A boundary value problem for a semilinear second order ODE (Q2767720)

The scalar boundary value problem NEWLINE\[NEWLINE y''=f(x,y,y'), \quad x\in (a,b), \qquad y(a)=\alpha, \quad y(b)=\beta,\tag{1}NEWLINE\]NEWLINE with \(f:[a,b]\times\mathbb{R}^2\to\mathbb{R}\) continuous, is considered. The authors, after presenting some conditions for the existence and uniqueness of solutions, consider the following ``continuation'' problem: NEWLINE\[NEWLINEy''=tf(x,y,y'), \quad x\in (a,b), \qquad y(a)=\alpha, \quad y(b)=\beta,\tag{\(1_t\)}NEWLINE\]NEWLINE with \(t\in [0,1]\) as a parameter. Under suitable assumptions, they show the existence of a positive number \(\varepsilon\), depending only on the derivatives of \(f\), with the property that for any \(0\leq\varepsilon'\leq\varepsilon\) a solution to (\(1_{t_0+\varepsilon'}\)) can be obtained from a known solution to (\(1_{t_0}\)) via an iterative method. In this way, equation (1) can be solved considering a sufficiently refined partition \(0=t_0<t_1<\ldots<t_n=1\) of \([0,1]\).