NONLINEAR EIGENVALUE PROBLEMS FOR HIGHER ORDER LIDSTONE BOUNDARY VALUE PROBLEMS

In this paper, we consider the Lidstone boundary value problem y (2m) (t) = a (t)f(y(t); : : : ; y (2j) (t); : : : y (2(m 1)) (t)), 0 0 and a is nonnegative. Growth conditions are imposed on f and inequalities involving an associated Green's function are employed which enable us to apply a well-known cone theoretic xed point theorem. This in turn yields a interval on which there exists a nontrivial solution in a cone for each in that interval. The methods of the paper are known. The emphasis here is that f depends upon higher order derivatives. Appli- cations are made to problems that exhibit superlinear or sublinear type growth.

In all the above cited works, the nonlinear term, f , only depends on position.The primary interest of this paper is that the nonlinear term, f , depends on position, acceleration and other even order derivatives of the unknown function.Recent related works in which dependence on higher order derivatives is allowed can be found in [4] or [9].
The Lidstone boundary value problem (BVP) was first studied by Lidstone [21]; Agarwal and Wong's work [3] has generated renewed interest in the problem.Recently, Davis,Henderson,and Lamar ([6], [7], [10], [19]) have studied the problem intensely.A feature of the Lidstone BVP that is exploited in this paper is that it can be analyzed as a nested family of second order conjugate BVPs.This feature has been employed by Davis, Eloe, Henderson, Islam and Thompson ( [14], [8], and [13]).The primary contribution of this paper is that this nested feature is exploited so that the methods employed by Erbe and Wang [15] can be be applyed to the BVP, (1.1), (1.2).Moreover, we indicate that the contribution is of interest by exhibiting applications to problems that exhibit superlinear or sublinear type growth.
We close the introduction with one open question.Can the methods employed here apply to a Lidstone BVP with nonlinear dependence on odd order derivatives of the unknown function?That question is completely open.The problem is that large in norm does not imply large componentwise; by exploiting the nested feature of Lidstone BVPs in this paper, large in norm will, in fact, imply large in the appropriate components.

The Fixed Point Operator
The method developed by Erbe and Wang [15] employs an application of the cone theoretic fixed point theorem that we credit to Krasnosel'skii [18].Also see [16].For simplicity we state the theorem here.
Theorem 2.1.Let B be a Banach space, and let P ⊂ B be a cone in B. Assume

and let
We now construct the fixed point operator upon which we apply the above fixed point theorem.To do so, we exploit that the Lidstone BVP, (1.1), (1.2), can be constructed as a nested sequence of second order conjugate EJQTDE, 2000 No. 2, p. 2 type BVPs.In particular, we shall construct a second order BVP that is equivalent to (1.1), (1.2).In Section 3 we shall apply the above fixed point theorem to the equivalent second order BVP.

Existence of Positive Solutions
We remind the reader of two fundamental bounds involving the Green's function, G 1 .
We remark that if f is superlinear (i.e., f 0 = 0, f ∞ = ∞) then the proof of Theorem 3.1 is readily adapted to show that the BVP, (1.1), (1.2), has a nontrivial solution, y, such that v = y (2(m−1)) belongs to P, for each 0 < λ < ∞.To illustrate that this observation is of interest, set m = 2 and consider the fourth order Lidstone BVP that relates to the cantilever beam problem.Note that each of satisfy conditions (A), (C1) and (D).
We will state without proof a second application of Theorem 2.1.The proof, when f depends only on position is standard (see [15]) and the extension to the problem addressed here is completely analogous to the extension illustrated in the proof of Theorem 3. there is at least one nontrivial solution, y, of the BVP, (1.1), (1.2), such that v = y (2(m−1)) belongs to P.