Orbital Stability and Counterexamples for Nonexpansive-Type Iteration

This paper studies the distinction between contractive inequalities, convergence of powers and the actual orbital behavior of iterative processes in metric fixed point theory. Motivated by the counterexample philosophy of Reich and Zaslavski, we use the orbital quantities \[ D_n(C;p)=x∈ C d(A^n x,p) \] to measure uniform convergence on prescribed sets of initial data. We prove a radial convergence principle, a finite-cover criterion for uniform convergence of nonexpansive powers, quantitative finite-cover bounds, and stability results for inexact Picard iterations. We also revisit standard and elementary counterexamples, emphasizing what they show from the orbital point of view rather than presenting them as new contractions. Further extensions are given for multivalued nonexpansive maps, cyclic best-proximity dynamics and ordered metric spaces. Numerical illustrations show that the quantities Dₙ(C;p) capture algorithmic decay even when no strict contractive power is available.