In this paper, a common fixed point theorem for pair of weakly compatible mappings in the setting digital metric space is proved with supportive example. Our result extends and improves Ozgur Ege et al. [2].

We present new fixed point theorems for inward and weakly inward Urysohn type maps. Also we discuss Mönch Kakutani and contractive type maps.

Recently, Zhang and Song [Q. Zhang, Y. Song, Fixed point theory forgeneralized $varphi$-weak contractions,Appl. Math. Lett. 22(2009) 75-78] proved a common fixed point theorem for two mapssatisfying generalized $varphi$-weak contractions. In this paper, we prove a common fixed point theorem fora family of compatible maps. In fact, a new generalization of Zhangand Song's theorem is given.

In this paper we prove a unique common coupled fixed point theorem for two pairs of $w$-compatible mappings in $S_b$-metric spaces satisfying a contrctive type condition. We furnish an example to support our main theorem. We also give a corollary for Junck type maps.