But the accumulation of new defensive initiatives leaves open the question of how forcefully Mr. Obama is committed to reversing the suspicion, from Europe to the Middle East to Asia, that the United States is in an era of retrenchment. In his travels in Europe this week and a lengthy tour of Asia planned this fall, the president faces a dual challenge: convincing American allies and partners that he has no intention to leave power vacuums around the globe for adversaries to fill, while convincing Americans that he can face each of these brewing conflicts without plunging them back into another decade of large military commitments and heavy casualties.

“There is a growing mismatch between the rhetoric and the policy,” said Richard N. Haass, the president of the Council on Foreign Relations and a senior national security official as the war with Iraq loomed a dozen years ago. “If you add up the resources needed to implement the Asian pivot, recommit to the Middle East and increase our presence in Europe, you can’t do it without additional money and capacity. The world has proved to be a far more demanding place than it looked to this White House a few years ago.”…

In China, the president faces the opposite challenge: a rising power with growing resources and a sense that this is China’s moment to reassert influence in Asia in a way it has not in hundreds of years. Here, the surprise to Mr. Obama has been the aggressiveness shown by Xi Jinping, China’s president, in embracing efforts to press territorial claims against Japan, South Korea, Vietnam and the Philippines, rather than focusing on the domestic economy.

“We didn’t see this coming,” one former member of Mr. Obama’s national security team said this summer, “and there’s a lot of debate about how to counter it.”