New fixed point theorems in metric spaces with applications to iterative learning and artificial intelligence

Abstract

 Fixed point theory provides a fundamental framework for understanding the stability of iterative processes. However, many learning algorithms in modern artificial intelligence involve update rules that are not contractive in a single step, alternate between multiple operators, or vary across iterations, placing them outside the scope of classical results. In this paper, we present three new fixed point theorems in complete metric spaces that address these limitations. The first theorem establishes convergence when an operator becomes contractive only after several compositions. The second theorem proves convergence of alternating two-phase update schemes whenever their composite mapping is contractive. The third theorem provides a convergence criterion for time-varying iterative systems under blockwise contractive behaviour. All results are proved purely in metric spaces without assuming linear structure. Detailed examples and applications to multi-step optimization, GAN training,  actor--critic methods, adaptive algorithms, and curriculum learning  demonstrate the broad relevance of the proposed theorems.