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Successive iteration and positive solution for nonlinear second-order three-point boundary value problems. (English) Zbl 1096.34015
This paper is concerned with the existence of positive solutions to a boundary value problem of the type
\[ u''(t) + f(t, u(t), u'(t))=0, t \in [0,1] \] with \( u(0)=0,\) \(u(1)= \alpha u(\eta)\) and \( 0 < \alpha <1/2,\) \( \eta <1\).
By transforming the differential problem into an equivalent integral problem \( u = T u, \) where \(T\) is a suitable integral operator, the author proves two theorems (for \(f(t,u)\) and \(f(t,u,u')\)) providing sufficient conditions on the nonlinear function \(f\) and the constants to ensure the existence of a concave solution. Further, a fixed-point iteration \(u_{n+1}= T u_n\) with given \(u_0\) is proposed to approximate the solution of the integral equation. Finally, two examples with some polynomial functions \(f\) are presented to show the performance of the proposed iterative scheme.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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