International Journal of Advanced and Applied Sciences, 4(2) 2017, Pages: 35-37 Contents lists available at Science-Gate International Journal of Advanced and Applied Sciences Journal homepage: http://www.science-gate.com/IJAAS.html 35 On the generalized  ∗ - valued metric spaces related with Banach fixed point theory Özen Özer 1, *, Saleh Omran 2, 3 1 Department of Mathematics, Faculty of Science and Arts, Kırklareli University 39100, Kirklareli, Turkey 2 Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia 3 Department of Mathematics, South Valley University, Quena, Egypt ARTICLE INFO ABSTRACT Article history: Received 24 November 2016 Received in revised form 27 January 2017 Accepted 28 January 2017 The Banach contraction principle, which shows that every contractive mapping has a unique fixed point in a complete metric space, has been extended in many directions. One of the branches of this theory is devoted to the study of fixed points. Especially, Fixed point theory in C ∗ - algebra valued metric spaces has greatly developed in recent times. Also, we study on generalized C ∗ - algebra valued metric space and give some examples, the idea of this metric is to replace the set of real numbers by the positive cone C ∗ - algebras, the set of positive elements on the C ∗ - algebras the notation introduced recently. Also, we prove certain fixed-point theorem for a single- valued mapping in such spaces. The mapping we consider here is assumed to satisfy certain D-metric conditions with generalized fixed-point theorem. Moreover, the paper provides an application to prove the existence and uniqueness of fixed points. Keywords: Fixed point theory  ∗ - algebra Positive cone © 2017 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction * The study of fixed points of mappings satisfying certain contractive conditions has got big importance. The notion of D-metric space is a generalization of usual metric spaces. Dhage (1992) introduced the notion of generalized metric and claimed that D-metric convergence define a Hausdorff topology and that D -metric is sequentially continuous in all the three variables (Dhage, 1992; 1999). Then, many of authors generalized Dhages contractive proved the existence of unique fixed point of a self map in generalized metric. El-Sayed et al. (2014) introduced the concept of quaternion metric spaces which generalizes both real and complex metric spaces and proved the fixed point theorem in normal cone metric spaces for four self-maps satisfying a general contraction condition. Huang and Zhang (2007) reviewed cone metric spaces. Rezapour and Hamlbarani (2008) obtained generalizations of the results for metric spaces and fixed point theorems of contractive mappings by providing non-normal cones and omitting the assumption of normality. Ma and Jiang (2015) * Corresponding Author. Email Address: ozenozer39@gmail.com (Ö. Özer) https://doi.org/10.21833/ijaas.2017.02.006 2313-626X/© 2017 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) established the notion of C ∗ - algebra valued metric spaces, and proved some fixed point theorems for contractive and expansive mappings. Ma and Jiang (2015) introduced a concept of C ∗ - algebra-valued b metric spaces which generalizes the concept of C ∗ - algebra valued metric spaces. They also proved the existence and uniqueness results for a type of operator equation and an integral equation were given. For more details and basic definitions of the fixed point theory and C ∗ - algebra we refer (Blackadar, 1986; Davidson, 1996; Gelfand and Neumark, 1943; Murphy, 1990; Pedersen, 1979). Besides, Özer and Omran (2016) have demonestrated the existence and uniqwness of the common fixed point theorem for self maps in C ∗ - algebra-valued b metric space. This work is motivated by the recent works on generalized metric spaces and C ∗ - algebra. In this paper, we introduce the notion of generalized metric space in the C ∗ - algebra valued metric space. Our results can be used to investigate a large class of nonlinear problems. As an application, we discuss the existence and uniqueness for fixed point. Definition 1.1: Let X denote a non empty set and the set off all non negative real numbers. Then X together with the function D:X×X×X→ ℝ + is called a D –metric space if satisfies the following properties: (i) D(x, y, z) = 0 ⟺ x = y = z