Fixed Points of Strongly Asymptotically Bounded Mappings

A selfmap T on a metric space (X, d) is said to be asymptotically regular [4] at x in X if d(Tⁿx, Tⁿ⁺¹x) → 0 as n → ∞ and is said to be asymptotically regular on X if it is asymptotically regular for each x in X. T is said to be asymptotically bounded [1] on a Banach space X if ||Tⁿ|| ≥ M for some constant M≥ 0 and for all n≥ 1.