Successive iteration and positive solution for nonlinear second-order three-point boundary value problems.

*(English)*Zbl 1096.34015This 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.

\[ u''(t) + f(t, u(t), u'(t))=0, t \in [0,1] \] 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.

Reviewer: Manuel Calvo (Zaragoza)

##### 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 |

##### Keywords:

Second-Order Ordinary Differential Equations; Multipoint Boundary Value Problems; Existence of Positive Solutions; Convergence of successive iteration
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\textit{Q. Yao}, Comput. Math. Appl. 50, No. 3--4, 433--444 (2005; Zbl 1096.34015)

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##### References:

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