The solvability of nonhomogeneous boundary value problems with ϕ-Laplacian operator

We treat the nonhomogeneous boundary value problems with ϕ-Laplacian operator (ϕ(u′(t)))′=−f(t,u(t),u′(t)), t∈(0,T), u(0)=A, ϕ(u′(T))=τu(T)+∑i=1kτiu(ζi), where ϕ:(−a,a)→(−b,b) (0<a,b≤+∞) is an increasing homeomorphism such that ϕ(0)=0, τ,τi∈R, ζi∈(0,T), i=1,2,…,k, A≥0, and f:[0,T]×R×R→R is continuous. We will show that even if some of the τ and τi are negative, the boundary value problem with singular ϕ-Laplacian operator is always solvable, and the problem with a bounded ϕ-Laplacian operator has at least one positive solution.