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Positive solutions of nonhomogeneous boundary value problems for some nonlinear equation with ϕ-Laplacian

Liang-Gen Hu* and Jing Xu

Author Affiliations

Department of Mathematics, Ningbo University, Ningbo, 315211, P.R. China

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Boundary Value Problems 2012, 2012:130  doi:10.1186/1687-2770-2012-130

Published: 12 November 2012

Abstract

We will consider the nonhomogeneous ϕ-Laplacian differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M3">View MathML</a>) is an increasing homeomorphism such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M6">View MathML</a> are continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M10">View MathML</a>. Based on the Krasnosel’skii fixed point theorem, the existence of a positive solution is obtained, even if some of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/130/mathml/M11">View MathML</a> coefficients are negative. Two examples are also given to illustrate our main results.

Keywords:
nonhomogeneous; ϕ-Laplacian; positive solution; fixed point; negative coefficient