2025, Vol. 10, Issue 10, Part A

We study the second order differential equation

(ϕ(v'(s)))'= k(s,v(s),v'(s)),a.e.s∈[0,ξ]

Submitted to nonlinear Neumann-Steklov boundary conditions on [0,ξ] where k:[0,ξ]×R²→R a L¹-Carathéodory function. ϕ:R→R, is initially considered as an increasing homeomorphism such that ϕ(0)=0. In a second step ϕ is considered as a continuous function on Dom(ϕ)⊂R and strictly increasing on [a,b]⊂ Dom(ϕ). We show the existence of at least one solution using some sign conditions and lower and upper solution method. No Nagumo-like growth condition for the dependence of f(s,w,z) with respect to v is required.