Answer: The Rema (YD 286:1) writes that one
who owns a house in partnership with a non-Jew is exempt from affixing a mezuza.
Different reasons are given for this. The Bach (YD 286:1) writes that it could
be dangerous to leave the mezuza exposed. Alternatively, it may become
mistreated by non-Jewish people. The Shach (YD 286:6) explains that the house needs
to belong fully to a Jewish person for it to require a mezuza.
R’ Akiva Eiger (1:66) argues that while a Jew is living in a
property that he owns as a timeshare together with a non-Jew, it is considered
to be totally his. As such, he is required to affix a mezuza then.
The Chayei Adam (Kuntres Hamezuza 289:59-60) sides with R’ Akiva
Eiger, though writes that there is a difference as to whether they have
properly split the times between them. Only if they have predetermined dates,
can the Jewish owner affix the mezuza with a beracha. However, if
their agreement is more fluid, and they each move in whenever they want, then
he affixes it without a beracha. Ideally, one should remove the mezuza
when leaving, and replace it each time they arrive. R’ Moshe Zvi Landau (Mezuzos
Melachim 165) maintains that one does not recite a beracha, regardless.
In conclusion, one should affix mezuzos when one is
staying in a timeshare, though one should not recite a beracha.