Remarks on contractive conditions of integral type

We show that most contractive conditions of integral type given recently by many authors coincide with classical ones. This is done with the help of a geometric lemma on subsets of the quadrant [0,∞)2[0,∞)2. In particular, we extend a recent result of Suzuki who observed that an integral version of the Banach contraction principle given by Branciari could be derived from a classical fixed point theorem of Meir and Keeler. Nevertheless, we also give a new contractive condition of integral type which is independent of classical ones since it does not force the nonexpansiveness of a mapping.