A note on the convergence of Mann Type Iterations on Hadamard Manifolds with Applications

In this paper, we propose a new iterative method inspired by recent developments in inertial techniques. The scheme is constructed by modifying the Mann-type iteration and incorporating inertial terms to improve the speed of convergence. The idea is to use information from both the current and the previous iterates so that the generated sequence approaches the solution more efficiently than the standard Mann process. We establish the strong convergence of the proposed algorithm under mild and natural assumptions on the underlying mappings. The proof relies on suitable auxiliary results and careful analysis of the sequence generated by the method. The assumptions imposed are standard in the literature and do not restrict the applicability of the algorithm to special cases. As a result, the convergence result holds for a broad class of nonlinear problems in Hilbert spaces. To highlight the usefulness of the method, we apply our main theorem to equilibrium problems, variational inclusion problems, and convex minimization problems. These applications show that the proposed iteration provides a unified framework for treating several important models in nonlinear analysis and optimization. In the final section, we include a numerical example to illustrate the convergence behavior of the algorithm and to support the theoretical results. The numerical outcome confirms that the method converges as predicted and demonstrates its practical effectiveness.