FIXED POINT THEOREMS FOR CONTRACTIVE MAPPINGS OF PEROV TYPE

[Perov, A. I., On Cauchy problem for a system of ordinary diferential equations, (in Russian), Priblizhen. Metody Reshen. Difer. Uravn., 2 (1964), 115-134] used the concept of a vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article we study fixed point results for new extensions of Park's contraction condition [S. Park,A unified approach to fixed points of contractive maps, J. Korean Math. Soc., 16 (1980), 95-106] to a cone metric space, and give some generalized versions of the fixed point theorem of Perov. As corollaries we generalized some results of [Zima, M., A certain fixed point theorem and its applications to integral-functional equations, Bull. Austral. Math. Soc. 46 (1992), 179-186] and [Borkowski, M., Bugajewski, D. and Zima, M.,On some fixed-point theorems for generalized contractions and their perturbations, J. Math. Anal. App!. 367 (2010), 464-475] for a Banach space with a non-normal cone.