We present a generalized Ishikawa iterative algorithm with an error term and a variable generalized Ishikawa iterative algorithm. Leveraging the geometric symmetry inherent in uniformly convex Banach spaces, we establish their respective weak convergence theorems for nonexpansive mappings. As applications, we extend several recent results in the literature related to the proximal point algorithm and the split feasibility problem. Consequently, we propose a hyper-generalized proximal point algorithm and a hyper-generalized perturbation CQ algorithm. Our work not only broadens the application scope of these methods but also highlights the foundational role of symmetric space properties in ensuring algorithmic convergence.