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 $\mathrm{\partial }D$ 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 the main results, 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.