Monotone convergence of iterative methods for (n,p) and (p,n) boundary value problems (Q1096353)

This paper presents a monotone iterative method for solving two-point boundary value problems of type: \(x^{(n)}=f(t,x,x',...,x^{(q)})\) \(x^{(i)}(a)=A_ i\) \((i=0,...,n-2)\), \(x^{(p)}(b)=B,\) where p and q are fixed integers \(0\leq p,q\leq n-1\). In order to define the iterative method, the author introduces in the space \(C^{(q)}[a,b]\) a partial order and an integral operator T which is proved to be isotone under suitable assumptions on the function f of the differential equation. Thus, starting with a lower and upper solutions \(x_ 0\) and \(y_ 0\) respectively, the sequences defined by \(x_ n=Tx_{n-1}\) and \(y_ n=Ty_{n-1}\) satisfy \(x_ n\leq y_ n\) and both converge to lower and upper solutions of the problem under consideration. The paper ends applying the proposed method to \(x\prime''=e^ x\), \(x'(0)=x(1)=x'(1)=0.\) Here it is found numerically that this problem has a unique solution between 0 and \((t-1)^ 2(t+1)/2)\).