After pressure from their backers, Russia and Turkey, the ceasefire officially started on Sunday.
But both the UN-backed government and forces loyal to Russian ally Gen Khalifar Haftar say there has been fighting around the capital, Tripoli.
Turkey last week sent troops to help Government of National Accord forces.
Gen Haftar is also backed by Egypt, Saudi Arabia, the United Arab Emirates (UAE), and Jordan, raising fears that oil-rich Libya could become the theatre of a regional conflict, or even a “second Syria”.
Amid the chaos, both Islamist militant groups and migrant smugglers have become well-established, causing particular concern in European countries just across the Mediterranean Sea.
Libya has been wracked by conflict since the 2011 uprising which ousted long-time strongman Muammar Gaddafi.
The Libyan National Army (LNA) forces loyal to Gen Haftar control most of eastern Libya. They launched an offensive on the capital in April 2019 but have been unable to take the city. Last week, however, they did take the country’s third-biggest city, Sirte.
Turkish President Recep Tayyip Erdogan and his Russian counterpart Vladimir Putin called for a ceasefire when they met in Istanbul last week.
What are the two sides saying?
LNA commander al-Mabrouk al-Gazawi said GNA “militias have breached the truce in several areas with all kinds of weapons”.
The GNA said it had “documented breaches by Haftar’s militias on the battlefronts of Salah al-Din and Wadi al-Rabie” around Tripoli.
Reuters news agency reports that from early on Sunday morning, exchanges of fire could be heard.
The LNA initially rejected the calls for a ceasefire before announcing late on Saturday that it accepted a truce in GNA-controlled western Libya “provided that the other party abides by the ceasefire”.
A statement from the GNA, led by Fayez al-Sarraj, later also announced a ceasefire, starting at 00:00 on Sunday (22:00 GMT on Saturday).
It says it remains committed to the truce and urged “the sponsors of the ceasefire and the UN mission in Libya to ensure its implementation and not to take lightly any breaches”.