In this manuscript, we first introduce the concept of \((\alpha ,\beta )\) -generalized hybrid mappings within the framework of convex metric spaces. We then prove that if S is an \((\alpha ,\beta )\) -generalized hybrid self-mapping of a nonempty closed convex subset D of a uniformly convex complete metric space \((M,\rho , W)\) , with \(\alpha \in \mathbb {R}\setminus (0,1)\) and \(\beta \in [0,1]\) , then S has at least a fixed point if and only if there exists some z in D such that the sequence \(\{S^{n}(z)\}_{n=1}^{\infty }\) is bounded. In particular, if D is bounded, then the fixed points set of S is nonempty, closed, and convex. Finally, under appropriate conditions on control sequences of the Ishikawa iteration process, S and its domain, we verify the convergence of the Ishikawa iteration process to a fixed point of S. Furthermore, the convergence of the Mann iteration process to a fixed point of S is investigated. The results given in this paper extend and improve some results in the recent literature. Also, they are usable in the more specific spaces, such as Hilbert spaces, uniformly convex Banach spaces and CAT(0) spaces.
