# Finding fixed points for BVP

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.

Fixed-point is the name of a method, not of a problem (so far we speak of nonlinear differential equations). Its starts from the observation that, if your equation was $x''=-g(t)$, then the solution would be $$x(t)=\int_0^t(1-t)sg(s)ds+\int_t^1t(1-s)g(s)ds,$$ which we write as $$x(t)=\int_0^1K(t,s)g(s)ds.$$ Therefore, your nonlinear problem is equivalent to the integral equation $$x(t)=-\int_0^1K(t,s)f(s,x(s))ds=:(Tx)(t).$$ This can be viewed as a fixed-point problem $Tx=x$. Hence the first idea of constructing a sequence of of 'approximate solutions' by $x^0\equiv0$ and then $x^{k+1}=Tx^k$, that is $$x^{k+1}(t)=-\int_0^1K(t,s)f(s,x^k(s))ds.$$ If $x^k$ converges pointwise boundedly to an $x$, then the convergence is uniform, thanks to the iteration, and $x$ is a fixed point. However, it is not always true that the sequence converges (it could oscillate, for instance), hence the second idea, which is of topological nature. Because $f$ is bounded, $T$ maps ${\mathcal C}(0,1)$ into a compact subset. In addition, $T$ is continuous. Then Schauder's fixed-point theorem ensures that $T$ admits a fixed point (not necessarily unique). Schauder's Theorem is a generalization of Brouwer's to infinite dimension (Banach spaces).