%0 Journal Article
%T Weakly Compatible Maps Using E.A. and (CLR) Properties in Complex Valued -Metric Spaces
%A Balbir Singh
%A Vishal Gupta
%A Sanjay Kumar
%J Journal of Complex Systems
%D 2013
%R 10.1155/2013/939607
%X We introduce the notion of complex valued -metric spaces and prove common fixed point theorems for weakly compatible maps along with E.A. and (CLR) properties in complex valued -metric spaces. 1. Introduction The study of fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Recently, Mustafa and Sims [1, 2] have shown that most of the results concerning Dhage¡¯s -metric spaces are invalid; therefore, they introduced an improved version of the generalized metric space structure and called it -metric spaces. In 2006, Mustafa and Sims [2] introduced the concept of -metric spaces as follows. Definition 1. Let be a nonempty set, and let be a function satisfying the following properties:(G1) if ;(G2) for all , with ;(G3) for all with ;(G4) = = (symmetry in all three variables);(G5) for all (rectangle inequality); Then the function is called a generalized metric or, more specially a -metric on , and the pair ( ) is called a -metric space. The idea of complex metric space was initiated by Azam et al. [3] to exploite the idea of complex valued normed spaces and complex valued Hilbert spaces. Definition 2. Let be the set of complex numbers and , . Define a partial order on as follows: That is, if one of the following holds:(C1) Re( ) Re( ) and Im( ) Im( );(C2) Re( ) Re( ) and Im( ) Im( );(C3) Re( ) Re( ) and Im( ) Im( );(C4) Re( ) Re( ) and Im( ) Im( ). In particular, we will write that if and one of (C2), (C3), and (C4) is satisfied, and we will write if only (C4) is satisfied. Remark 3. We noted that the following statements hold:(i) and for all ;(ii) ;(iii) and . Now we introduce the notion of complex valued -metric space akin to the notion of complex valued metric spaces [3] as follows. Definition 4. Let be a non-empty set. Let be a function satisfying the following properties:(CG1) if ;(CG2) for all with ;(CG3) for all with ;(CG4) (symmetry in all three variables);(CG5) for all . Then the function is called a complex valued generalized metric or more specially, a complex valued -metric on , and the pair is called a complex valued -metric space. 2. The Complex Valued -Metric Topology A point is called interior point of a set , whenever there exists such that A point is called limit point of a set , whenever there exists : The set is called open, whenever each element of is an interior point of . A subset is called closed, whenever each limit point of belongs to . Proposition 5. Let ( ) be complex valued -metric space, then for any and 0, one has the following:(1)if , then ; (2)if , £¿then there
%U http://www.hindawi.com/journals/jcs/2013/939607/