Eigenvalues of boundary value problems for higher order differential equations

Abstract

We shall consider the boundary value problem y(n)+λQ(t,y,y1,⋅⋅⋅,y(n−2))=λP(t,y,y1,⋅⋅⋅,y(n−1)),n≥2,t∈(0,1),y(i)(0)=0,0≤i≤n−3,αy(n−2)(0)−βy(n−1)(0)=0,γy(n−2)(1)+δy(n−1)=0, where λ>0,α,β,γ and δ are constants satisfying αγ+αδ+βγ>0,β,δ≥0,β+α>0 and δ+γ>0 to characterize the values of λ so that it has a positive solution. For the special case λ=1, sufficient conditions are also established for the existence of positive solutions.