Weakly Picard mappings: Retraction-displacement condition, quasicontraction notion and weakly Picard admissible perturbation

Let (X, d) be a metric space, f : X → X be a mapping and G(·, f (·)) be an admissible perturbation of f. In this paper we study the following problems: In which conditions imposed on f and G we have the following: (DDE) data dependence estimate for the mapping f perturbation; (UH) Ulam-Hyers stability for the equation, x = f (x); (WP) well-posedness of the fixed-point problem for f; (OP) Ostrowski property of the mapping f. Some research directions are suggested. Mathematics Subject Classification (2010): 47H25, 54H25, 47H09, 65J15, 37N30, 39A30.