WASHINGTON — With his troops anxious for a way forward, Speaker John A. Boehner began a closed-door meeting of House Republicans on Friday morning with a recitation of letters from local schoolchildren on how they deal with stress.

A shower helps, one child counseled. So does a nap.

But on the matter causing all the Congressional stress, the speaker offered no clue as to how he expected Congress to get out of the dead end it has found itself in, with the government shut for a fourth day and no clear path to raise the federal debt limit to avoid the nation’s first default. “We are locked in an epic battle,” the speaker told his rank and file, those who attended the meeting said, urging them to “hang tough.”

The overarching problem for the man at the center of the budget fight, say allies and opponents, is that he and his leadership team have no real idea how to resolve the fiscal showdown.

They are only trying to survive another day, Republican strategists say, hoping to maintain unity as long as possible so that when the Republican position collapses, they can capitulate on two issues at once — financing the government and raising the debt ceiling — and head off any internal party backlash. Republican lawmakers say Mr. Boehner has assured them privately that he will not permit a default.