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Open Access Research Article

Eigenvalues of complementary Lidstone boundary value problems

Ravi P Agarwal12* and Patricia JY Wong3

Author Affiliations

1 Department of Mathematics, Texas A&M University - Kingsville, Kingsville, TX 78363, USA

2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

3 School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

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Boundary Value Problems 2012, 2012:49  doi:10.1186/1687-2770-2012-49

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/49


Received:9 December 2011
Accepted:24 April 2012
Published:24 April 2012

© 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.

Abstract

We consider the following complementary Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M1">View MathML</a>

where λ > 0. The values of λ are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of λ such that for any λ 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M1">View MathML</a>

(1.1)

where m ≥ 1, λ > 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 C(2m+1)(0, 1) satisfying (1.1) and y(t) ≥ 0 for t ∈ (0, 1). If, for a particular λ the boundary value problem (1.1) has a positive solution y, then λ 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.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M2">View MathML</a>

The focus of this article is eigenvalue problem, as such we shall characterize the values of λ 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 (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+ 1)th order) differential equation together with boundary data at the odd order derivatives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M3">View MathML</a>

(1.2)

The boundary conditions (1.2) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [4-7] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an even order (2mth order) differential equation and the Lidstone boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M4">View MathML</a>

(1.3)

There is a vast literature on Lidstone interpolation and boundary value problems. The Lidstone interpolation has a long history from 1929 when Lidstone [8] introduced a generalization of Taylor's series that approximates a given function in the neighborhood of two points instead of one. Further characterization can be found in the study of [9-16]. More research on Lidstone interpolation as well as Lidstone spline is seen in [1,17-23]. On the other hand, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4,18,24-37] and the references cited therein. It is noted that in most of these studies the nonlinear terms considered do not involve derivatives of the dependent variable, only a handful of articles [30,31,34,35] tackle nonlinear terms that involve even order derivatives. In the present study, our study of the complementary Lidstone boundary value problem (1.1) where F depends on a derivative certainly extends and complements the rich literature on boundary value problems and in particular on Lidstone boundary value problems.

The plan of the article is as follows. In Section 2, we shall state a fixed point theorem due to Krasnosel'skii [38], and develop some inequalities for certain Green's function which are needed later. The characterization of the set E is presented in Section 3. Finally, in Section 4, we establish explicit subintervals of E.

2 Preliminaries

Theorem 2.1. [38] Let B be a Banach space, and let C(⊂ B) be a cone. Assume Ω1, Ω2 are open subsets of B with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M5">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M6">View MathML</a> be a completely continuous operator such that, either

(a) ∥Sy∥ ≤ ∥y∥, y C1, and Sy∥ ≥ ∥y, y C 2, or

(b) ∥Sy∥ ≥ ∥y∥, y C1, and Sy∥ ≤ ∥y∥, y C 2.

Then, S has a fixed point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M7">View MathML</a>.

To tackle the complementary Lidstone boundary value problem (1.1), let us review certain attributes of the Lidstone boundary value problem. Let gm(t, s) be the Green's function of the Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M8">View MathML</a>

(2.1)

The Green's function gm(t, s) can be expressed as [4,5]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M9">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M10">View MathML</a>

(2.3)

Further, it is known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M11">View MathML</a>

(2.4)

We also have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M12">View MathML</a>

(2.5)

The following two lemmas give the upper and lower bounds of |gm(t, s)|, they play an important role in subsequent development.

Lemma 2.1. For (t, s) ∈ [0, 1] × [0, 1], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M13">View MathML</a>

(2.6)

Proof. For (t, s) ∈ [0, 1] × [0, 1], it is clear from (2.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M14">View MathML</a>

(2.7)

Using (2.7), (2.4), and (2.5) in (2.2) yields for (t, s) ∈ [0, 1] × [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M15">View MathML</a>

(2.8)

By induction, we can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M16">View MathML</a>

(2.9)

Now (2.6) is immediate by applying (2.9) to (2.8).

