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On the existence of weak solutions for singular strongly nonlinear boundary value problems on the half-line

  • Stefano BiagiEmail author
Article
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Abstract

In the present paper, we consider boundary value problems on the real half-line \({\varLambda }:= [0,\infty )\) of the following form
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \Big ({\varPhi }\big (a(t,x(t))\,x'(t)\big )\Big )' = f(t,x(t),x'(t)) \quad \text {a.e.}\,\text {on} \, {\varLambda }, \\ \,\,x(0) = \nu _1,\quad x(\infty ) = \nu _2, \end{array}\right. } \end{aligned}$$
where \({\varPhi }:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing homeomorphism, \(a\in C({\varLambda }\times {\mathbb {R}},{\mathbb {R}})\) is nonnegative which can vanish on a set of zero Lebesgue measure and f is a Caratheódory function on \({\varLambda }\times {\mathbb {R}}^2\). Under very general assumptions on the functions a and f, including an appropriate version of the well-known Nagumo–Wintner growth condition, we prove the existence of at least one solution of the above problem in a suitable Sobolev space. Our approach combines a fixed-point technique with the method of lower/upper solutions.

Keywords

Singular ODEs Heteroclinic solutions \({\varPhi }\)-Laplace operators Nonlinear ODEs BVPs on unbounded intervals Nagumo–Wintner condition 

Mathematics Subject Classification

34C37 34B16 34B40 34L30 

Notes

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Università Politecnica della MarcheAnconaItaly

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