Positive Solutions for Nonlinear th-Order Singular Nonlocal Boundary Value Problems
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Research Article Positive Solutions for Nonlinear nth-Order Singular Nonlocal Boundary Value Problems Xin’an Hao, Lishan Liu, and Yonghong Wu Received 23 June 2006; Revised 16 January 2007; Accepted 26 January 2007 Recommended by Ivan Kiguradze
We study the existence and multiplicity of positive solutions for a class of nth-order singular nonlocal boundary value problems u(n) (t) + a(t) f (t,u) = 0, t ∈ (0,1), u(0) = 0, u (0) = 0,...,u(n−2) (0) = 0, αu(η) = u(1), where 0 < η < 1, 0 < αηn−1 < 1. The singularity may appear at t = 0 and/or t = 1. The Krasnosel’skii-Guo theorem on cone expansion and compression is used in this study. The main results improve and generalize the existing results. Copyright © 2007 Xin’an Hao et al. 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. 1. Introduction In this paper, we study the existence and multiplicity of positive solutions for the following nth-order nonlinear singular nonlocal boundary value problems (BVPs): u(n) (t) + a(t) f (t,u) = 0, u(0) = 0,
t ∈ (0,1),
u (0) = 0,...,u(n−2) (0) = 0,
αu(η) = u(1),
(1.1)
where 0 < η < 1, 0 < αηn−1 < 1, a may be singular at t = 0 and/or t = 1. We call a(t) singular if limt→0+ a(t) = ∞ or limt→1− a(t) = ∞. The BVPs for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory. Many authors have discussed the existence of solutions of second-order or higher-order BVPs, for instance, [1–4]. Singular BVPs have also been widely studied because of their importance in both practical and theoretical aspects. In many practical problems, it is frequent that only positive solutions are useful. There have been many papers available in literature concerning the positive solutions of singular BVPs, see [5–9] and references therein. The
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Boundary Value Problems
study of singular nonlocal BVPs for nonlinear differential equations was initiated by Kiguradze and Lomtatidze [10] and Lomtatidze [11, 12]. Since then, more general nonlinear singular nonlocal BVPs have been studied extensively. Recently, Eloe and Ahmad [13] studied the positive solutions for the nth-order differential equation u(n) (t) + a(t) f (u) = 0,
t ∈ (0,1),
(1.2)
subject to the nonlocal boundary conditions u(0) = 0,
u (0) = 0,...,u(n−2) (0) = 0,
αu(η) = u(1),
(1.3)
where 0 < η < 1, 0 < αηn−1 < 1. For the case in which a is nonsingular, Eloe and Ahmad established the existence of one positive solution for BVPs (1.2) and (1.3) if f is either superlinear (i.e., limu→0+ ( f (u)/u) = 0, limu→∞ ( f (u)/u) = ∞) or sublinear (i.e., limu→0+ ( f (u)/u) = ∞, limu→∞ ( f (u)/u) = 0) by applying the fixed point theorem on cones duo to Krasnosel’skii and Guo. However, research for existence of multiple positive solutions for higher-order singular nonlocal BVPs has proceeded very slowly and the related results are very limited. Motivated by the
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