THE EXISTENCE OF POSITIVE SOLUTIONS FOR A SEMIPOSITONE SECOND-ORDER m-POINT BOUNDARY VALUE PROBLEM

In this paper, we study the existence of positive solutions to boundary value problem

$$ \left\{\begin{array}{ll} u^{\prime\prime} + \lambda f(t,u) = 0, & t \in (0, 1),\\ u(0) = \sum_{i=1}^{m-2}\alpha_i u(\xi_i), & u^{\prime}(1) = \sum_{i=1}^{m-2}\beta_i u^{\prime}(\xi_i), \end{array} \right. $$

where $\xi_i \in (0, 1)$, $0 < \xi_1 < \xi_2 <\cdots < \xi_{m−2} < 1$, $\alpha_i , \beta_i \in [0, ∞)$, $λ$ is positive parameter. By using Krasnosel’skii’s fixed point theorem, we provide sufficient conditions for the existence of at least one positive solution to the above boundary value problem.