The fixed-point property for simply connected plane continua

We answer a question of R. Ma\'{n}ka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let $M$ be a plane continuum with the property that every simple closed curve in $M$ bounds a disk in $M$. Then every map of $M$ that sends each arc component into itself has a fixed point. Hence every deformation of $M$ has a fixed point. These results are corollaries to the following general theorem. If $M$ is a plane continuum, $\mathcal {D}$ is a decomposition of $M$, and each element of $\mathcal {D}$ is simply connected, then every map of $M$ that sends each element of $\mathcal {D}$ into itself has a fixed point.