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Ukraine optimism amid fierce Donetsk fighting

The difficult future that lies ahead for Ukraine was laid bare Tuesday following the results of the relatively smooth presidential elections, as Russian stocks tumbled and fierce fighting took place in the east.

Confectionery magnate Petro Poroshenko has claimed victory in Ukraine's presidential elections after exit polls on Sunday gave him 55 percent of the vote. Meanwhile, Europe's Energy Commissioner signaled that Ukraine and Russia have made further progress in an ongoing dispute over gas prices.

Brendan Hoffman | Getty Images

The two events look to be putting the country back on track following bloody protests, the ousting of former president Viktor Yanukovych and the annexation of Crimea by Russia. However, fierce battles continue to rage in the eastern part of the country.

Read MoreUkraine forces oust rebels from airport in east

A military offensive against pro-Russian rebels was launched on Monday at the strategically important location of Donetsk airport. Ukrainian forces launched air strikes and a paratrooper assault against the separatists, according to news agency Reuters. The separatists had seized the airport in response to the change of power at Kiev back in February.

Gun battles continued on Tuesday with Reuters reporting that more than 50 rebel forces' lives may have been lost during the fighting. In the city center a fire at a four-storey hockey stadium had to be controlled by local firefighters. By midday local time, Interior Minister Arsen Avakov told reporters in Kiev that Ukrainian forces had the airport "completely under control."

U.S. Ambassador to Ukraine, Geoffrey Pyatt, told CNBC that he believed the Ukrainian government is acting within its rights.

"(The unrest) seems to be fueled, at least in part, by people coming over the Russian border," he said. He added that the government had also made an amnesty available to the rebels should they opt to down their weapons and cease fighting. The U.S. has made $20 million available to Ukraine, but Pyatt said that this initially meant that non-lethal equipment was being provided to border guards. He added that while the U.S. would be "accommodative" to the military, it is not provide training at this point in time.

Meanwhile, a political debate raged between representatives from both Kiev and Moscow regarding the deal surrounding the gas that Russia supplies to Ukraine. An expensive $485 per 1,000 cubic meters agreement under the old regime was reduced after Yanukovych rejected further ties with the European Union.

Read MorePutin cuts Western ties in shift to the East

However, after his ousting Russia has sought to receive the full amount once more from cash-strapped Ukraine. Talks in Berlin on Monday appeared to go well, according to Europe's Energy Commissioner Guenther Oettinger, but this seems to be at odds with mutterings from Ukraine, with state-owned Naftogaz indicating that talks had made no progress.

The current proposal on the table appears to be for Ukraine to pay $2 billion on Thursday to Russia and a further $500 million by June 7. Policymakers on both sides of the deal have been very vocal in the lead-up to this deadline. Ukrainian Prime Minister Arseniy Yatsenyuk said at a cabinet meeting that he hoped the deal would be signed or suggested more talks for the two sides in Stockholm, according to the Interfax news agency.

Pyatt said that he has been encouraged by statements by the Ukrainian government acknowledging the need to pay its bills to Russia. However, he added that the country should only pay for the gas on market based prices, not on some "political penalty" that Russia imposes.

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