Eigenvalues of complementary Lidstone boundary value problems
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RESEARCH ARTICLE
Open Access
Eigenvalues of complementary Lidstone boundary value problems Ravi P Agarwal1,2* and Patricia JY Wong3 * Correspondence: [email protected] 1 Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363, USA Full list of author information is available at the end of the article
Abstract We consider the following complementary Lidstone boundary value problem (−1)m y(2m+1) (t) = λF(t, y(t), y (t)), y(0) = 0,
y
(2k−1)
(0) = y
(2k−1)
t ∈ (0, 1) 1≤k≤m
(1) = 0,
where l > 0. The values of l are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of l such that for any l in the interval, the existence of a positive solution of the boundary value problem is guaranteed. Some examples are also included to illustrate the results obtained. Note that the nonlinear term F depends on y’ and this derivative dependence is seldom investigated in the literature. AMS Subject Classification: 34B15. Keywords: eigenvalues, positive solutions, complementary Lidstone boundary value problems
1 Introduction In this article, we shall consider the complementary Lidstone boundary value problem (−1)m y(2m+1) (t) = λF(t, y(t), y (t)), y(0) = 0,
y
(2k−1)
(0) = y
(2k−1)
(1) = 0,
t ∈ (0, 1) 1≤k≤m
(1:1)
where m ≥ 1, l > 0, and F is continuous at least in the interior of the domain of interest. Note that the nonlinear term F involves a derivative of the dependent variable–this is seldom studied in the literature and most research articles on boundary value problems consider nonlinear terms that involve y only. We are interested in the existence of a positive solution of (1.1). By a positive solution y of (1.1), we mean a nontrivial y Î C(2m+1)(0, 1) satisfying (1.1) and y(t) ≥ 0 for t Î (0, 1). If, for a particular l the boundary value problem (1.1) has a positive solution y, then l is called an eigenvalue and y is a corresponding eigenfunction of (1.1). We shall denote the set of eigenvalues of (1.1) by E, i.e., E = {λ > 0|(1.1) has a positive solution}.
The focus of this article is eigenvalue problem, as such we shall characterize the values of l so that the boundary value problem (1.1) has a positive solution. To be specific, we shall establish criteria for E to contain an interval, and for E to be an interval © 2012 Agarwal and Wong; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Agarwal and Wong Boundary Value Problems 2012, 2012:49 http://www.boundaryvalueproblems.com/content/2012/1/49
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(which may either be bounded or unbounded). In addition explicit subintervals of E are derived. The complementary Lidstone interpolation and boundary value problems are very recently introduced in [1], and studied by Agarwal et. al. [2,3] where they consider an odd order ((2m+
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