Let be the set of positive inteqers and be the set of all nonnegative reals. Let be a complete metric space with the metric . The function from into is called a -distance on if there exists a function from into satisfying the following conditions:
for all ;
and for all and and is concave and continuous in its second variable;
and imply for all ;
and imply ;
and imply .
It is shown that the given concept of -distance is a generalization of the concept of -distance introduced by Kada et al. and in the same time a generalization of the concept of generalized distance introduced by Tataru. The properties of the defined -distance are analyzed and the generalization and improvement of the Banach contraction principle, Caristi’s fixed point theorem, Ekeland’s variational principle and Takahashi’s nonconvex minimalization principle are given.