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A monotone iterative method for boundary value problems of parametric differential equations. (English) Zbl 1013.34005
Consider the initial value problem for the DAE-system (\(\ast\)) \(dx/dy = f(t,x,\lambda)\), \(0 = g(x,\lambda)\), \(x(0) = x_0\), for \(0 \leq t \leq T\), \(x, \lambda \in \mathbb{R}\). The authors derive conditions on f and g such that there exist monotone sequences of lower and upper solutions to (\(\ast\)) \(\overline{x}_n\), \({\overline{\lambda}}_n\), \({\underline{x}}_n\), \({\underline{\lambda}}_n\), \({\underline{x}}_n \leq {\underline{x}}_{n+1} \leq\dots\leq \overline{x}_{n+1} \leq \overline{x}_n\) converging to minimal and maximal solutions to (\(\ast\)).

MSC:
34A09 Implicit ordinary differential equations, differential-algebraic equations
34A45 Theoretical approximation of solutions to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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