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

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