International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 4, Issue 1, January 2016, PP 97-112 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org ©ARC Page | 97 Fixed Point Theorems for Non-compatible, Discontinuous Hybrid Pairs of Mappings on 2-Metric Spaces by Using Implicit Relation R. D. Daheriya 1* , Motiram Likhitker 1 , Manoj Ughade 2 1 Department of Mathematics, Government J. H. P.G. College, Betul, P.O. Box 460225, India 2 Department of Mathematics, SRK University, Bhopal, P.O. Box 462026, India * Correspondingauthor, e-mail: mrlikhitker950@gmail.com Abstract: In this article, we prove a number of common fixed point theorems for hybrid pairs of mappings satisfying an implicit contraction relation by using weak commutativity of type (KB) in the setting of a 2-metric space. Also, we present an example to illustrate the effectiveness of our results. Keywords: Coincidence point, Common fixed point, D-maps, Weak commutativity of type (KB), implicit contraction relation. 2010 Mathematics Subject Classification: 47H10, 54H25. 1. DEFINITION AND NOTATION The concept of a 2-metric space is a natural generalization of a metric space. It has been introduced by ([3]-[5]) and extensively studied by some mathematicians such as ([3]-[5]), White [18], [6]. Moreover, a number of authors ([1], [10], [13], [17]) have studied the contractive, non- expansive and contraction type mapping in 2-metric spaces. On the other hand, Jungck [7] studied the common fixed points of commuting maps. Then Sessa [16] generalized the commuting maps by introducing the notion of weakly commuting and proved a common fixed point theorem for weakly commuting maps. Jungck [8] further made a generalization of weakly commuting maps by introducing the notion of compatible mappings. Moreover, Jungck and Rhoades [9] introduced the notion of coincidentally commuting or weakly compatible mappings. Several authors used these concepts to prove some common fixed point theorems on usual metric, as well as on different kinds of generalized metric spaces ([1], [2], [11], [15]). In this paper, the existence and approximation of a unique common fixed point of two families of weakly compatible self maps on a 2-metric space are proved. Pant ([20]-[23]) initiated the study of non-compatible maps and introduced pointwise R-weak commutativity of mappings in [20]. He also showed that point wise R-weak commutativity is a necessary, hence minimal, condition for the existence of a common fixed point of contractive type maps [21]. Pathak et al. [24] introduced the concept of R-weakly commuting maps of type (A), and showed that they are not compatible. Kubiaczyk and Deshpande [19] extended the concept of R- weakly commutativity of type (A) for single valued mappings to set valued mappings and introduced weak commutativity of type (KB) which is a weaker condition than -compatibility. In fact, - compatibility maps are weakly commuting of type (KB) but converse is not true. For example we can see [19], [25 and [26]. Recently, Sharma and Deshpande [25] proved a common fixed point theorem for two pairs of hybrid mappings by using weak commutativity of type (KB) on a non-complete metric space without assuming continuity of any mapping. In this paper, we present a number of common fixed point theorems for hybrid pairs of mappings satisfying an implicit contraction relation in the setting of a 2-metric space by using weak commuting of type (KB). In Section 2.4, we give an example to illustrate the effectiveness of our results. 2. Preliminaries Throughout this paper, we will adopt the following notations: is the set of all natural numbers, is the set of all non-negative real numbers. For mappings , we denote