Lemma 2.2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M17">View MathML</a> be given. For (t, s) ∈ [δ, 1-δ] × [0, 1], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M18">View MathML</a>

(2.10)

Proof. For (t, s) ∈ [δ, 1-δ] × [0, 1], from (2.3) we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M19">View MathML</a>

(2.11)

Then, using (2.11), (2.4), and (2.5) in (2.2), we get for (t, s) ∈ [δ, 1 - δ ] × [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M20">View MathML</a>

which, in view of (2.9), gives (2.10) immediately.

Remark 2.1. The bounds in Lemmas 2.1 and 2.2 are sharper than those given in the literature [4,5,35,37].

3 Eigenvalues of (1.1)

To tackle (1.1) we first consider the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M21">View MathML</a>

(3.1)

whose solution is simply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M22">View MathML</a>

(3.2)

Taking into account (3.1) and (3.2), the complementary Lidstone boundary value problem (1.1) reduces to the Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M23">View MathML</a>

(3.3)

If (3.3) has a positive solution x*, then by virtue of (3.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M24">View MathML</a> is a positive solution of (1.1). Hence, the existence of a positive solution of the complementary Lidstone boundary value problem (1.1) follows from the existence of a positive solution of the Lidstone boundary value problem (3.3). It is clear that an eigenvalue of (3.3) is also an eigenvalue of (1.1), thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M25">View MathML</a>

With the lemmas developed in Section 2 and a technique to handle the nonlinear term F, we shall study the eigenvalue problem (1.1) via (3.3).

For easy reference, we list below the conditions that are used later. In these conditions, f, α, and β are continuous functions with f : (0, ∞) × (0, ∞) → (0, ∞) and α, β : (0, 1) → [0, ∞).

(A1) f is nondecreasing in each of its arguments, i.e., for u, u1, u2, v, v1, v2 ∈ (0, ∞) with u1 u2 and v1 v2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M26">View MathML</a>

(A2) for t ∈ (0, 1) and u, v ∈ (0, ∞),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M27">View MathML</a>

(A3) α(t) is not identically zero on any nondegenerate subinterval of (0, 1) and there exists a0 ∈ (0, 1] such that α(t) ≥ a0β(t) for all t ∈ (0, 1);

(A4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M28">View MathML</a>;

(A5) for t ∈ (0, 1) and u, u1, u2, v, v1, v2 ∈ (0, ∞) with u1 u2 and v1 v2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M29">View MathML</a>

We shall consider the Banach space B = C[0, 1] equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M30">View MathML</a>

For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M31">View MathML</a>, let the cone Cδ be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M32">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M33">View MathML</a> (a0 is defined in (A3)). Further, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M34">View MathML</a>

Let the operator S : Cδ B be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M35">View MathML</a>

(3.4)

To obtain a positive solution of (3.3), we shall seek a fixed point of the operator S in the cone Cδ.

Further, we define the operators U, V : Cδ B by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M36">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M37">View MathML</a>

If (A2) holds, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M38">View MathML</a>

(3.5)

Lemma 3.1. Let (A1)-(A4) hold. Then, the operator S is compact on the cone Cδ.

Proof. Let us consider the case when α(t) is unbounded in a deleted right neighborhood of 0 and also in a deleted left neighborhood of 1. Clearly, β(t) is also unbounded near 0 and 1.

For n ∈ {1, 2, 3, ...}, let αn, βn : [0, 1] → [0, ∞) be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M39">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M40">View MathML</a>

Also, we define the operators Un, Vn : Cδ B by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M41">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M42">View MathML</a>

It is standard that for each n, both Un and Vn are compact operators on Cδ. Let M > 0 and x Cδ(M). For t ∈ [0, 1], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M43">View MathML</a>

By the monotonicity of f (see (A1)), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M44">View MathML</a>

(3.6)

Coupling with Lemma 2.1, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M45">View MathML</a>

The integrability of β(t) sin πt (see (A4)) ensures that Vn converges uniformly to V on Cδ(M). Hence, V is compact on Cδ. By a similar argument, we see that Un converges uniformly to U on Cδ(M) and therefore U is also compact on Cδ. It follows immediately from inequality (3.5) that the operator S is compact on Cδ.

