Hi. While learning BVP, I came across a problem. It was mentioned that given the below left-focal BVP, a ﬁxed–point problem can be formed for the system.

x'' = f (t, x), x (0) = 0, x(1) = 0.

where, f : [0, 1] × R → R is continuous and uniformly bounded. It was also mentioned that Schauder’s theorem can be applied to ensure the corresponding existence.

Can someone please help me in understanding this proof? Please let me know if the problem seems vague.

From this, can we infer that, in general, if

x'' = f (t, x), x(a) = x(b), x (a) = x (b) and f : [a, b] × R → R is continuous and uniformly bounded, then Schauder’s theorem can be applied to ensure that the system has at least one solution?

Regards, Salil