Boundary value problems associated with singular strongly nonlinear equations with functional terms

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type (φ (k (t) x ′ (t))) ′ + f (t, G x (t)) ρ (t, x ′ (t)) = 0, on a compact interval [a, b]. These equations are quite general due to the presence of a strictly increasing homeomorphism φ, the so-called φ-Laplace operator, of a non-negative function k, which may vanish on a set of null measure, and moreover of a functional term Gx. We look for solutions, in a suitable weak sense, which belong to the Sobolev space W1,1([a, b]). Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.