North Korean leader Kim Jong Un has sought "a new future" in relations with the United States, saying he believes efforts to improve ties will succeed, according to a letter addressed to U.S. President Donald Trump.

But the letter, which Trump posted in an image with an English translation on Twitter on Thursday, made no reference to denuclearization, which Kim committed to achieve in a joint statement he signed with Trump in their meeting last month in Singapore.

"I firmly believe that the strong will, sincere efforts and unique approach of myself and Your Excellency Mr. President aimed at opening up a new future between the DPRK and the U.S. will surely come to fruition," Kim said, according to the translation of the letter dated last Friday.

DPRK is the acronym of North Korea's official name, the Democratic People's Republic of Korea.

Kim said he wished that "the invariable trust and confidence in Your Excellency Mr. President will be further strengthened in the future process of taking practical actions," but did not specify what such actions would entail.

"I extend my conviction that the epochal progress in promoting the DPRK-U.S. relations will bring our next meeting forward," he said.

In the Twitter post, Trump hailed the letter, saying, "A very nice note from Chairman Kim of North Korea. Great progress being made!"

Kim is chairman of the Workers' Party of Korea, as well as of the North's State Affairs Commission.

Earlier Thursday, Trump expressed optimism over the course of bilateral ties, despite concern in Washington and beyond that Kim may not abandon his nuclear weapons.

"I really think that we established a very good relationship," Trump said at a news conference in Brussels. "We'll see where it all ends. But there have been no missile tests. There has been no research."

The U.S. leader also cited Pyongyang's release of three American detainees, the North's claim of having destroyed its only known nuclear test site, and a plan to dismantle a missile test site in the country.