On a third-order boundary value problem at resonance on the half-line

In this paper, we establish existence of solutions for the following boundary value problem on the half-line: (q(t)u′′(t))′=g(t,u(t),u′(t),u′′(t)),t∈(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q(t)u''(t))' = g(t, u(t), u'(t), u''(t)),\;\;\; t \in (0, \infty )$$\end{document} subject to the boundary conditions u′(0)=∑i=1mαi∫0ξiu(t)dt,u(0)=0,limt→∞q(t)u′′(t)=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u'(0) = \sum ^{m}_{i=1}\alpha _i\int ^{\xi _i}_0 u(t)\mathrm{d}t, u(0) = 0,\; \lim _{t\rightarrow \infty }q(t)u''(t)=0.$$\end{document} We establish sufficient conditions for the existence of at least one solution using coincidence degree arguments. An example is provided to validate our result.