The upsurge of violence around Gaza follows a familiar pattern: militants fire rockets into Israel which brings air strikes, or vice versa.
The death toll has pushed past two dozen on the Palestinian side, with many more wounded, most of them civilians.
The Israeli military said its first air strike on Friday killed Zuheir al-Qessi, who Palestinians said headed the Popular Resistance Committees, armed activists from different factions. Israel named a second militant leader killed as Mahmoud al-Hanani.
Israel released video purporting to be of what it calls preventive targeting to kill militants.
Nightfall brought separate Israeli air strikes against the coastal territory ruled by Hamas.
When rockets were fired for a fourth day in a row on Monday, Israel said its short-range air defence interception system called “Iron Dome” is protecting its southern cities, downing rockets with a 75 percent success rate.
With the storm of violence came a storm of funerals in Gaza.
Most of the dead are said to have been militants.
Gaza’s Hamas rulers and Israel both seem to want to avoid the kind of all-out war that erupted three years ago.
But Hamas can not restrain smaller Gaza factions from attacking Israel.
Prime Minister Benjamin Netanyahu said: “My policy and instructions are clear: we will strike anybody that plans to strike us, tries to strike us or actually does hit us. The Israeli Army is dealing these terror organisations very heavy blows.”
Islamic Jihad, the faction behind most of the rocket fire, said there will be no ceasefire.
The group said: “We will not take part in any calm with the enemy, who impose their unfair rules. We will not allow a calm that disregards our people’s blood.”
Calls by Hamas asking the rest of the Palestinian factions to stop their rocket fire have gone unheeded.
Senior Hamas official Mahmoud Al-Zahar in Cairo said he expects matters will calm down.
He said: “Hamas has not taken any decision to escalate. It is trying to reach a truce, conditioned on the Israeli enemy halting aggression.”
Egyptian efforts to broker a ceasefire appeared to be stuck over a demand by Islamic Jihad that Israel first promise not to target militant leaders for future attack.