The Common Fixed Point Theorem For Family Of Mappings Satisfying Expansive And Contractive Conditions On Complex Metric Space

Banach fixed point theorem is the most important fixed point theory, and one of the most basic theoretical tools. Many scholars through various ways to continually extend and improve the Banach fixed point theorem, greatly extend and improve the Banach fixed point theory. Later, many scholars begin to discuss and study the existence and uniqueness of common fixed point of mapping, changing the shape of deflating, considering Quasi contract conditions, considering the set-valued mappings, considering the integral type shrinkage or quasi contract condition, etc are the main ways. Make the common fixed point theory to get a great promotion and development.Some authors by introducing the concept of three kinds of expansion mappings on re-al metric space, and the weakly expand condition or change some conditions, discussed the existence of fixed point problems of expansion mapping, promoted and improved the corre-sponding results in some articles. There are also many results of Banach fixed point theorem, especially in the more general spaces. For example, cone metric space, topological linear space cone valued space, W-space, 2-metric space, D-metric space, etc., also got a lot of good results, well extend and improve the Banach fixed point theorem, thus completing and developing the Banach fixed point theorem.Recently, Azam A through creating a partial order on the complex-valued set, established complex-value measure and complex-valued metric space of non-empty set, and give the co-incidence point theorems and fixed point theorem of two maps satisfying a type of contract conditions. Later, some authors extend and improve the author’s conclusion.In this paper, we use the given expansive or contractive conditions to construct the Cauchy Sequences on complex valued metric space, and use the completeness of spaces or the related conditions to obtain the limits of the sequences. Finally, we prove that the unique limit is the point of coincidence or common fixed points of the given mappings.We mainly discuss the existence problems of unique common fixed points for two map-pings satisfying two expansive conditions respectively on complex valued metric spaces, and provide some fixed point theorems. On the other hand, we also discuss the existence problem of common fixed points for two mappings satisfying A-contractive or A*-contractive conditions on a set having two related complex metrics.