New existence results for some periodic and Neumann-Steklov boundary value problems with ϕ-Laplacian

We study the existence of solutions of the quasilinear equation ( ϕ ( u ′ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) ) , a.e. t ∈ [ 0 , T ] , $$\bigl(\phi\bigl(u'(t)\bigr)\bigr)'= f \bigl(t,u(t),u'(t)\bigr), \quad\mbox{a.e. } t\in[0,T], $$ with periodic or nonlinear Neumann-Steklov boundary conditions, where ϕ : ] − a , a [ → R $\phi: \,]{-}a, a[\rightarrow\mathbb{R}$ with 0 < a < + ∞ $0 < a< +\infty$ is an increasing homeomorphism such that ϕ ( 0 ) = 0 $\phi(0)=0$ . Combining some sign conditions and the lower and upper solution method, we obtain the existence of solutions when there exists one lower solution or one upper solution.