Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

Abstract

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition \gather u''+g(t)f(t,u)=0, \quad t\in (0,1),\nonumber u(0)=\alpha u(\xi )+\lambda ,\quad u(1)=\beta u(\eta )+\mu .\nonumber \endgather Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term $f(t,x)$ may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of $f(t, x)/x$ for $x$ near $0$ and $\pm \infty $, and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.