2.10). Further, we discuss the analogous results for monotone mappings on preordered metric spaces, where a preordered binary relation is weaker than a partial order. Next, we will prove a random version of the presented deterministic fixed-point theorems.

A remark on asymptotic regularity and fixed point property

“…Then z = t and y n = x n+1 = f x n → z. Using 10 Some generalizations of the above proved theorems are due to Kannan [18], Geobel [10], Reich [29], Jungck [16], Górnicki [12,13] Park and Rhoades [27], Proinov [28], Pant and Pant [23] Pant, [25], Sastry et al [33], Singh et al [34,35] and many others. [31], p.242) on the existence of contractive mappings which admit discontinuity at the common fixed point.…”

Contractive Inequalities for Some Asymptotically Regular Set-Valued Mappings and Their Fixed Points

“…An open problem was posed by Rhoades [15] about the availability of contractive conditions that guarantee the existence of a fixed point but the mapping is not necessarily continuous at that fixed point. In [16], Górnicki considered a special class of mappings R : X → X satisfying the condition…”

A Note on the Górnicki-Proinov Type Contraction

“…(2) belongs to Kannan [2]. By using the "asymptotic regularity" concept, Górnicki [3] proved an extension of Kannan Theorem 1.2. Before giving this interesting result, we recollect the interesting concepts: Let T be a self-mapping on a metric space ðX, dÞ and f T n ug be the Picard iterative sequence, for an initial point u ∈ X.…”

“…In this first generalization, Kannan [2] removed the necessity of the continuity of the contraction mapping. Recently, Górnicki [3] expressed an extension of Kannan type of contraction but the continuity condition was assumed. After then, Bisht [4] refined the result of Górnicki [3] by replacing the continuity condition for the considered mapping with orbitally continuity or p-continuity.…”

“…Recently, Górnicki [3] expressed an extension of Kannan type of contraction but the continuity condition was assumed. After then, Bisht [4] refined the result of Górnicki [3] by replacing the continuity condition for the considered mapping with orbitally continuity or p-continuity. Very recently, Górnicki [5] improved these two mentioned results by introducing new contractions, "Geraghty-Kannan type" and "ϕ-Kannan type."…”