Perov's theorem is one of many different extensions of the famous Banach fixed point theorem, one which is notable for its wide spectra of applications. It claims existence and uniqueness of a fixed point for a new class of contractive mappings. Russian mathematician A. I. Perov introduced a concept of generalized metric with values in RnRn and defined a contractive condition that includes a matrix with nonnegative elements instead of a contractive constant.
We give extensions of this result in various settings and include different type of contractions. A contraction of Perov type on a cone metric space involves an operator instead of a matrix. Other extensions of Perov's theorem are inspired by quasi-contraction along with some results for common and coupled fixed point problems. We will discuss applications of the presented results in solving differential equations and Ulam-Hyers stability of functional equations.
(This is joint work with Vladimir Rako\v cevi\'c)