Multiple Positive Solutions for Singular Quasilinear Multipoint BVPs with the First-Order Derivative
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Research Article Multiple Positive Solutions for Singular Quasilinear Multipoint BVPs with the First-Order Derivative Weihua Jiang,1, 2 Bin Wang,3 and Yanping Guo1 1
College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018 Hebei, China College of Mathematics and Science of Information, Hebei Normal University, Shijiazhuang, 050016 Hebei, China 3 Department of Basic Courses, Hebei Professional and Technological College of Chemical and Pharmaceutical Engineering, Shijiazhuang, 050031 Hebei, China 2
Correspondence should be addressed to Weihua Jiang, [email protected] Received 28 November 2007; Accepted 1 April 2008 Recommended by Wenming Zou The existence of at least three positive solutions for differential equation φp u t gtft, ut, u t 0, under one of the following boundary conditions: u0 m−2 i1 ai uξi , ϕp u 1 m−2 m−2 m−2 i1 bi ϕp u ξi or ϕp u 0 i1 ai ϕp u ξi , u1 i1 bi uξi is obtained by using the H. Amann fixed point theorem, where ϕp s |s|p−2 s, p > 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, ai > 0, bi > 0, m−2 0 < m−2 i1 ai < 1, 0 < i1 bi < 1. The interesting thing is that gt may be singular at any point of 0,1 and f may be noncontinuous. Copyright q 2008 Weihua Jiang 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, of multiple positive solutions for differential equation ϕp u t gtf t, ut, u t 0, a. e. t ∈ 0, 1,
1.1
subject to boundary conditions: u0
m−2
ai u ξi ,
m−2 bi ϕ u ξi , ϕp u 1
i1
1.2
i1
m−2 ϕp u 0 ai ϕp u ξi , i1
u1
m−2 i1
bi u ξi ,
1.3
2
Boundary Value Problems
respectively, where ϕp s |s|p−2 s, p > 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, ai > 0, bi > 0, 0 < m−2 m−2 i1 ai < 1, 0 < i1 bi < 1, gt may be singular at any point of 0,1. The multipoint boundary value problems for ordinary differential equations arise in a variety of different areas of applied mathematics and physics. The study of the multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev 1, 2. Since then, nonlinear second-order multipoint boundary value problems have been studied by several authors. We refer the reader to 3–9 and references cited therein. Recently, in 10, Liang and Zhang studied the existence of positive solutions for differential equation
ϕu atf ut 0,
0 < t < 1,
1.4
under the boundary conditions 1.2 by using the fixed point index theory. Wang and Hou 11 investigated the multiplicity of solutions for the differential equation
ϕp u t f t, ut 0,
t ∈ 0, 1,
1.5
under the boundary conditions 1.3 by utilizing the fixed point theorem for operators on a cone. Guo et a
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