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Generalized distance and existence theorems in complete metric spaces. (English) Zbl 0983.54034
Let $$\mathbb{N}$$ be the set of positive inteqers and $$\mathbb{R}_+$$ 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 $$\mathbb{R}_1$$ is called a $$\tau$$-distance on $$X$$ if there exists a function $$\eta$$ from $$X\times \mathbb{R}_+$$ into $$\mathbb{R}_+$$ satisfying the following conditions:
$$(\tau 1)$$ $$p(x,z)\leq p(x,y)+ p(y,z)$$ for all $$x,y,z\in X$$;
$$(\tau 2)$$ $$\eta(x,0)=0$$ and $$\eta(x,t)\geq t$$ for all $$x\in X$$ and $$t\in\mathbb{R}_+$$ and $$\eta$$ is concave and continuous in its second variable;
$$(\tau 3)$$ $$\lim_nx_n=x$$ and $$\limsup_n\{ \eta(z_n,p (z_n, x_m)): m\geq n\}=0$$ imply $$p(w,x)\leq \liminf_n p(w,x_n)$$ for all $$w\in X$$;
$$(\tau 4)$$ $$\limsup_n \{p(x_n,y_m): m\geq n\}=0$$ 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 (topological aspects)
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