NEW YORK – Sprint, the No. 3 cellphone company in the U.S., is selling a controlling stake to Japan’s Softbank for $20.1 billion.

The deal, announced Monday in Tokyo, positions Sprint Nextel Corp. as a stronger competitor to U.S. market leaders Verizon Wireless and AT&T, but it doesn’t solve all of the company’s underlying problems.

Sprint, based in Overland Park, Kan., has been limping along since 2005, when it bought Nextel. The merger quickly turned sour, saddling Sprint with the cost of running two incompatible networks while customers fled.

Softbank Corp., a holding company with investments in Internet and telecom businesses, made its own venture into the wireless world in 2005 with the acquisition of Vodafone Japan.

It turned that business around, giving President Masayoshi Son the confidence that he can make Sprint a profitable company again after five straight years of losses.

Sprint CEO Dan Hesse has laid the groundwork for a turnaround — the company’s reputation for customer service has improved during his tenure. But his efforts haven’t had an immediate impact on profitability.

On its own, the company would have a hard road ahead, as it pays for both a network revamp and $15.5 billion in iPhones from Apple.

Under the deal, Sprint shareholders can turn in 55 percent of their shares to Softbank in exchange for $7.30 per share. Sprint shares were up just 3 cents at $5.76 in morning trading Monday, suggesting that investors had accurately pegged the value of the transaction last week, when they sent the stock up 14 percent based on reports of talks between Softbank and Sprint.

Softbank is paying $12.1 billion for the 55 percent stake. It’s buying an additional $8 billion worth of shares from the company, for a total stake of 70 percent. That investment will dilute the value of existing shares, and is the reason Sprint’s stock didn’t trade higher Monday.

“While we believe it will take far more than capital for Sprint Nextel to effectively compete with Verizon Wireless and AT&T Mobility, we believe the deal announced today, without question, strengthens Sprint’s position in the long-run,” said Christopher King at Stifel Nicolaus.

Kevin Smithen at Macquarie Capital said the deal doesn’t improve Sprint’s access to space on the airwaves, which is critical to improving its wireless data network, nor does it provide a path to improving its profitability. A merger with T-Mobile USA, the No. 4 carrier, might still be needed to deal with those problems, he said.

T-Mobile USA has its own plan: Two weeks ago, it struck a deal to buy MetroPCS Communications Inc., the No. 5 carrier in the U.S.