On characterizations of \(\leq\)-completeness and metric completeness
In [Trans. Am. Math. Soc. 226, 257--290 (1977; Zbl 0365.54023)], \textit{B. E. Rhoades} has shown that every complete metric space \((X,d)\) has the following property: (FP1) Every continuous mapping \(f: X\to X\) for which there exists a \(c\in (0,1)\) such that \(d(f(x),f(y))\leq c\max\{d(x,y), d(x,f(x))\), \(d(y, f(y),{1\over 2} [d(y, f(x))+ d(x, f(y))]\}\) for each pair \(x,y\in X\) has a fixed point. Theorem 1. A metric space is complete if and only if it has property (FP1). Recall that for a subset \(A\) of a metric space \((X,d)\) the measure \(\alpha(A)\) of non-compactness of \(A\) is the infimum of all \(\varepsilon> 0\) such that \(A\) admits a finite covering by sets with diameters less than \(\varepsilon\). By a theorem of \textit{W. A. Kirk} [J. Lond. Math. Soc. 44, 107--111 (1969; Zbl 0162.54902)] every complete metric space \((X,d)\) has the following property: (FP2) For every closed bounded subset \(A\) of \(X\) with \(\alpha(A)= 0\) every continuous mapping \(f: A\to A\) with diminishing orbital diameters has a fixed point in \(A\). Theorem 2. A metric space is complete if and only if it has property (FP2). Following \textit{M. Turinici} [Math. Jap. 25, 511--517 (1980; Zbl 0452.54025)] a metric space \((X,d)\) with an order \(\leq\) is called \(\leq\)-complete if every \(\leq\)-monotone Cauchy sequence in \(X\) is convergent, where a sequence \((x_n)_{n<\omega}\) is called \(\leq\)-monotone if \(x_m\leq x_n\) whenever \(m\leq n\). The following property is a natural generalization of a classical characterization of metric completeness due to G. Cantor: (C) Whenever \((x_n)_{n<\omega}\) is a \(\leq\)-monotone sequence in \(X\) and \((A_n)_{n<\omega}\) is a sequence of closed subsets of \(X\) such that \(x_n\in A_n\supset A_{n+1}\) for each \(n<\omega\) and \(\lim\delta(A_n)_{n<\omega}= 0\), then \(\bigcap\{A_n\mid n<\omega\}\). Theorem 3. A metric space \((X,d)\) with order \(\leq\) is \(\leq\)-complete if and only if it has property (C). A chain \(C\) in a metric space \((X,d)\) with order \(\leq\) is said to be a Cauchy chain if for each \(\varepsilon> 0\) there exists an \(x\in C\) such that \(d(y,z)< \varepsilon\) for all \(y,z\in X\) with \(x\leq y\) and \(x\leq z\). A chain \(C\) converges to a point \(p\in X\), if for each \(\varepsilon> 0\) there exists an \(x\in C\) such that \(d(y,p)< \varepsilon\) for all \(y\in X\) such that \(x\leq y\). Theorem 4. A metric space \((X,d)\) with order \(\leq\) is \(\leq\)-complete if and only if every Cauchy chain in \(X\) converges. This result generalizes a similar theorem of \textit{V. Conserva} and \textit{S. Ricco} [Math. Jap. 38, No. 5, 901--903 (1993; Zbl 0811.54027)].
scientific article; zbMATH DE number 2101799
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