In recent years, many researchers have studied the fixed points of generalized $\alpha$-nonexpansive (GAN) mappings, yet there has been limited research on multi-valued GAN mappings. In this paper, we focused on the class of multi-valued GAN mappings in the context of Banach spaces. We introduced a novel iterative process to find fixed points of these mappings and established weak and strong convergence theorems under mild conditions. To demonstrate the numerical efficiency of our new approach, we presented a supportive example and compare the accuracy of our method with existing iteration processes. To show practical usability of the research, we considered a larger class of boundary value problems (BVPs) and proved a convergence result with a supportive example. Our results are new and unify several results of the current literature.