Remark 3.1. From the proof of Lemma 3.1, we see that if the functions α and β are continuous on the close interval [0, 1], then the conditions (A1) and (A4) are not needed in Lemma 3.1.

The first result shows that E contains an interval.

Theorem 3.1. Let (A1)-(A4) hold. Then, there exists c > 0 such that the interval (0, c] ⊆ E.

Proof. Let M > 0 be given. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M46">View MathML</a>

(3.7)

Let λ ∈ (0, c]. We shall prove that S(Cδ(M)) ⊆ Cδ(M). Let x Cδ (M). First, we shall show that Sx Cδ. It is clear from (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M47">View MathML</a>

(3.8)

Further, from (3.5) and Lemma 2.1 we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M48">View MathML</a>

which leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M49">View MathML</a>

(3.9)

Now, applying (3.5), Lemma 2.2, (A3) and (3.9) successively, we find for t ∈ [δ, 1-δ],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M50">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M51">View MathML</a>

(3.10)

Inequalities (3.8) and (3.10) imply that Sx Cδ.

Next, we shall verify that ∥Sx∥ ≤ M. For this, an application of (3.5), Lemma 2.1, (3.6) and (3.7) provides

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M52">View MathML</a>

or equivalently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M53">View MathML</a>

Hence, S(Cδ(M)) ⊆ Cδ(M). Also, the standard arguments yield that S is completely continuous. By Schauder fixed point theorem, S has a fixed point in Cδ(M). Clearly, this fixed point is a positive solution of (3.3) and therefore λ is an eigenvalue of (3.3). Since λ ∈ (0, c] is arbitrary, it follows immediately that the interval (0, c] ⊆ E.

Remark 3.2. From the proof of Theorem 3.1, we see that (A2) and (A3) lead to S : Cδ Cδ.

Theorem 3.2. Let (A1)-(A5) hold. Suppose that λ* ∈ E, for any λ ∈ (0, λ*), we have λ E, i.e., (0, λ*] ⊆ E.

Proof. Let x* be the eigenfunction corresponding to the eigenvalue λ*. Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M54">View MathML</a>

(3.11)

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M55">View MathML</a>

Let λ ∈ (0, λ*) and x K*. Using (A5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M56">View MathML</a>

where the last equality follows from (3.11). This immediately implies that the operator S maps K* into K*. Moreover, the operator S is continuous and completely continuous. Schauder's fixed point theorem guarantees that S has a fixed point in K*, which is a positive solution of (3.3). Hence, λ is an eigenvalue, i.e., λ E.

The following result shows that E is an interval.

Corollary 3.1. Let (A1)-(A5) hold. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M57">View MathML</a>, then E is an interval.

Proof. Suppose E is not an interval. Then, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M59">View MathML</a> with τ E. However, this is not possible as Theorem 3.2 guarantees that τ E. Hence, E is an interval.

The following two results give the upper and lower bounds of an eigenvalue in terms of some parameters of the corresponding eigenfunction.

Theorem 3.3. Let (A1) and (A2) hold. Assume that m is odd. Let λ be an eigenvalue of (3.3) and x Cδ be a corresponding eigenfunction. If x(i) (0) = bi, i = 1, 3, ..., 2m - 1, where b2m-1 > 0, then λ satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M60">View MathML</a>

(3.12)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M61">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M62">View MathML</a>

Proof. For n ∈ {1, 2, 3, ...}, we define fn = f * ωn, where ωn is a standard mollifier [25] such that fn is Lipschitz and converges uniformly to f.

