The Generalizations Of Two Fixed Point Theorems Of The Plane

In this paper, we give two plane fixed point theorems which are the generalizations of the Cartwright-Littlewood’s classical theorem and Guillou’s result. In the second result, we use the tool of prime ends to analysis the orientation preserving continuous map on them.Here are the precise statements:Theorem A Let f be an orientation preserving continuous map of R2, and let A be a non separating continuum in R2. If f (Λ) c A, then f admits a fixed point in A.Theorem B Let f be an orientation preserving continuous map of R2, and let M be a closed, compactly connected, non separating subset of R2without interior. If f(M)=M, then f admits a fixed point in M.