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 (Φ(k(t)v′(t)))′=ft,∫0tv(s)ds,v(t),v′(t),a.e.onR0+:=[0,+∞[,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (\\Phi (k(t)v'(t)))'=f\\left( t,\\int _0^t v(s)\\, \ extrm{d}s,v(t),v'(t) \\right) , \ ext{ a.e. } \ ext{ on } {\\mathbb {R}}^{+}_0 := [0, + \\infty [, \\end{aligned}$$\\end{document}where f is a Carathéodory function, Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} 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∈Llocp(R0+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/k \\in L^p_\ extrm{loc}({\\mathbb {R}}^{+}_0)$$\\end{document} for a certain p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}. 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.