Common fixed point theorems under (R,S)-contractivity conditions

Abstract

Very recently, Roldan-Lopez-de-Hierro and Shahzad introduced the notion of R-contractions as an extension of several notions given by different researchers (for instance, R-contractions generalize Meir-Keeler contractions, Z-contractions - involving simulation functions - by Khojasteh et al., manageable contractions by Du and Khojasteh, Geraghty’s contractions, Banach contractions, etc.). In this manuscript, we use R-functions to present existence and uniqueness coincidence (and common fixed) point results under a contractivity condition that extend some celebrated contractive mappings. In our main theorems, we employ a binary relation on the metric space, which does not have to be a partial order. Finally, we illustrate our technique with an example in which other previous statements cannot be applied: in fact, we show how to apply our main results to a new kind of contractivity conditions which cannot be expressed in separate variables.