The purpose of this paper is to establish some fixed point results on the complete and orbitally complete metric spaces for Ćirić type contractive mappings in the context of weakly orbital continuity. Also, the concept of asymptotically regular mapping is used for the existence of fixed points. However, our results generalize several well known results in the literature. Additionally, we have presented our theorems through some non-trivial examples, and an application to integral equations has also provided.