Fixed Points Theorems in Hausdorff M-distance Spaces

Abstract

We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including metric and uniform spaces. On the other hand, compared to the so-called cone metric spaces and K-metric spaces, we do not require that the distance function range has a linear structure. We also consider several applications of the obtained fixed point theorems. In particular, we consider the questions of the existence of solutions of the Fredholm integral equation in L-spaces.

Fixed sets and fixed points in $\Lim$--spaces]{Fixed sets and fixed points for mappings in generalized $\Lim$--spaces of Fréchet

“…which is consistent with the definition of convergence in almost all previously considered spaces, in particular in usual metric spaces, in partial metric spaces [16], dualistic partial metric spaces [17], dislocated metric spaces [18], metric-like spaces [19], distance spaces [3], metric and distance spaces with more general than R sets of values of a distance function (K-metric spaces, cone metric spaces, M-distance spaces, probabilistic metric spaces, fuzzy metric spaces, and others -see [20] and references therein). Example 2.5.…”