Existence of solutions for second-order $\phi$-Laplacian nonlinear impulsive bvps on the half-line

The present paper is concerned with the existence of solutions for a $\phi-$Laplacian nonlinear boundary value problem (bvp) set on the half-line of second-order differential equation: \begin{equation*} (\phi (r'(t)))' +g(t,r(t),r'(t))=0 ,\quad\text{a.e. } t\in [ 0,+\infty )\backslash \{t_{1},t_{2},\ldots\}, \end{equation*} with the boundary conditions, \begin{equation*} r(0)=A,\quad r'(+\infty )=B, \end{equation*} where $g:[0,+\infty )\times \mathbb{R}^{2}\to \mathbb{R}$ is an $L^{1}$-Carath\'eodory function, and the impulsive conditions are Carathéodory sequences. We use the Schauder fixed point theorem to get our result.