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

$$\begin{aligned} (\Phi (k(t)v'(t)))'=f\left( t,\int _0^t v(s)\, \textrm{d}s,v(t),v'(t) \right) , \text{ a.e. } \text{ on } {\mathbb {R}}^{+}_0 := [0, + \infty [, \end{aligned}$$

where f is a Carathéodory function, \(\Phi \) 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 \(1/k \in L^p_\textrm{loc}({\mathbb {R}}^{+}_0)\) for a certain \(p>1\). 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.