Fixed point theorems for a ph contractive mappings of rational type in complex valued metric spaces with applications
A very interesting approach in the theory of contractive mappings was recently given by Samet et al. (Nonlinear Anal. 75, 2012, 2154 2165) by using the concept of a ps contractive type mappings in metric spaces. The purpose of this paper is to give the generalized version of a ps contractive type mappings in complex valued metric spaces. For this, we define the notion of (a, ph) contractive mappings of rational type and obtain fixed point theorems for such mappings in the framework of complex valued metric spaces. This extends and subsumes the recent results of Azam et al. (Numerical Functional Analysis and optimization 32, 2011, 243 253) and Rouzkard and Imdad (Computers and Mathematics with Applications 2012, doi:10.1016/j.camwa.2012.02.063) regarding common fixed point theorems in complex valued metric spaces. Also, we derive common coupled fixed point theorems in complete complex valued metric spaces. Our results generalize many previously obtained results. Some examples are presented to verify the effectiveness and applicability of our main results. Towards the end, an application to integral equations has also been presented to support the usability of the obtained results.