Arbitrary binary relations, contraction mappings and b -metric spaces

UDC 517.9We prove some results on the existence and uniqueness of fixed points defined on a b -metric space endowed with an arbitrary binary relation.  As applications, we obtain some statements on coincidence points involving a pair of mappings.  Our results generalize, extend, modify and unify several well-known results especially those obtained by Alam and Imdad [J. Fixed Point Theory and Appl., 17 , 693–702 (2015); Fixed Point Theory, 18 , 415–432 (2017); Filomat, 31 , 4421–4439 (2017)] and Berzig [J. Fixed Point Theory and Appl., 12 , 221–238 (2012)].  Also, we provide an example to illustrate the suitability of results obtained.