Positive Solutions of a Nonlinear Three-Point Integral Boundary Value Problem
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Research Article Positive Solutions of a Nonlinear Three-Point Integral Boundary Value Problem Jessada Tariboon1, 2 and Thanin Sitthiwirattham1 1
Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand 2 Centre of Excellence in Mathematics, CHE, Sri Ayutthaya Road, Bangkok 10400, Thailand Correspondence should be addressed to Jessada Tariboon, [email protected] Received 4 August 2010; Accepted 18 September 2010 Academic Editor: Raul F. Manasevich Copyright q 2010 J. Tariboon and T. Sitthiwirattham. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the existence of positive solutions to the three-point integral boundary value problem η u atfu 0, t ∈ 0, 1, u0 0, α 0 usds u1, where 0 < η < 1 and 0 < α < 2/η2 . We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones.
1. Introduction The study of the existence of solutions of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev 1. Then Gupta 2 studied three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, nonlinear second-order three-point boundary value problems have also been studied by several authors. We refer the reader to 3–19 and the references therein. However, all these papers are concerned with problems with three-point boundary condition restrictions on the slope of the solutions and the solutions themselves, for example, u0 0, αu η u1, u0 βu η , αu η u1, αu η u1, 1.1 u 0 0, u0 − βu 0 0, αu η u1, αu0 − βu 0 0, u η u 1 0, and so forth.
2
Boundary Value Problems In this paper, we consider the existence of positive solutions to the equation u atfu 0,
t ∈ 0, 1,
1.2
usds u1,
1.3
with the three-point integral boundary condition
u0 0,
η α 0
where 0 < η < 1. We note that the new three-point boundary conditions are related to the area under the curve of solutions ut from t 0 to t η. The aim of this paper is to give some results for existence of positive solutions to 1.21.3, assuming that 0 < α < 2/η2 and f is either superlinear or sublinear. Set
f0 lim u→0
fu , u
f∞ lim
u→∞
fu . u
1.4
Then f0 0 and f∞ ∞ correspond to the superlinear case, and f0 ∞ and f∞ 0 correspond to the sublinear case. By the positive solution of 1.2-1.3 we mean that a function ut is positive on 0 < t < 1 and satisfies the problem 1.2-1.3. Throughout this paper, we suppose the following conditions hold: H1 f ∈ C0, ∞, 0, ∞; H2 a ∈ C0, 1, 0, ∞ and there exists t0 ∈ η, 1 such that at0 > 0. The proof of the main theorem is based upon an a
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