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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.

54E50 Complete metric spaces
49J53 Set-valued and variational analysis
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI
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