A Fixed-Point Theorem for Mappings Satisfying a Contractive Condition Involving Tρ

We prove that a self-mapping T on a complete metric space (X,D), satisfying a generalized contractive condition involving its ρ-th iterate Tρ (ρ∈N,ρ≥2) and a uniform coefficient bound L<1, possesses a unique fixed point. The proof establishes that the Picard iteration sequence is Cauchy with the further assumption of continuity of T, and fixed point uniqueness follows directly from the contractive inequality.