Israeli forces shot and killed two Palestinians on Saturday, saying the two men were militants who had been armed and were approaching the Gaza border.
The military said Israeli forces entered Gaza and fired on them.
A Palestinian health official reported a third man in the area at the time, but it was not clear if he was wounded.
Also on Saturday, a woman opened fire on a police station in southern Israel before officers shot back and killed her, in an apparent Palestinian militant attack.
The shooting took place near the desert town of Beersheba, said a police spokesman.
The woman fired at a paramilitary police post but did not hit anyone before she was gunned down, he said.
Authorities have not yet identified the woman, but police said the shooting appeared to be a militant attack.
If so it would mark the second attack of the week and pose an early test for the new, hard-line government of Prime Minister Benjamin Netanyahu.
Netanyahu has promised a firm hand against Palestinian militants and lowered expectations on prospects for peace.
On Thursday, a day after the prime minister took office, a Palestinian killed an Israeli teenager with a pick-axe and seriously injured a 7-year-old boy in a rampage through a West Bank Jewish settlement.
Sporadic violence along the border with Gaza also threatens to re-ignite clashes after Israel's three-week offensive in the territory earlier this year.
There are fears, however, that occasional attacks could develop into larger clashes between Israel and Palestinian militants.
In the West Bank town of Hebron, Palestinians gathered on Saturday to mark Land Day, a day of protest against the confiscation of their lands by the Israeli government.
Dozens of people gathered, waving flags and chanting slogans calling for an end to Israeli occupation.
During the demonstration protestors scuffled with Israeli security forces who hurled stun grenades among the protesters in order to disperse them.
