On some generalized contraction type mappings

This paper presents some local results of a fixed point theorem proved by Walter [W. Walter, Remarks on a paper by F. Browder, about contractions, Nonlinear Anal. 5 (1981) 21–25]. We also give a negative answer to an open question in connection with this theorem raised by Kirk and Saliga [W.A. Kirk, L.M. Saliga, Some results on existence and approximation in metric fixed point theory, J. Comput. Appl. Math. 113 (2000) 141–152]. At the same time we formulate additional conditions that provide an affirmative answer to this problem.