### The Lower and Upper Solution Method for Three-Point Boundary Value Problems with Integral Boundary Conditions on a Half-Line

#### Abstract

This paper deal with the following second-order three-point boundary value problem with integral boundary condition on a half-line

\begin{eqnarray*}

\left.

\begin{array}{ll}

u''(x)+q(x)f(x, u(x), u'(x))=0, \ \ \hbox{$x\in(0,+\infty)$,} \\

u(0)=\lambda\ds\int_{0}^{\eta}u(s)ds,\ \ u'(+\infty)=C,

\end{array}

\right.

\end{eqnarray*}

where $\lambda>0, \ 0<\lambda\eta<1$ and $f:[0,+\infty)\times \mathbb{R}^2\rightarrow\mathbb{R}$ satisfies a Nagumo's condition which plays an important role in the nonlinear term depend on the first-order derivative explicitly. By using Schauder's fixed point theorem, the upper and lower solution method and topological degree theory, first we give sufficient conditions for the existence of at least one solution and next at least three solutions of the above problem. Moreover, an example is included to demonstrate the efficiency of the main results.

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