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

Abstract

In this paper, we consider the solvability of a third-order three-point boundary value problem on a half-line of the form: $$\begin{aligned} {\left\{ \begin{array}{ll} x^{\prime \prime \prime }(t)=f\left( t,x(t),{x^{\prime }}(t),{x^{\prime \prime }}(t)\right) ,\quad 0< t<+\infty , \\ x(0)=\alpha x(\eta ),\quad \mathop {\lim }\nolimits _{t\rightarrow +\infty }x^{(i)}(t)=0,\quad i=1,2, \end{array}\right. } \end{aligned}$$ where $$\alpha \ne 1$$ and $$\eta \in (0,+\infty )$$ , while $$f:[0,+\infty )\times {\mathbb {R}}^3 \rightarrow {\mathbb {R}}$$ is $$S^2$$ —Caratheodory 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.