Need help to design voltage regulator circuit with LT1959
You're in luck. The datasheet for this part goes into extensive detail about the design considerations for the input and output caps. I won't quote it here, because it's several paragraphs. But if you follow those recommendations, and test the design then I'm sure it will be fine.

One problem I'm seeing immediately is that your ESR is going to be too LOW with the ceramic output caps. Most regulators come with a maximum and minimum output ESR. The datasheet goes into more detail about this, but the first thing they say is that for typical applications it will be between 0.05 and 0.2 ohms; you're going to be below that.

Edit: I think you're jumping the gun worrying about this. By default you should:


Design according to the datasheet. Read it carefully and make sure you're within the specs.
Test it.


If you do that, and it still has problems, then you might have to dig into the theory of switching regulators, and what might be going wrong.