n past week, Kenya has conducted air strikes in Somalia against al Shabaab. Israel has undertaken airstrikes in Syria against Syrian military targets in response to a cross-border attack that killed an Israeli teenager.  And the Syrian air force reportedly has carried out air strikes in western Iraq against ISIS.  Each of these actions seems to have been taken in reliance on a theory of self-defense, and at least the first two strikes took place without the consent of the host state.  (It is not clear whether Iraq consented to Syrian air strikes against ISIS.)  But in none of these cases did the state using force submit an Article 51 letter to the U.N. Security Council.  Nor does it appear that these states provided a less formal form of notification to the UNSC (though if readers know otherwise, I'd welcome correction).
Article 51 states, in relevant part, “Nothing in the present Charter shall impair the inherent right of individual or collective self-defence if an armed attack occurs against a Member of the United Nations, until the Security Council has taken measures necessary to maintain international peace and security.  Measures taken by Members in the exercise of this right of self-defence shall be immediately reported to the Security Council.”
Ashley S. Deeks, A Call for Article 51 Letters, Lawfare (June 25, 2014).