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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 (*).

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