Correspondence of Fixed-Point Theorem in 𝑻𝟐, 𝑻𝟑 − 𝑺𝑷𝑨𝑪�

Abstract

Fixed-point theory (FPT) has lot of applications not only in the field of mathematics but also in various other disciplines. Fixed Point Theorem presents that if 𝑇: 𝑋 → 𝑋 is a contraction mapping on a complete metric space (𝑋, 𝑑) then there exists a unique fixed point in 𝑋. FPT is also essential in game theory, in this case Brower Fixed Point has an application in game theory specifically in non-cooperative games and existence of Equilibrium. In particular, a game is a set of actions done by the participants defined by a set of rules. This is commonly described using mathematical concepts, which offers a concrete model to describe a variety of situations. On the other hand, the separation axioms 𝑇𝑖 , 𝑖 = 0,1,2,3,4 are vital properties that describes the topological spaces 𝑇0 , 𝑇1 , 𝑇2, 𝑇3 and 𝑇4 . It is noted that a 𝑇3 − 𝑠𝑝𝑎𝑐𝑒 is a generalized version of 𝑇2-space and since various results on application of fixed point theory in game theory on an arbitrary locally convex 𝑇2 − space has been established, in this study we sort to extend this concept to the general 𝑇3 − 𝑠𝑝𝑎𝑐𝑒. The utilization of a symmetric property of Hausdorff space established that if two continuous commutative mappings are defined on a 𝑇3 − 𝑠𝑝𝑎𝑐𝑒, then the two maps achieves unique fixed points.