Positive solutions of some three-point boundary value problems via fixed point index for weakly inward \(A\)-proper maps (Q2495069)

This paper deals with a nonlinear three-point boundary value problem of the form \[ -u''=f(t,u,u',u''), \quad u(0)=0, \quad \alpha u(\eta)=u(1), \] with \(0<\eta <1\), \(\alpha \eta <1\). Under a set of technical conditions (which does not include the positivity of \(f\)), the existence of at least one positive solution is proved. The proof is performed by using a version of Krasnosel'skii's fixed-point theorem in cones suitable for weakly inward \(A\)-proper maps. This framework is useful when \(f\) is not positive.