We originate some existence results on periodic and fixed point using generalized $(\psi\circ\varphi)$-contractions and generalized $(\psi\circ\varphi)$-contractive mappings with weaker Meir-Keeler function in the setup of a complete generalized metric space in sense of Branciari without Hausdor assumption. Our results generalize the results of several well-known comparable results in the literature. To illustrate our results, we conclude the paper with some examples.