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Generalized distance and existence theorems in complete metric spaces. (English) Zbl 0983.54034
Let $\bbfN$ be the set of positive inteqers and $\bbfR_+$ be the set of all nonnegative reals. Let $X$ be a complete metric space with the metric $d$. The function $p$ from $X\times X$ into $\bbfR_1$ is called a $\tau$-distance on $X$ if there exists a function $\eta$ from $X\times \bbfR_+$ into $\bbfR_+$ satisfying the following conditions: $(\tau 1)$ $p(x,z)\le p(x,y)+ p(y,z)$ for all $x,y,z\in X$; $(\tau 2)$ $\eta(x,0)=0$ and $\eta(x,t)\ge t$ for all $x\in X$ and $t\in\bbfR_+$ and $\eta$ is concave and continuous in its second variable; $(\tau 3)$ $\lim_nx_n=x$ and $\limsup_n\{ \eta(z_n,p (z_n, x_m)): m\ge n\}=0$ imply $p(w,x)\le \liminf_n p(w,x_n)$ for all $w\in X$; $(\tau 4)$ $\limsup_n \{p(x_n,y_m): m\ge n\}=0$ and $\lim_n\eta (z_n,p(z_n,y_n)) =0$ imply $\lim_n \eta(y_n,t_n) =0$; $(\tau 5)$ $\lim_n\eta (z_n,p(z_n,x_n))=0$ and $\lim_n\eta(z_n,p(z_n,y_n))=0$ imply $\lim_n d(x_n,y_n) =0$. It is shown that the given concept of $\tau$-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 $\tau$-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:
 54E50 Complete metric spaces 49J53 Set-valued and variational analysis 54H25 Fixed-point and coincidence theorems in topological spaces
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##### References:
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