For a fixed n, let λn be an eigenvalue and xn(t), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M63">View MathML</a> be a corresponding eigenfunction of the following boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M64">View MathML</a>

(3.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M65">View MathML</a>

(3.14)

where Fn converges uniformly to F, and for u, v ∈ (0, ∞),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M66">View MathML</a>

(3.15)

(see the proof of Lemma 3.1 for the definitions of αn(t) and βn(t)).

It is clear that xn(t) is the unique solution of the initial value problem (3.13),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M67">View MathML</a>

(3.16)

First, we shall establish an upper bound for xn. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M68">View MathML</a>

we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M69">View MathML</a> is nonincreasing and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M70">View MathML</a>

(3.17)

In view of the initial conditions (3.16) and also (3.17), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M71">View MathML</a>

(3.18)

Next, an application of (3.18) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M72">View MathML</a>

By repeating the process, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M73">View MathML</a>

(3.19)

By the monotonicity of fn, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M74">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M75">View MathML</a>

Coupling with (3.13) and (3.15), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M76">View MathML</a>

(3.20)

Once again, using the initial conditions (3.16), repeated integration of (3.20) from 0 to t provides

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M77">View MathML</a>

(3.21)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M78">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M79">View MathML</a>

In order to satisfy the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M80">View MathML</a>, from inequality (3.21) it is necessary that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M81">View MathML</a>

This readily implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M82">View MathML</a>

(3.22)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M83">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M84">View MathML</a>

From (3.20) it is observed (by using the initial conditions (3.16) and repeated integration) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M85">View MathML</a> is a uniformly bounded sequence on [0, 1]. Thus, there exists a subsequence, which can be relabeled as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86">View MathML</a>, that converges uniformly (in fact, in C(2m-1)-norm) to some x on [0, 1]. We note that each xn(t) can be expressed as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M87">View MathML</a>

(3.23)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M88">View MathML</a> is a bounded sequence (from (3.22)), there is a subsequence, which can be relabeled as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M88">View MathML</a>, that converges to some λ. Then, letting n → ∞ in (3.23) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M89">View MathML</a>

This means that x(t) is an eigenfunction of (3.3) corresponding to the eigenvalue λ. Further, x(i)(0) = bi, i = 1, 3, ..., 2m - 1 and inequality (3.12) follows from (3.22) immediately.

Theorem 3.4. Let (A1)-(A4) hold. Let λ be an eigenvalue of (3.3) and x Cδ be a corresponding eigenfunction. Further, let ∥x∥ = p. Then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M90">View MathML</a>

(3.24)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M91">View MathML</a>

(3.25)

where t1 is any number in (0, 1) such that x(t1) ≠ 0.

Proof. Let t0 ∈ [0, 1] be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M92">View MathML</a>

Then, using (3.5), Lemma 2.1 and the monotonicity of f, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M93">View MathML</a>

which gives (3.24) readily.

Next, we employ (3.5), the monotonicity of f and the fact that mint∈[δ, 1-δ] x(t) ≥ γp to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M94">View MathML</a>

from which (3.25) is immediate.

The following result gives the criteria for E to be a bounded/unbounded interval.

Theorem 3.5. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M95">View MathML</a>

(a) Let (A1)-(A5) hold. If f WB, then E = (0, c) or (0, c] for some c ∈ (0, ∞).

(b) Let (A1)-(A5) hold. If f W0, then E = (0, c] for some c ∈ (0, ∞).

(c) Let (A1)-(A4) hold. If f W, then E = (0, ∞).

Proof. (a) This is immediate from (3.25) and Corollary 3.1.

(b) Since W0 WB, it follows from Case (a) that E = (0, c) or (0, c] for some c ∈ (0, ∞).

