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

Abstract

AbstractThis work is devoted to the study of singular strongly non-linear integro-differential equations of the type $$\begin{aligned} (\Phi (k(t)v'(t)))'=f\left( t,\int _0^t v(s)\, \textrm{d}s,v(t),v'(t) \right) , \text{ a.e. } \text{ on } {\mathbb {R}}^{+}_0 := [0, + \infty [, \end{aligned}$$ ( 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, $$\Phi $$ 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 $$1/k \in L^p_\textrm{loc}({\mathbb {R}}^{+}_0)$$ 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.