Remarks on a Recent General Even-Tupled Coincidence Theorem

Abstract

In this note, we present a sharpened form of a recent general even-tupled coincidence theorem due to Imdad et al: (Journal of Operators, Volume 2013 (2013) Art. ID 532867, 8 pp.) and also furnish a proof which also oers consolidation to the proof of the theorem of Imdad et al. which contains some gaps. Unfortunately, certain earlier core results on coupled coincidence points due to Lakshmikantham and Ciric (Nonlinear Anal. 70 (2009), 4341-4349) and quadruple coincidence points due to Karapinar and Berinde (Banach J. Math. Anal. 6 (1) (2012), 74-89) also contain some errors. Besides consolidating the proofs of earlier mentioned results, we also prove a corresponding unique common even tupled fixed point theorem. Finally, we demonstrate that such theorems can not be extended to odd natural numbers besides furnishing an illustrative example for our results.