• COMMON FIXED POINTS IN COMPLEX-VALUED b-METRIC SPACES SATISFYING A SET OF RATIONAL NEQUALITIES

Abstract

In 1989, Bakhtin introduced the notion of b-metric space (I. A. Bakhtin, The contraction principal in quasi-metric spaces, Functional Analysis 30(1989), 26-37) as a generalization of metric space in which the triangle inequality was relaxed. Further, in 1993, Czerwik first proved a contraction mapping theorem for this space (S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1(1993), 5-11) which generalized the well known Banach contraction mapping principal. In 2011, Azam et al. introduced the notion of complex-valued metric space (A. Azam, B. Fisher and M. S. Khan, Common fixed point theorems in complex-valued metric spaces, Numerical Functional Analysis & Optimization 32(3)(2011), 243-253) to obtained a common fixed point result for a pair of self-mappings satisfying a rational inequality. Meanwhile, Jungck relaxed the concept of commutativity of a pair of mappings by compatibility [9], and further by weakly compatibility [10]. In this paper, we will prove some common fixed point theorems in complex-valued b-metric spaces for two pairs of self-mappings satisfying a set of rational inequalities using the weakly compatible mappings. Our result generalizes many results in the existing literature.