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M149">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M88">View MathML</a> be a monotonically increasing sequence in E which converges to c, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86">View MathML</a> be a corresponding sequence of eigenfunctions in the context of (3.3). Further, let pn = ∥xn∥. Then, (3.25) together with f W0 implies that no subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M96">View MathML</a> can diverge to infinity. Thus, there exists R > 0 such that pn R for all n. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86">View MathML</a> is uniformly bounded. This implies that there is a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M86">View MathML</a>, relabeled as the original sequence, which converges uniformly to some x, where x(t) ≥ 0 for t ∈ [0, 1]. Clearly, we have Sxn = xn, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M97">View MathML</a>

(3.26)

Since xn converges to x and λn converges to c, letting n → ∞ in (3.26) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M98">View MathML</a>

Hence, c is an eigenvalue with corresponding eigenfunction x, i.e., c = sup E E. This completes the proof for Case (b).

(c) Let λ > 0 be fixed. Choose ε > 0 so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M99">View MathML</a>

(3.27)

By definition, if f W, then there exists M = M(ε) > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M100">View MathML</a>

(3.28)

We shall prove that S(Cδ(M)) ⊆ Cδ(M). Let x Cδ (M). As in the proof of Theorem 3.1, we have (3.8) and (3.10) and so Sx Cδ. Thus, it remains to show that ∥Sx∥ ≤ M. Using (3.5), Lemma 2.1, (3.6), (3.28), and (3.27), we find for t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M101">View MathML</a>

It follows that ∥Sx∥ ≤ M and hence S(Cδ(M)) ⊆ Cδ(M). Also, S is continuous and completely continuous. Schauder's fixed point theorem guarantees that S has a fixed point in Cδ(M). Clearly, this fixed point is a positive solution of (3.3) and therefore λ is an eigenvalue of (3.3). Since λ > 0 is arbitrary, we have proved that E = (0, ∞).

Example 3.1. Consider the complementary Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M102">View MathML</a>

(3.29)

where λ > 0 and q ≥ 0.

Here, m = 2 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M103">View MathML</a>

Clearly, F(t, u, v) is nondecreasing in u and v, thus (A5) is satisfied.

Choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M104">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M105">View MathML</a>

We see that (A1)-(A4) are satisfied.

Case 1. 0 ≤ q < 1. Clearly f W. It follows from Theorem 3.5(c) that the set E = (0, ∞). As an example, when λ = 24, the boundary value problem (3.29) has a positive solution given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M106">View MathML</a>.

Case 2. q = 1. Here f WB. By Theorem 3.5(a) the set E is an open or a half-closed interval. Further, from Case 1 and Theorem 3.2 we note that E contains the interval (0, 24].

Case 3. q > 1. Clearly f W0. By Theorem 3.5(b) the set E is a half-closed interval. Again, as in Case 2 we note that (0, 24] ⊆ E.

4 Eigenvalue intervals

In this section, we shall establish explicit subintervals of E. Here, the functions α and β in (A2)-(A4) are assumed to be continuous on the closed interval [0, 1]. Hence, noting Remark 3.1, we shall not require conditions (A1) and (A4) to show the compactness of the operator S. For the function f in (A2), we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M107">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M108">View MathML</a> be given. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M109">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M110">View MathML</a>

(4.1)

Theorem 4.1. Let (A2)-(A4) hold. Then, λ E if λ satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M111">View MathML</a>

(4.2)

Proof. We shall use Theorem 2.1. Let λ satisfy (4.2) and let ε > 0 be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M112">View MathML</a>

(4.3)

First, we pick p > 0 so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M113">View MathML</a>

(4.4)

Let x Cδ be such that ∥x∥ = p. Note that for s ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M114">View MathML</a>

Then, using (3.5), Lemma 2.1, (4.4) and (4.3) successively, we find for t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M115">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M116">View MathML</a>

(4.5)

If we set Ω1 = {x B |∥x∥ < p}, then (4.5) holds for x Cδ 1.

