Solvability of a class of multi-point boundary value problem with integral boundary conditions on a half-line at resonance

By using Mawhin coincidence degree theorey, the solvability of a class of third-order multi-point boundary value problems　　　　　　　　　　　　　　　 \begincases (q(t)u''(t))'=g(t,u(t),u'(t),u''(t)), t\in 0,\infty ), \\ u\left(0\right)=\displaystyle\sum \limits_i=1^m\alpha _i\displaystyle\int \nolimits_0^\xi _iu\left(t\right)\mathrmdt, \\ u' \left(0\right)=\displaystyle\sum \limits_j=1^n\beta _j\displaystyle\int \nolimits_0^\eta _ju' \left(t\right)\mathrmdt, \\ \undersett\rightarrow +\infty \lim \;q(t)u''(t)=0\\ \endcases at resonance on half-line is discussed, where g: 0,1\times \bfR^3\rightarrow \bfR satisfies L^10,~\mathrm\infty ) -Carathéodory conditions, \alpha _i, \beta _j, \xi _i, \eta _j\in \bfR(1\leqslant i\leqslant m, 1\leqslant j\leqslant n,m,n\in \bfZ^+), q(t)> 0, q(t)\in C0,\mathrm\infty )\cap C^2(0,\mathrm\infty ), \dfrac1q(t)\in L^10,\mathrm\infty ) , and obtained sufficient conditions for the existence of at least one solution to this boundary value problem．