Existence of positive solutions for multi-point semi-positive boundary value problems (Q2026823)

The author considers the existence of positive solutions for the nonlinear multi-point boundary value problem \[ - \big(u''(t) + a(t)u'(t) \big ) = \lambda f(t, u) + \mu g(t, u), \, t \in (0, 1), \] \[ u'(0) = 0, \, \, u(1) = \sum_{i=1}^k \alpha_i u(\eta_i) - \sum_{i = k+1}^{m-2} \alpha_i u(\eta_i). \] Using cone theoretic techniques, the author shows the existence of at least one positive solution when \(\lambda\) and \(\mu\) are small under various suitable conditions on \(f\) and \(g\). They also establish the existence of at least three positive solutions under other conditions on \(f\) and \(g\) when \(\lambda\) and \(\mu\) are small.