Next, let q > 0 be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M117">View MathML</a>

(4.6)

Let x Cδ be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M118">View MathML</a>

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M119">View MathML</a>

(4.7)

Then, an application of (3.5), (4.7), and (4.6) gives for t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M120">View MathML</a>

Taking supremum both sides and using (4.3) then provides (see (4.1) for the definition of t*)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M121">View MathML</a>

Therefore, if we set Ω2 = {x B| ∥x∥ < q0}, then for x Cδ 2 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M122">View MathML</a>

(4.8)

Now that we have obtained (4.5) and (4.8), it follows from Remark 3.2 and Theorem 2.1 that S has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M123">View MathML</a> such that p ≤ ∥x∥ ≤ q0. Obviously, this x is a positive solution of (3.3) and hence λ E.   □

Theorem 4.2. Let (A2)-(A4) hold. Then, λ E if λ satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M124">View MathML</a>

(4.9)

Proof. We shall apply Theorem 2.1 again. Let λ satisfy (4.9) and let ε > 0 be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M125">View MathML</a>

(4.10)

First, we choose r > 0 so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M126">View MathML</a>

(4.11)

Let x Cδ be such that ∥x∥ = r. Then, on using (3.5), (4.11), and (4.10) successively, we have for t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M127">View MathML</a>

Taking supremum both sides and using (4.10) then yields (see (4.1) for the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M128">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M129">View MathML</a>

Hence, if we set Ω1 = {y B| ∥x∥ < r}, then (4.8) holds for x Cδ 1.

Next, pick w > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M130">View MathML</a>

(4.12)

We shall consider two cases - when f is bounded and when f is unbounded.

Case 1. Suppose that f is bounded. Then, there exists some M > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M131">View MathML</a>

(4.13)

Let x Cδ be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M132">View MathML</a>

From (3.5), Lemma 2.1 and (4.13), it is clear for t ∈ [0, 1] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M133">View MathML</a>

Hence, (4.5) holds.

Case 2. Suppose that f is unbounded. Then, there exists w0 > max {r + 1, w} such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M134">View MathML</a>

(4.14)

Let x Cδ be such that ∥x∥ = w0. Then, applying (3.5), Lemma 2.1, (4.14), (4.12), and (4.10) successively gives for t ∈ [0, 1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M135">View MathML</a>

Thus, (4.5) follows immediately.

In both Cases 1 and 2, if we set Ω2 = {x B| ∥x∥ < w0}, then (4.5) holds for x Cδ 2.

Now that we have obtained (4.8) and (4.5), it follows from Remark 3.2 and Theorem 2.1 that S has a fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M123">View MathML</a> such that r ≤ ∥x∥ ≤ w0. It is clear that this x is a positive solution of (3.3) and hence λ E.

Remark 4.1. In (4.2) and (4.9), although t* and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M128">View MathML</a> can be computed from (4.1), we can circumvent the computation by giving further bounds. Indeed, applying Lemma 2.2 we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M136">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M137">View MathML</a>

The following corollary is immediate from Theorems 4.1, 4.2 and Remark 4.1.

Corollary 4.1. Let (A2)-(A4) hold. Then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M138">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M139">View MathML</a>

Remark 4.2. If f is superlinear (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M141">View MathML</a>) or sublinear (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M143">View MathML</a>), then we conclude from Corollary 4.1 that E = (0, ∞), i.e., the boundary value problem (3.3) (or (1.1)) has a positive solution for any λ > 0.

Example 4.1. Consider the complementary Lidstone boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M144">View MathML</a>

(4.15)

where λ, a, b, c > 0 and r ≤ 1.

Here, m = 2. It is clear that (A2)-(A4) are satisfied with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M145">View MathML</a>

Case 1. r < 1. It is clear that f is sublinear. Therefore, by Remark 4.2 the boundary value problem (4.15) has a positive solution for any λ > 0. In fact, we note that when λ = 120, (4.15) has a positive solution given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M146">View MathML</a>.

Case 2. r = 1, a = b = 0.5, c = 10. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M142">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M147">View MathML</a>. It follows from Corollary 4.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M148">View MathML</a>

Once again we note that when λ = 120 ∈ (0,528.99), the corresponding eigenfunction is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/49/mathml/M146">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

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