Quasi-intermediate value theorem and outflanking arc theorem for plane maps

For a disk $ D $ in the plane $ \mathbb R^2 $ and a plane map $ f $, we give several conditions on the restriction of $ f $ to the boundary $ \partial D $ of $ D $ which imply the existence of a fixed point of $ f $ in some specified domain in $ D $. These conditions are similar to those appeared in the intermediate value theorem for maps on the real line. As an application of this result, we establish a fixed point theorem for plane maps having an outflanking arc, which extends the famous theorem due to Brouwer: if $ f $ is an orientation-preserving homeomorphism on the plane and has a periodic point, then it has a fixed point.