Existence results for singular strongly non-linear integro-differential BVPs on the half line

Abstract This work is devoted to the study of singular strongly non-linear integro-differential equations of the type aligned ( (k (t) v' (t) ) ) '=f (t, ₀ᵗ v (s) \, ds, v (t), v' (t) ), a. e. on {R}^+₀: = [0, + [, aligned (Φ (k (t) v ′ (t) ) ) ′ = f t, ∫ 0 t v (s) d s, v (t), v ′ (t), a. e. on R 0 +: = [ 0, + ∞ [, where f is a Carathéodory function, Φ is a strictly increasing homeomorphism, and k is a non-negative integrable function, which is allowed to vanish on a set of zero Lebesgue measure, such that 1/k Lᵖₗoc ({R}^+₀) 1 / k ∈ L loc p (R 0 +) for a certain p>1 p > 1. By considering a suitable set of assumptions, including a Nagumo–Wintner growth condition, we prove existence and non-existence results for boundary value problems associated with the non-linear integro-differential equation of our interest in the sub-critical regime on the real half line.