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The purpose of this paper is to present the definition of $$(\phi ,\varphi )$$ -contractive mapping and to discuss the relation of $$\psi$$ -contractive mappings and $$(\phi ,\varphi )$$ -contractive mappings. Furthermore, the generalized $$(\phi ,\varphi )$$ -contraction mapping principle has been proved without the uniqueness condition. Meanwhile, the generalized $$\psi$$ -contraction mapping principle has been obtained by using an ingenious method.

The purpose of this article is to establish a kind of non-convex hybrid iteration algorithms and to prove relevant strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Meanwhile, the main result is applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. It is worth pointing out that a non-convex hybrid iteration algorithm is first presented in this article, a new technique is applied in our process of proof. Finally, an example is given which is a uniformly closed asymptotically family of countable quasi-Lipschitz mappings. The results presented in this article are interesting extensions of some current results.