Georgian Math. J. 20 (2013), 213 – 221 DOI 10.1515 / gmj-2013-0013 © de Gruyter 2013 Fixed points of asymptotically regular mappings in complex-valued metric spaces Mujahid Abbas, Muhammad Arshad and Akbar Azam Abstract. In [Numer. Funct. Anal. Optim. 32 (2011), 243–253], Azam, Fisher, and Khan introduced a notion of complex-valued metric spaces and obtained common fixed point results for mappings satisfying rational inequalities. In the present paper, employing the concept of asymptotically regular mappings, the existence of a fixed point is obtained in a complex-valued metric space. Keywords. Common fixed point, well-posedness, periodic point, complex-valued metric spaces. 2010 Mathematics Subject Classification. 47H10, 54H25. 1 Introduction Fixed point theory is one of the famous and traditional theories in mathematics and has a broad set of applications. In this theory, contraction is one of the main tools to prove the existence and uniqueness of a fixed point. Banach’s contraction prin- ciple, which gives an answer as to the existence and uniqueness of a solution of an operator equation Tx D x, is the most widely used fixed point theorem in all of the analysis. This principle is constructive in nature and is one of the most useful tools in the study of non-linear equations. There are many generalizations of Banach’s contraction mapping principle in the literature [3, 4, 8, 11, 14, 19]. These general- izations were made either by using the contractive condition or by imposing some additional conditions on an ambient space. There have been a number of general- izations of metric spaces such as rectangular metric spaces, pseudometric spaces, fuzzy metric spaces, quasi-metric spaces, quasi-semi-metric spaces, probabilistic metric spaces, D-metric spaces and cone metric spaces. A. Azam, B. Fisher, and M. Khan [2] obtained the generalization of Banach’s contraction principle intro- ducing the concept of a complex-valued metric space. Recently, R. Chugh and M. Aggarwal [5] and L. Ibeni and Y. Rohen [12] discussed some fixed point theo- rems in uniform spaces. The purpose of the present paper is to study the fixed points of a mapping sat- isfying a generalized inequality involving the product of complex numbers in the