Singular multipoint impulsive boundary value problem with \(p\)-Laplacian operator (Q1031998)

The authors consider the singular multipoint boundary value problem with \(p\)-Laplacian operator \[ \begin{aligned} &-(\phi_p(u'(t)))' + q(t)f(t,u(t)) = 0, \;t \not= t_k, \;0 < t < 1,\\ &\triangle u|_{t=t_k} = I_k(u(t_k)),\;k = 1,\ldots,m,\\ &au(0) - bu'(0) = \sum_{i=1}^{l} \alpha_iu(\xi_i), \;u'(1) = \sum_{i=1}^l \beta_iu'(\xi_i), \end{aligned} \] where \(\phi_p\) is the \(p\)-Laplacian operator, \(p > 1\), \(0 < t_1 < \cdots < t_m < 1\), \(\xi_i \neq t_i\), \(q\) may be singular at \(t = 0\), \(t = 1\), \(f\) may be singular in the phase variable at \(u = 0\), \(I_k\) are continuous. Sufficient conditions ensuring the existence of at least one and two positive solutions are obtained. The existence results are obtained via fixed point index theory.