Coincidence and Common Fixed Point Results under Generalized (A, S)f-Contractions

Abstract

Very recently, Shahzad et al. [RACSAM (2017) 111:307324] introduced the notion of (A, S)-contractions which unify several well known nonlinear type contractions (e.g. R-contraction, Z-contraction, L-contraction etc.) in one go. In this paper, we introduce the notion of generalized (A, S)f-contraction and utilize the same to present some coincidence and common fixed point results for a pair of self-mappings (T, f) defined on a metric space endowed with a binary relation S. In the process we ought to introduce some new notions namely: (I, S)-continuity, (I, S)-compatibility and local (T, f)-transitivity. Consequently, several results involving R-contraction and Z-contraction are deduced. Finally, we furnish examples to demonstrate the utility of our main results.