A common property to generalized contractions and existence of fixed points

The definitions of several types of generalized contractions for an application T from a complete metric space (X,d) in itself are recalled and reviewed. A common property [H] to all these concepts is put in light. We observe that assumption [H] is fulfilled in many cases and we prove that assumption [H] and lower semi-continuity of the function x maps to d(x,T(x)) ensure existence of a fixed point along with a sharp estimate for the distance to the fixed-points set.