Is one allowed to pump air into an inflatable mattress on Shabbos? Is this a problem of tikkun cli, makeh b'patish, etc? (Or any other problem.) What about if the mattress was inflated Erev Shabbos and some air came out and now one wants to add some more air to make it "harder/stronger" on Shabbos. Would this be any different (if the first case is ossur.)
One may not inflate a mattress for the first time on Shabbos, as this would violate makeh b'patish. However, one may inflate or reinflate a mattress on Shabbos as long as it is not the first time the mattress was inflated, since deflating a mattress that is not being used is the standard mode of use. Therefore, the mattress is not considered broken when it is deflated (Shmiras Shabbos K'Hilchasah, 34:24). Note that this does not apply when you seal the air in with a knot, as that would violate kosheir, although you may use a rubber plug or the like to seal the opening (ibid., 16:8).
See also this article that cites opinions variously agreeing and dissenting with this view; some opinions forbid inflating mattresses because "new air" is used each time, which they maintain introduces a makeh b'patish problem any time it is reinflated. R' Shlomo Zalman Auerbach, disagrees with this rationale and writes that all of the air in the world is considered one entity (see footnotes 2,5, and 7 in the article, as well as Shmiras Shabbos K'Hilchasah, 34:24, footnote 95, for citations of various opinions).