Existence of Positive Solutions for Multipoint Boundary Value Problem with -Laplacian on Time Scales
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Research Article Existence of Positive Solutions for Multipoint Boundary Value Problem with p-Laplacian on Time Scales Meng Zhang,1 Shurong Sun,1 and Zhenlai Han1, 2 1 2
School of Science, University of Jinan, Jinan, Shandong 250022, China School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
Correspondence should be addressed to Shurong Sun, [email protected] Received 11 March 2009; Accepted 8 May 2009 Recommended by Victoria Otero-Espinar We consider the existence of positive solutions for a class of second-order multi-point boundary value problem with p-Laplacian on time scales. By using the well-known Krasnosel’ski’s fixedpoint theorem, some new existence criteria for positive solutions of the boundary value problem are presented. As an application, an example is given to illustrate the main results. Copyright q 2009 Meng Zhang 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 The theory of time scales has become a new important mathematical branch since it was introduced by Hilger 1. Theoretically, the time scales approach not only unifies calculus of differential and difference equations, but also solves other problems that are a mix of stop start and continuous behavior. Practically, the time scales calculus has a tremendous potential for application, for example, Thomas believes that time scales calculus is the best way to understand Thomas models populations of mosquitoes that carry West Nile virus 2. In addition, Spedding have used this theory to model how students suffering from the eating disorder bulimia are influenced by their college friends; with the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks 2. By using the theory on time scales we can also study insect population, biology, heat transfer, stock market, epidemic models 2–6, and so forth. At the same time, motivated by the wide application of boundary value problems in physical and applied mathematics, boundary value problems for dynamic equations with p-Laplacian on time scales have received lots of interest 7–16.
2
Advances in Difference Equations
In 7, Anderson et al. considered the following three-point boundary value problem with p-Laplacian on time scales:
ϕp uΔ t
∇
ctfut 0,
ua − B0 uΔ v 0,
t ∈ a, b,
1.1
uΔ b 0,
where v ∈ a, b, f ∈ Cld 0, ∞, 0, ∞, c ∈ Cld a, b, 0, ∞, and Km x ≤ B0 x ≤ KM x for some positive constants Km , KM . They established the existence results for at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type. For the same boundary value problem, He in 8 using a new fixed point theorem due to Avery and Henderson obtained the existence results for at least two positive solutions. In 9, Sun
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