Existence and Uniqueness of Solutions for Singular Higher Order Continuous and Discrete Boundary Value Problems
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Research Article Existence and Uniqueness of Solutions for Singular Higher Order Continuous and Discrete Boundary Value Problems Chengjun Yuan,1, 2 Daqing Jiang,1 and You Zhang1 1 2
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China School of Mathematics and Computer, Harbin University, Harbin 150086, Heilongjiang, China
Correspondence should be addressed to Chengjun Yuan, [email protected] Received 4 July 2007; Accepted 31 December 2007 Recommended by Raul Manasevich By mixed monotone method, the existence and uniqueness are established for singular higher-order continuous and discrete boundary value problems. The theorems obtained are very general and complement previous known results. Copyright q 2008 Chengjun Yuan 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 recent years, the study of higher-order continuous and discrete boundary value problems has been studied extensively in the literature see 1–17 and their references. Most of the results told us that the equations had at least single and multiple positive solutions. Recently, some authors have dealt with the uniqueness of solutions for singular higherorder continuous boundary value problems by using mixed monotone method, for example, see 6, 14, 15. However, there are few works on the uniqueness of solutions for singular discrete boundary value problems. In this paper, we state a unique fixed point theorem for a class of mixed monotone operators, see 6, 14, 18. In virtue of the theorem, we consider the existence and uniqueness of solutions for the following singular higher-order continuous and discrete boundary value problems 1.1 and 1.2 by using mixed monotone method. We first discuss the existence and uniqueness of solutions for the following singular higher-order continuous boundary value problem y n tλqtgyhy 0, y i 0 y n−2 1 0,
0 < t < 1, λ > 0,
0 ≤ i ≤ n − 2,
1.1
2
Boundary Value Problems
where n ≥ 2, qt ∈ C0, 1, 0, ∞, g : 0, ∞ → 0, ∞ is continuous and nondecreasing; h : 0, ∞ → 0, ∞ is continuous and nonincreasing, and h may be singular at y 0. Next, we consider the existence and uniqueness of solutions for the following singular higher-order discrete boundary value problem Δn yiλqin−1gyin−1 hyin−1 0, Δk y0 Δn−2 yT 1 0,
i ∈ N {0, 1, 2, . . . , T − 1}, λ > 0,
0 ≤ k ≤ n − 2, 1.2
where n ≥ 2, N {0, 1, 2, . . . , T n}, qi ∈ CN , 0, ∞, g : 0, ∞ → 0, ∞ is continuous and nondecreasing; h : 0, ∞ → 0, ∞ is continuous and nonincreasing, and h may be singular at y 0. Throughout this paper, the topology on N will be the discrete topology. 2. Preliminaries Let P be a normal cone of a Banach space E, and e ∈ P with e ≤ 1, e / θ. Define Qe {x ∈ P | x / θ, there exist constants m, M >
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