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Let (X, d) be a metric space, Y ⊂ X a nonempty closed subset of X and let f : Y→ X be a non self operator. In this paper we study the following problem: under which conditions on f we have all of the following assertions: 1. The operator f has a unique fixed point; 2. The operator f satisfies a retraction-displacement condition; 3. The fixed point problem for f is well posed; 4. The operator f has the Ostrowski property. Some applications and open problems related to these questions are also presented. To the memory of Professor Lj. Ćirić (1935–2016)

Saturated contraction principles for non self operators, generalizations and applications