Commun. Korean Math. Soc. 27 (2012), No. 4, pp. 709–720 http://dx.doi.org/10.4134/CKMS.2012.27.4.709 COINCIDENCES AND FIXED POINT THEOREMS FOR MAPPINGS SATISFYING CONTRACTIVE CONDITION OF INTEGRAL TYPE ON d-COMPLETE TOPOLOGICAL SPACES Ramesh Chandra Dimri and Amit Singh Abstract. In this paper, we prove some fixed point theorems for some weaker forms of compatibility satisfying a contractive condition of integral type on d-complete Hausdorff topological spaces. Our results extend and generalize some well known previous results. 1. Introduction Branciari [7] obtained a fixed point result for a single mapping satisfying an analogue of Banach’s contraction principle for an integral type inequality. The authors in [3], [4], [6], [22], [28] and [30] proved some fixed point theorems involving more general contractive conditions. Recently ([10]) some fixed point theorems have been proved in non-metric setting wherein the distance function used need not satisfy triangle inequality. The purpose of this paper is to inves- tigate some new result of fixed points in non-metric settings. In the sequel, we use contractive condition of integral type on d-complete Hausdorff topological spaces. Sessa [24] generalized the concept of commuting mappings by calling self- mappings A and S on metric space (X, d) a weakly commuting pair if and only if d(ASx, SAx) ≤ d(Ax, Sx) for all x ∈ X . He and others proved some com- mon fixed point theorems of weakly commuting mappings [24, 25, 26]. Then, Jungck [13] introduced the concept of compatibility and he and others proved some common fixed point theorems using this concept [13, 14, 15, 29]. Clearly, commuting mappings are weakly commuting and weakly commuting mappings are compatible. Examples in [13, 24] show that neither converse is true. Re- cently, Jungck and Rhoades [15] defined the concept of weak compatibility. Received August 19, 2011. 2010 Mathematics Subject Classification. 54H25, 47H10. Key words and phrases. coincidences and fixed points, d-complete topological spaces, contractive conditions of integral type. c 2012 The Korean Mathematical Society 709