$E$-Compactness and Frum-Ketkov Operators in $E$-Metric Spaces

This paper studies Frum-Ketkov operators in $E$-metric spaces, that is, operators whose contractive condition involves the distance from a point to a prescribed compact subset rather than the distance between two points. Two properties that hold in the real-valued case fail in this setting. An $E$-compact set need not be $E$-bounded when $E$ has no order unit, and the infimum defining the distance from a point to a set, being taken in the lattice order, need not be attained. The notion of $E$-compactness is introduced and its relations to $E$-closedness and $E$-boundedness are established. It is then shown that an $E$-compact subset need not be proximinal, and sufficient conditions for proximinality are obtained. Under the proximinality assumption, the fixed-point set and the $\omega$-limit sets of a Frum-Ketkov operator are located in the prescribed compact set, the invariance of this set is proved, and contractive Frum-Ketkov operators are shown to be Picard operators. For $E=\mathbb{R}$, these results reduce to the corresponding results in metric spaces.