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Generalized distance and existence theorems in complete metric spaces. (English) Zbl 0983.54034

Let be the set of positive inteqers and + be the set of all nonnegative reals. Let X be a complete metric space with the metric d. The function p from X×X into 1 is called a τ-distance on X if there exists a function η from X× + into + satisfying the following conditions:

(τ1) p(x,z)p(x,y)+p(y,z) for all x,y,zX;

(τ2) η(x,0)=0 and η(x,t)t for all xX and t + and η is concave and continuous in its second variable;

(τ3) lim n x n =x and lim sup n {η(z n ,p(z n ,x m )):mn}=0 imply p(w,x)lim inf n p(w,x n ) for all wX;

(τ4) lim sup n {p(x n ,y m ):mn}=0 and lim n η(z n ,p(z n ,y n ))=0 imply lim n η(y n ,t n )=0;

(τ5) lim n η(z n ,p(z n ,x n ))=0 and lim n η(z n ,p(z n ,y n ))=0 imply lim n d(x n ,y n )=0.

It is shown that the given concept of τ-distance is a generalization of the concept of w-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 τ-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.


MSC:
54E50Complete metric spaces
49J53Set-valued and variational analysis
54H25Fixed-point and coincidence theorems in topological spaces