Global structure of positive solutions for three-point boundary value problems
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RESEARCH
Open Access
Global structure of positive solutions for three-point boundary value problems Jia-Ping Gu1 , Liang-Gen Hu1 and Huai-Nian Zhang2* *
Correspondence: [email protected] 2 Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing, 102617, P.R. China Full list of author information is available at the end of the article
Abstract In this paper, we are concerned with the three-point boundary value problem for second-order differential equations
u (t) + w(t)f (u(t)) = 0, 0 < t < 1, u(1) = α u(η), u(0) = β u (0),
where β ≥ 0, 0 < η < 1, 0 < αη < 1 and 1 + β – αη – αβ > 0; w ∈ C([0, 1], (0, +∞)) and f ∈ C(R+ , R+ ), R+ = [0, ∞) satisfies f (u) > 0 for u > 0. The existence of the continuum of a positive solution is established by utilizing the Leray-Schauder global continuation principle. Furthermore, the interval of α about the nonexistence of a positive solution is also given. MSC: 34B10; 34B18; 34G20 Keywords: positive solution; global continuous theorem; continuum; differential equation
1 Introduction In this paper, we consider the following three-point boundary value problem for secondorder differential equations: ⎧ ⎨u (t) + w(t)f (u(t)) = , ⎩u() = βu (),
< t < ,
u() = αu(η),
(.)
where β ≥ , < η < , < αη < and + β – αη – αβ > ; w ∈ C([, ], (, +∞)) and f ∈ C(R+ , R+ ), R+ = [, ∞) satisfies f (u) > for u > . The existence and multiplicity of positive solutions for multi-point boundary value problems have been studied by several authors and many nice results have been obtained; see, for example, [–] and the references therein for more information on this problem. The multi-point boundary conditions of ordinary differential equations arose in different areas of applied mathematics and physics. In addition, they are often used to model many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems, infectious diseases as well as concentration in chemical or biological problems. In all these problems, only positive solutions are very meaningful. © 2013 Gu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Gu et al. Boundary Value Problems 2013, 2013:174 http://www.boundaryvalueproblems.com/content/2013/1/174
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In , Sun et al. [] studied the three-point boundary value problem ⎧ ⎨u (t) + μa(t)f (t, u(t)) = , ⎩u() = βu (),
< t < ,
u() = αu(η),
(.)
where μ > is a parameter, β ≥ , < η < , < αη < and + β – αη – αβ > . Based on Krein-Rutmann theorems and the fixed point index theory, they not only established the criteria of the existence and multiplicity of a positive solution, but also obtained the parameter μ in relation with the nonlinear term f a
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