Opuscula Math. 36, no. 3 (2016), 399-407
http://dx.doi.org/10.7494/OpMath.2016.36.3.399

Opuscula Mathematica

# Existence theorems of nonlinear asymptotic BVP for a homeomorphism

Katarzyna Szymańska-Dębowska

Abstract. In this work, we are concerned with the existence of solutions for the following $$\varphi$$-Laplacian boundary value problem on the half-line $(\varphi (x'))' =f(t,x,x'),\quad x(0)=0,\quad x'(\infty)=0,$ where $$f:\mathbb{R}_+\times\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}^k$$ is continuous. The results are proved using the properties of the Leray-Schauder topological degree.

Keywords: half-line, nonlinear, asymptotic boundary value problem, $$\varphi$$-Laplacian, Leray-Schauder degree.

Mathematics Subject Classification: 34B15, 34B40.

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• Katarzyna Szymańska-Dębowska
• Lodz University of Technology, Institute of Mathematics, 90-924 Łódź, ul. Wólczańska 215, Poland
• Communicated by Giovanni Molica Bisci.
• Accepted: 2015-11-26.
• Published online: 2016-02-21.