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Two iterative schemes for an \(H\)-system. (English) Zbl 1092.34509

The authors consider the nonlinear boundary value problem \[ \begin{aligned} f'- f=-2H(f, g)f g',\quad x\in I,\qquad &g''= 2H(f,g)f f',\quad x\in I,\\ f(0)= \alpha_0,\quad f(L)=\alpha_L,\qquad &g(0)= \beta_0,\quad g(L)= \beta_L,\end{aligned}\tag{1} \] where \(f,g\in C^2(\overline I)\); \(f> 0\), \(g'> 0\) in \(I= (0,L)\subset \mathbb{R}\); \(H: \mathbb{R}^2\to \mathbb{R}\) is a given continuous function; and \(\alpha_0,\alpha_L> 0\), \(\beta_0< \beta_L\) are fixed real numbers.
Two iterative schemes for the solution of problem (1) are studied: a Newton imbedding-type procedure, and a scheme based on the method of upper and lower solutions.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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