Positive solutions to boundary value problems with nonlinear boundary conditions

In this paper, we consider the boundary value problem yΔΔ(t)=−λf(t,yσ(t))yΔΔ(t)=−λf(t,yσ(t)) subject to the boundary conditions y(a)=ϕ(y)y(a)=ϕ(y) and y(σ2(b))=0y(σ2(b))=0. In this setting, ϕ:Crd([a,σ2(b)]T,R)→R is a continuous functional, which represents a nonlinear nonlocal boundary condition. By imposing sufficient structure on ϕϕ and the nonlinearity ff, we deduce the existence of at least one positive solution to this problem. The novelty in our setting lies in the fact that ϕϕ may be strictly nonpositive for some y>0y>0. Our results are achieved by appealing to the Krasnosel’skiĭ fixed point theorem. We conclude with several examples illustrating our results and the generalizations that they afford.