Implicit contractive maps in ordered metric spaces

TL;DR

This paper extends fixed point results for implicit contractions in ordered metric spaces, connecting recent advances with historical approaches and related statements in the field.

Contribution

It provides new extensions to fixed point theorems for implicit contractions, linking recent results with classical and related fixed point statements.

Findings

Extended fixed point results for implicit contractions

Connections established with prior related fixed point theorems

Historical approach from Turinici revisited

Abstract

Further extensions are given to the fixed point result (for implicit contractions) due to Altun and Simsek [Fixed Point Th. Appl., Volume 2010, Article ID 621469]. Some connections with related statements in the area due to Agarwal, El-Gebeily and O'Regan [Appl. Anal., 87 (2008), 109-116] are also discussed. Finally, the old approach in Turinici [An. St. Univ. "A. I. Cuza" Iasi, 22 (1976), 177-180] is presented, for historical reasons.