Marcin Borkowski**Fixed point theorems with a boundary condition**

In 1998, Espínola and López proved a certain fixed point theorem for a mapping transforming an admissible subset of a hyperconvex metric space into the whole space, assuming that this mapping sends the boundary of into . We will show how to simplify their proof considerably, at the same time relaxing its assumptions. In particular, this will enable proving a new fixed point theorem, generalizing a classical result of Baillon from 1988.