2016/04/05 by Mortaza Abtahi, Abtahi, Mortaza
Mathematics · #47H10 #FOS: Mathematics #Fixed Point Theorems Analysis #General Topology (math.GN) #Primary 54H25 #Secondary 54E50
paper · pdf · doi:10.48550/arxiv.1604.01296
openalex publication_date 2016/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a simple and powerful lemma that provides a criterion for sequences in metric spaces to be Cauchy. Using the lemma, it is then easily verified that the Picard iterates \Tnx\, where T is a contraction or asymptotic contraction of Meir-Keeler type, are Cauchy sequences. As an application, new and simple proofs for several known results on the existence of a fixed point for continuous and asymptotically regular self-maps of complete metric spaces satisfying a contractive condition of Meir-Keeler type are derived. These results include the remarkable fixed point theorem of Proinov in [Petko D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 46 (2006) 546--557], the fixed point theorem of Suzuki for asymptotic contractions in [Tomonari Suzuki, A definitive result on asymptotic contractions, J. Math. Anal. Appl. 335 (2007) 707--715], and others. We also prove some new fixed point theorems.