The existence of countably many positive solutions for some nonlinear singular three-point impulsive boundary value problems (Q838056)

The authors consider the impulsive three-point boundary value problem \[ \begin{aligned} &(\varphi(u'(t)))' + a(t)f(u(t)) = 0,\quad t \in J', \\ &\triangle u|_{t = t_k} = I_k(u(t_k)),\;k = 1,\ldots,n, \\ &x(0) - B_0(u'(\eta)) = 0,\quad u'(1) = 0, \end{aligned} \] where \(0 < t_1 < \ldots < t_n < 1\), \(J' = [0,1]\setminus\{t_1,\ldots,t_n\}\), \(\eta \in (0,1)\), \(\varphi\) is an increasing and positive homomorphism, \(a\) has countably many singularities. Sufficient conditions for the existence of countably many positive solutions to this problem are found. The existence results are obtained via fixed point index theory. Examples illustrating the application of the results are presented.