Multiple positive solutions for singular quasilinear multipoint BVPs with the first-order derivative.

Jiang, Weihua; Wang, Bin; Guo, Yanping

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The paper deals with the existence of multiple positive solutions for the boundary value problem $\{\begin{matrix} {(ϕ (p (t) u^{(n - 1)}) (t))}^{'} + a (t) f (t, u (t), u^{'} (t), ..., u^{(n - 2)} (t)) = 0, 0 < t < 1, \\ u^{(i)} (0) = 0, i = 0, 1, ..., n - 3, \\ u^{(n - 2)} (0) = \sum_{i = 1}^{m - 2} α_{i} u^{(n - 2)} (ξ_{i}), u^{(n - 1)} (1) = 0, \end{matrix}$ where $ϕ : ℝ \to ℝ$ is an increasing homeomorphism and a positive homomorphism with $ϕ (0) = 0$ . Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.