The Existence of Zeros of Multivalued Functionals, Coincidence Points, and Fixed Points in $f$-Quasimetric Spaces

The notion of a $\lambda$-generalized-search multivalued functional on an $f$-quasimetric space is introduced. An existence theorem for zeros of such functionals is proved. As corollaries, theorems on coincidence and fixed points of multivalued mappings of $f$-quasimetric spaces are proved. In particular, Nadler's well-known theorem on fixed points of multivalued contraction mappings is generalized to the case of an $f$-quasimetric space. For a large class of single-valued mappings, including generalized contractions, a theorem on the existence of a (not necessarily unique) fixed point is proved. This theorem extends the existence part of E. S. Zhukovskii's recent fixed-point theorem for generalized contractions, which is a generalization to $f$-quasimetric spaces of Krasnosel'skii's well-known fixed-point theorem and Browder's fixed-point theorem (equivalent to Krasnosel'skii's theorem).