Fixed point theorems for single- and set-valued \(F\)-contractions in \(b\)-metric spaces
This paper is concerned with fixed point theorems for certain mappings on semi-metric and \(b\)-metric spaces. The definition of a semi-metric space is similar to the one of a metric space with the difference that a semi-metric need not satisfy the triangle inequality. \(b\)-metric spaces are somewhat in between semi-metric spaces and metric spaces in the sense that there is a constant \(K\geq 1\) such that the semi-metric \(d\) satisfies the \(K\)-relaxed triangle inequality \[ d(x,z) \leq K(d(x,y)+d(y,z)). \] The main result for set-valued mappings is the following: Let \((X,d,K)\) be a complete \(b\)-metric space and \(T: X\to\mathrm{CB}(X)\), where \(\mathrm{CB}(X)\) denotes the space of non-empty bounded and closed subsets of \(X\) equipped with the Hausdorff-Pompeiu distance. Assume that there is a function \(F: (0,\infty)\to\mathbb{R}\) satisfying the conditions {\parindent=0.8cm\begin{itemize}\item[(F1)] \(y_n\to 0 \Leftrightarrow F(y_n)=-\infty\) for every sequence \((y_n)\) of positive numbers and \item[(F2)] \(\lim\limits_{t\searrow 0} t^kF(t) = 0\) for some \(k\in (0,1/(1+\log_2K))\) \end{itemize}} and a \(\tau\in(0,\infty)\) such that for all \(x,y\in X\), all \(u\in Tx\) there is a \(v\in Ty\) such that either \(v=u\) or \[ \tau + F(d(u,v)) \leq F(d(x,y)) \] then \(T\) has a fixed point. If \(F\) is increasing, upper semi-continuous and satisfies conditions (F1) and (F2) then the above condition on \(T\) can be replaced by \[ \tau + F(H(Tx,Ty)) \leq F(d(x,y)) \qquad\text{if} \quad Tx\neq Ty \] where \(H\) stands for the Hausdorff-Pompeiu distance. For single-valued mappings, fixed point theorems of similar type are presented. In addition, the paper contains a fixed point result for \(F\)-contractions on semi-metric spaces which satisfy the following condition: For every \(\varepsilon>0\) there exists \(\delta>0\) such that \(d(x,y)<\delta\) and \(d(y,z)<\delta\) together imply that \(d(x,y)<\varepsilon\). The main tool in the proofs of these results is a sufficient condition on the growth behaviour of \(d(x_n,x_{n+1})\) which implies that \(\{x_n\}\) is a Cauchy sequence. The necessity of this condition is also discussed.
scientific article; zbMATH DE number 6858727