A UN source says Israeli troops have opened fire on Palestinian farmers several times, causing injuries. Israel says its forces fired warning shots.
Palestinian militants have also broken the ceasefire, firing rockets and mortars into Israeli territory.
Meanwhile, Israel has kept the Gaza border closed for a third day.
The authorities have allowed fuel into the Gaza Strip, but blocked all other supplies including humanitarian and commercial goods.
Deal under strain
Despite the breaches, the ceasefire is officially holding, correspondents say.
Hamas, the main Palestinian group in Gaza, says it is still committed to the truce.
But the rocket attacks and mortar fire have come from other Palestinian groups, such as Islamic Jihad and the al-Aqsa Martyrs Brigades, which is affiliated to the Fatah faction of Palestinian Authority President Mahmoud Abbas.
The BBC's Wyre Davis, in Jerusalem, says it is not clear how much control Hamas has over these other groups.
He says gunfire from the Israelis and continued rocket fire from the Palestinian side are putting the ceasefire under strain.
Israel Radio reported that two mortar rounds landed in an open field on the Israeli side of the border on Friday.
Prior to Friday, four rockets had been fired into Israel by Palestinian militants since the truce was agreed on 19 June.
The militants say they are a response to Israeli violations of the truce.
Fuel delivery
Israel has kept border crossings closed for the third consecutive day in response to the rocket attacks.
An estimated 600,000 litres of industrial fuel will be allowed to pass through the Nahal Oz border terminal for Gaza's only power station, a European Union Official told the Reuters news agency. This will keep the plant running for several days.
The EU funds the deliveries of fuel to the power station.
Israeli officials confirmed the fuel delivery but said other border crossings remained closed.
Gaza's Hamas rulers have called on all groups to respect the ceasefire but say they will not be Israel's "police".
Israel says it will keep the crossings into Gaza closed while rocket attacks continue.