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Quadratic approximation of solutions for differential equations with nonlinear boundary conditions. (English) Zbl 1097.34008
Consider the boundary value problem \[ x'=f(t,x)\quad\text{for} \quad t\in J:=[0,T],\;T>0,\quad 0=g\bigl(x(0),x(T)\bigr),\tag{*} \] with \(f\in C(J\times \mathbb R,\mathbb R)\) and \(g\in C(\mathbb R\times \mathbb R,\mathbb R)\). The author applies the method of lower and upper solutions coupled with monotone iterative technique. He gives sufficient conditions on the functions \(f\) and \(g\) (monotonicity and convexity properties) implying the existence of monotone sequences converging quadratically to the unique solution of (*).

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