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Bulletin of the Malaysian Mathematical Sciences Society, Volume 38, Issue 3, 2015, pp. 909-926

#### Solvability of a Third-Order Three-Point Boundary Value Problem on a Half-Line

Shi H. 1, Pei M. * 1, Wang L. 1
Abstract :

In this paper, we consider the solvability of a third-order three-point boundary value problem on a half-line of the form: \begin{aligned}{\left\{ \begin{array}{ll}x^{\prime\prime\prime}(t)=f\left(t,x(t),{x^{\prime}}(t),{x^{\prime \prime}}(t)\right),\quad 0< t<+\infty,\\ x(0)=\alpha x(\eta),\quad \mathop {\lim }\nolimits_{t\rightarrow +\infty}x^{(i)}(t)=0,\quad i=1,2,\end{array}\right.}\end{aligned}x″′(t)=ft,x(t),x′(t),x″(t),0x(i)(t)=0,i=1,2,where $$\alpha\ne1$$α≠1 and $$\eta\in (0,+\infty)$$η∈(0,+∞),while $$f:[0,+\infty)\times{\mathbb{R}}^3\rightarrow{\mathbb {R}}$$f:[0,+∞)×R3→R is $$S^2$$S2—Carathéodory function. The existence and uniqueness of solutions for the boundary value problems are obtained by the Leray-Schauder continuation theorem. As an application, an example is given to demonstrate our results.

Keywords : Half-line,Leray-Schauder continuation theorem,Three-point boundary value problem
Subject Area : Mathematics(all)