Boundary value problems on the half-line with impulses and infinite delay (Q5945752)

The following problem is considered NEWLINE\[NEWLINE\begin{cases} (Lx)(t)+ f(t, x_t)= 0,\;t\neq t_k,\;\Delta x|_{t=t_k}= I_k(x_{t_k}),\;k= 1,2,\dots, m,\\ \lambda x(0)- \beta\lim_{t\to 0} p(t) x'(t)= a,\;\gamma x(\infty)+ \delta\lim_{t\to\infty} p(t) x'(t)= b,\\ x(t)\text{ is bounded on }[0,+\infty),\end{cases}\tag{1}NEWLINE\]NEWLINE where \(x_t\) is defined by \(x_t(s)= \begin{cases} x(t+ s),\;t\geq t+ s\geq 0;\\ \phi(t+ s),\;-\infty< t+ s< 0,\end{cases}\) NEWLINE\[NEWLINE(Lx)(t)= {1\over p(t)} (p(t) x'(t))',\;p\in C([0, +\infty), R)\cap C^1(0,+\infty),\;p(t)> 0\quad\text{for }t\in (0,\infty),NEWLINE\]NEWLINE \(\Delta x|_{t_k}= \lim_{\varepsilon\to 0^+} [x(t_k+ \varepsilon)- x(t_k- \varepsilon)]\) and \(\lambda\), \(\beta\), \(a\), \(\gamma\), \(\delta\), \(b\), \(\phi(t)\) are given. The existence and uniqueness of a solution to problem (1) are proved.