JERUSALEM — An informal lull between Israel and Hamas timed for the start of a Muslim holiday collapsed into deadly violence on Monday and Israel’s prime minister signaled no quick end to the three-week-old Gaza war, advising Israelis that “patience and determination” were needed.

Despite intensifying outside pressure for a cease-fire, no diplomatic way out was in sight.

A day of relatively low-level hostilities erupted by late afternoon into a series of bloody events in quick succession. An explosion at Gaza’s Shati refugee camp killed 10 Palestinians, mostly children playing outdoors, and another blast damaged an outside wall of Gaza’s main hospital compound.

Hamas blamed Israel for both strikes but the Israeli military insisted it had not carried out any attacks in those areas and said the blasts were caused by misfired Hamas rockets meant to hit Israel.

Mortar rounds fired from Gaza killed four Israeli soldiers at a staging area on the Israeli side of the border, and Palestinian gunmen emerged from a tunnel running beneath the border and clashed with Israeli soldiers. The military said the gunmen had intended to attack civilians in a nearby Israeli border community. One militant was killed in the clash.