Mathematica Moravica Vol. 19-2 (2015), 97–112 Some Fixed Point Theorems for (CAB)-contractive Mappings and Related Results Arslan Hojat Ansari, Maher Berzig, and Sumit Chandok * Abstract. In this paper, we introduced the concept of (CAB)-contractive mappings and provide sufficient conditions for the existence and unique- ness of a fixed point for such class of generalized nonlinear contractive mappings in metric spaces and several interesting corollaries are de- duced. Also, as application, we obtain some results on coupled fixed points, fixed point on metric spaces endowed with N -transitive binary relation and fixed point for cyclic mappings. The proved results gener- alize and extend various well-known results in the literature. 1. Introduction and Preliminaries Fixed point theory is one of the traditional branch of nonlinear analy- sis. The importance of fixed point theory has been increasing rapidly over the time as this theory provide useful tools for proving the existence and uniqueness of the solutions to various mathematical models (integral and partial differential equations, variational inequalities etc). Also, it has a broad range of application potential in various fields such as engineering, economics, computer science, and many others. It is well known that the contractive-type conditions are very indispens- able in the study of fixed point theory and Banach’s fixed point theorem [1] for contraction mappings is one of the pivotal result in analysis. This theorem that has been extended and generalized by various authors (see, e.g., [2],[7],[8],[9],[15],[17],[28]) and has many applications in mathematics and other related disciplines as well. In [26], Samet and Turinici extended and generalized the Banach contraction principle to spaces endowed with an arbitrary binary relation, and they unified many known results. Recently, there have been so many exciting developments in the field of existence of fixed point in partially ordered metric spaces and fixed point for cyclic map- pings. For more details, we refer the reader to the Bhaskar et al. [6], Berzig Received by the editors 29-4-2015. 2010 Mathematics Subject Classification. Primary: 47H10; Secondary: 54H25. Key words and phrases. fixed point, (CAB)-contraction mappings, binary relations, metric space. ∗ Corresponding author. c 2015 Mathematica Moravica 97