Perov-Type Contractions

Excerpt

Famous Banach fixed point theorem [13] was generalized in numerous ways by changing a setting, contractive condition, or both. Among large quantity of fixed point theorems, it is important to demand some applications of these results and independence of previously presented theorems. Russian mathematician A.I. Perov [61] in 1964 published a paper, in Russian, dealing with a Cauchy problem for a system of ordinary differential equations. In this paper, he presented a concept of generalized metric space (in a sense of Perov) and gave a proof of a new type of fixed point theorems. From that point of view, we can say that Perov theorem was created as a tool in the area of differential equations and therefore fulfilled the application goal. It was used once again in Perov’s paper in 1966, and then were no significant results on this topic till the 2000s. In the meantime, Polish mathematician S. Czerwik [31] in 1976 published a similar result as a generalization of Edelstein’s fixed point theorem. In 1992, M. Zima [86], who also works in the area of differential equations, published a paper, quoting different work of Czerwik, which gave fixed point result on Banach space that could be related to Perov fixed point theorem. G. Petruşel [65] in 2005 did some research on Perov contractions for multivalued operators that was followed by results for Perov multivalued operators by A. Petruşel and A.D. Filip in 2010 ([35]). This led to several published papers on this topic [6, 34, 39, 40, 78]. N. Jurja [50] proved version of Perov theorem for partially ordered generalized metric space. In 2014, M. Cvetković and V. Rakočević published a generalization of Perov fixed point theorem on cone metric spaces, and this result obtained many extensions such as quasi-contraction, Fisher contraction, θ-contraction, F-contraction, coupled fixed point problem, common fixed point problem, etc. [2, 3, 22‐30, 38, 41, 45, 63, 69, 73]. Many papers were published in the 2010s citing Perov work, adjusting and generalizing that idea for multivalued operators, spaces endowed with a graph, ω-distance, etc., but will not be the main topic of this chapter. We will focus on three different frameworks: metric space, generalized metric space, and cone metric space. Thus, we present some basic definitions and properties. As one of the examples, we will present a system of operatorial equations that transforms into coupled fixed point problem. …