Hong Kong gained 0.66 per cent and Seoul fell 0.56 per cent while Sydney, Shanghai and Taipei were all flat.
Tokyo was closed for a public holiday.
The European Banking Authority said on Friday that eight lenders -- including five from Spain and two from Greece -- out of 91 had failed to meet the capital requirements it deems essential to act as a buffer against any unforeseen financial shocks.
The EBA said the banks, which also included one from Austria, had a total shortfall of 2.5 billion euros ($3.5 billion), while the total number of failures was less than expected by markets.
Friday's announcement, however, was tainted by criticism that the tests did not include the possibility of a sovereign debt default, a prospect many believe is likely for bailed-out Greece, and a possibility for Spain and Italy.
"The stress tests came out and to all intents and purposes on the surface didn't look too bad, but the market is fairly and squarely focused on the risk of sovereign default and the risk of contagion in Europe," said HiFX Trading Director Mike Hollows in New Zealand.
Traders also have their eyes on the United States, where President Barack Obama and the Republican party struggled to make progress at the weekend in hammering out a deal to raise the debt ceiling.
If the two sides are unable to reach an agreement by the August 2 deadline the world's biggest economy will likely default on its debt obligations, a scenario Obama described as economic "Armageddon" that would have a devastating global effect.
The banking results gave some immediate respite to the euro but the single currency slipped back in Asia on Monday. It was at $1.4074 in early trade, from $1.4150 late Friday in New York, and at 111.26 yen, from 112.00.
In Sydney, Rupert Murdoch's News Corporation slumped more than five per cent in early trade as the phone hacking scandal that brought down the top-selling News of the World in Britain rumbled on.
The dive came after Rebekah Brooks, the former head of Murdoch's British newspaper wing, was arrested on Sunday.
Brooks was questioned for 12 hours over allegations that she conspired to intercept communications and that the News of the World Sunday tabloid, of which she was editor, paid police for stories.
At the same time, Britain's top police officer resigned due to speculation about his links to the Murdoch empire and the force's botched investigation into phone hacking.
The dollar fetched 79.03 yen from 79.14 yen.
Gold opened at $1,596.00-$1,597.00 in Hong Kong, up from Friday's close of $1,578.00-$1,579.00.
The precious metal is sitting at record highs as dealers buy into it for its safe-haven status.
On oil markets, New York's main contract, light sweet crude for delivery in August, fell nine cents to $97.15 per barrel while Brent North Sea crude for September gained 22 cents to $117.48.
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