The Ukrainian army has claimed that its soldiers came under artillery fire from the Russian side of the border overnight and were attacked by rebels in several other places in the country's restive east.

In a statement on Friday, the headquarters of the government's military operation in the east listed at least seven locations where rebels attacked Ukrainian troops.

They also claimed that attacks on two locations including a border crossing were supported by artillery fire from Russia.

In response, Russia claimed that around 40 mortar shells fired by Ukrainian forces fell in the province of Rostov near the border with eastern Ukraine.

"Around 40 mortar shells have fallen from Ukraine ... in Rostov province," Vasily Malayev, a representative of the region's Federal Security Service devoted to border security, was cited by the state Ria Novosti news agency as saying.

Moscow has vehemently denied a role in the conflict between pro-Russian separatists and government troops which has left more than 400 people dead and displaced tens of thousands.

As shellings and attacks continue, more Ukrainians are also fleeing the border region to escape the fighting.

According to the latest United Nations report, more than 230,000 Ukrainians have fled, many of them to Russia.

State Department spokeswoman Marie Harf said on Thursday that the US had evidence that Russia is firing artillery from within Russia to attack Ukraine military positions.

Al Jazeera's Rory Challands, reporting from Moscow, said the US is having "a bit of a credibility issue".

"It wasn't willing to reveal the intelligence that pointed it in that direction. The Kremlin can just ignore such accusations, which is what it has done so far."

Army captures Lysychansk

Ukrainian forces are trying to close in on the rebels, cutting them off from the border with Russia which Kiev believes is the source of arms and reinforcement.

Late on Thursday, Ukrainian troops entered the town of Lysychansk, which has been in rebel hands for several months, the military press office said.

Rebels on Friday morning admitted in comments carried by Interfax news agency that they had to flee the town which is 70km northwest of the regional capital Luhansk.

Also on Friday, Ukraine's parliament said it had yet to receive a resignation letter from Prime Minister Arseniy Yatsenyuk so it could not vote on whether to accept it or not, stalling work at the heart of government.

Yatsenyuk, Ukraine's point man for the West during much of the turmoil in the country since November, tendered his resignation on Thursday, saying parliament was betraying its people's demands for change by failing to pass legislation.

The move could hamstring decision-making as Ukraine struggles to fund a war against pro-Russian rebels, and deals with the aftermath of the Malaysian Airlines Flight 17 crash.