New violence has rocked Israel and the West Bank, including three stabbings and an Arab shot dead by police, as Israeli and Palestinian leaders tried to ease tensions.
In the occupied West Bank, men thought to be undercover Israeli police opened fire on Palestinian stone-throwers in a group they had infiltrated, wounding three of them.
Four masked men suddenly drew pistols and began firing before soldiers rushed to the scene and helped haul away the wounded Palestinians.
One of three shot was seriously wounded in the back of the head.
The Palestinians regularly accuse Israel of placing Arabic-speaking infiltrators among demonstrators.
Separately, in Kiryat Gat in central Israel, police shot dead an Arab man after he allegedly wounded a soldier with a knife and took his weapon, authorities said.
Israeli Prime Minister Benjamin Netanyahu postponed a visit to Germany set for Thursday to tackle the violence that has raged for three weeks despite appeals for calm.
Security officials from the two sides had met Tuesday evening after Abbas said he did not want an escalation in violence.
Israel lifted age restrictions from Wednesday on Muslims praying at east Jerusalem's sensitive Al-Aqsa mosque compound in an apparent bid to ease tensions.
Not far from the mosque, police said an 18-year-old Palestinian woman stabbed a 35-year-old Jewish man in the back and lightly wounded him.
The man, who was armed, managed to draw his gun, shooting and seriously wounding her, police said.
Later on Wednesday, a Jewish man was attacked with a knife outside a shopping centre in Petah Tikva, near Tel Aviv, police said, adding that the attacker had been overpowered.
Jewish settlers in the West Bank shot and seriously wounded an 18-year-old Palestinian near Bethlehem, the Red Crescent and witnesses said.
In the same area, a group of Palestinians had sought to seize a Jewish woman from her car and possibly kidnap her, Israeli military spokesman Arye Shalicar said.
Settlers fired in the air and the woman was able to escape, Shalicar said.
Netanyahu has spoken of "no limits" for the security forces in responding to the unrest, and the military on Tuesday demolished the homes of two Palestinians who carried out attacks last year.
UN Secretary General Ban Ki-moon criticised the home demolitions and urged Israel to fully investigate deadly clashes in Jerusalem and the West Bank.