A Note on Fixed Points for Asymptotic Nonlinear Contractions

Let (M,d) be a complete metric space and T be a self-mapping of M. W.A. Kirk proved a fixed point theorem for a continuous asymptotic contraction T in(4) .Y.Z. Chen extended Kirk's theorem in(2) by assuming weaker assump- tions on T. Also Chen introduced some other conditions to replace the assumption on the boundedness of the orbit. We introduce the weaker condition liminfn → ∞ (d(x,T n x)) = 0 for some x in M, and prove that this condition implies the exis- tence of a fixed point and the convergence of the Picard iterates to this fixed point.