They were the first direct clashes between Israeli soldiers and Hamas militants since the devastating Gaza war in 2014, and come as a new tunnel reaching into the Jewish state was discovered.
Since the confrontations broke out, Hamas and other militant groups have fired bullets and mortar rounds on at least 10 occasions, and the Israeli air force has carried out at least three bombing raids.
Israeli tanks stationed on the border have also fired multiple times at what the army said were Hamas targets.
A 21-year-old was also wounded in artillery shelling in the area.
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The shelling came after a night in which air raids wounded four people, three of them children, medical and security officials said.
The Palestinian attacks were targeting Israeli forces working near the border fence searching for tunnels crossing under the border, like those whose destruction it cited as one of the main achievements of the 2014 conflict.
The army announced a new tunnel was found on Thursday, after a first one was discovered in mid-April.
Hamas's armed wing, the Ezzedine al-Qassam Brigades, accused Israeli forces of encroaching into Gazan territory and said it was ready to respond.
Israeli army spokesman Lieutenant Colonel Peter Lerner said the new tunnel discovered Thursday had been about 30 metres deep.
Lerner did not say how far into Israel the tunnel stretched or when it was built, but confirmed the discovery was made by Israeli forces on the Gazan side of the border.
"We see Hamas are continuing to try and build this infrastructure into Israel and it is something we are not prepared to tolerate," said Lerner.
Israeli media have reported a major technological advance in Israel's ability to detect Hamas tunnels, with Prime Minister Benjamin Netanyahu last month hailing the "breakthrough in the ability to locate tunnels".
The army has kept the nature of the alleged technology secret and Lerner refused to elaborate.