On the existence of weak solutions for singular strongly nonlinear boundary value problems on the half-line

In the present paper, we consider boundary value problems on the real half-line Λ: = [0 , ∞) of the following form (Φ(a(t,x(t))x′(t)))′=f(t,x(t),x′(t))a.e.onΛ,x(0)=ν1,x(∞)=ν2,where Φ: R→ R is a strictly increasing homeomorphism, a∈ C(Λ× R, R) is nonnegative which can vanish on a set of zero Lebesgue measure and f is a Caratheódory function on Λ× R2. Under very general assumptions on the functions a and f, including an appropriate version of the well-known Nagumo–Wintner growth condition, we prove the existence of at least one solution of the above problem in a suitable Sobolev space. Our approach combines a fixed-point technique with the method of lower/upper solutions.