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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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