Dynamic Systems and Applications 15 (2006) 173-192

EIGENVALUE CHARACTERIZATION FOR A SYSTEM OF THIRD-ORDER GENERALIZED RIGHT FOCAL PROBLEMS

PATRICIA J. Y. WONG

School of Electrical and Electronic Engineering Nanyang Technological University 50 Nanyang Avenue, Singapore 639798, Singapore. E-mail: ejywong@ntu.edu.sg

Dedicated to John Baxley on the occasion of his retirement from Wake Forest University.

ABSTRACT. We consider the following system of third-order three-point generalized right focal boundary value problems,

\begin{matrix}u′′′_{i}(t) = λf_{i}(t, u_{1}(t), u_{2}(t), ⋯, u_{n}(t)), t ∈ [a, b], \\ u_{i}(a) = u′_{i}(t_{i}) = 0, γ_{i}u_{i}(b) + δ_{i}u′′_{i}(b) = 0, \end{matrix}

where λ > 0, i = 1, 2, ⋯, n, a < t_{i} < b, γ_{i} ≥ 0 and δ_{i} > 0. Criteria will be developed so that the above system has a constant-sign solution for values of λ that form an interval (bounded or unbounded). Explicit intervals for such λ will also be presented.

AMS (MOS) Subject Classification. 34B15, 34B18.

1. INTRODUCTION

In this paper we shall consider the following system of third-order three-point generalized right focal boundary value problems

\{ \begin{matrix}u′′′_{i}(t) = λf_{i}(t, u_{1}(t), u_{2}(t), ⋯, u_{n}(t)), t ∈ [a, b], \\ u_{i}(a) = u′_{i}(t_{i}) = 0,  γ_{i}u_{i}(b) + δ_{i}u′′_{i}(b) = 0, \end{matrix}  (1.1)

where λ > 0 and i = 1, 2, ⋯, n. For each 1 ≤ i ≤ n, it is assumed that a < t_{i} < b with t_{i} − a > b − t_{i}, and

γ_{i} ≥ 0,  δ_{i} > 0,  η_{i} ≡ 2δ_{i} + γ_{i}(b − a)(b − 2t_{i} + a) > 0.

A solution u = (u_{1}, u_{2}, ⋯, u_{n}) of (1.1) will be sought in (C[a, b])^{n} = C[a, b] × C[a, b] × ⋯ × C[a, b] (n times). We say that u is a solution of constant sign if, for each 1 ≤ i ≤ n, we have θ_{i}u_{i}(t) ≥ 0 for t ∈ [a, b], where θ_{i} ∈ {1, −1} is fixed.

Received October 26, 2004               1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

174 PATRICIA J. Y. WONG

If, for a particular λ, the system (1.1) has a constant-sign solution u = (u_{1}, u_{2}, ⋯, u_{n}), then λ is called an eigenvalue, and u a corresponding eigenfunction of the system. Let E be the set of eigenvalues, i.e.,

E = {λ | λ > 0 such that the system (1.1) has a constant-sign solution}.

We shall characterize those values of λ for which the system (1.1) has a constantsign solution. In particular, we shall establish criteria for E to contain an interval, and to be an interval (which may either be bounded or unbounded). Upper and lower bounds for an eigenvalue λ are also established. In addition, explicit subintervals of E are derived.

Eigenvalue characterization for the two-point right focal boundary value problem

\{ \begin{matrix}(−1)^{3−k}y′′′(t) = λf(t, y(t)), t ∈ [0, 1] \\ y^{(j)}(0) = 0, 0 ≤ j ≤ k − 1; y^{(j)}(1) = 0, k ≤ j ≤ 2 \end{matrix}  (1.2)

where k ∈ {1, 2}, has been well discussed in the literature [1, 4]. The related higher order discrete problem can be found in [10, 19–21]. Work on a three-point right focal problem, a special case of (1.1) when n = 1, δ_{i} = 1, γ_{i} = 0, is available in [7]. Recently, Anderson [6] considered (1.1) when n = λ = 1 and developed the Green’s function for the boundary value problem. Existence criteria for multiple constantsign solutions of (1.1) when λ = 1 have very recently been established in [18]. In our present work, we generalize the boundary value problem considered in [6] to a system, with a general nonlinear term, and consider the eigenvalue problem where the eigenfunctions concerned are of constant sign. It is noted that the definition of constant-sign solution includes the usual positive solution as a particular case, which has been the usual consideration in the literature; see for example [2–4, 8, 9, 11–14, 16, 19–21] and the references cited therein.

The plan of the paper is as follows. In Section 2, we shall state Krasnosel’skii’s fixed-point theorem which is crucial in establishing subintervals of E. In Section 3, we develop criteria for E to contain an interval, and for E to be an interval, which may either be bounded or unbounded. Moreover, upper and lower bounds are established for an eigenvalue λ. Explicit subintervals of E are derived in Section 4. Finally, we present an example in Section 5 to illustrate the importance of the results obtained.

2. PRELIMINARIES

The following theorem will be needed. It is usually called Krasnosel’skii’s fixed point theorem in a cone.

EIGENVALUE CHARACTERIZATION 175

Theorem 2.1. [15] Let B = (B, ∥ · ∥) be a Banach space, and let C ⊂ B be a cone in B. Assume Ω_{1}, Ω_{2} are open subsets of B with 0 ∈ Ω_{1}, \overline{Ω}_{1} ⊂ Ω_{2}, and let

S : C ∩ (\overline{Ω}_{2}\Ω_{1}) → C

be a completely continuous operator such that, either

(a) ∥Su∥ ≤ ∥u∥, u ∈ C ∩ ∂Ω_{1}, and ∥Su∥ ≥ ∥u∥, u ∈ C ∩ ∂Ω_{2}, or

(b) ∥Su∥ ≥ ∥u∥, u ∈ C ∩ ∂Ω_{1}, and ∥Su∥ ≤ ∥u∥, u ∈ C ∩ ∂Ω_{2}.

Then S has a fixed point in C ∩ (\overline{Ω}_{2}\Ω_{1}).

We also require the definition of a L^{q}-Carathéodory function.

Definition 2.1. [17] A function F : [a, b] × IR^{n} → IR is a L^{q}-Carathéodory function if the following conditions hold:

(a) The map t → F (t, u) is measurable for all u ∈ IR^{n}.

(b) The map u → F (t, u) is continuous for almost all t ∈ [a, b].

(c) For any r > 0, there exists μ_{r} ∈ L^{q}[a, b] such that |u| ≤ r implies that |F (t, u)| ≤ μ_{r}(t), for almost all t ∈ [a, b].

3. EIGENVALUE CHARACTERIZATION

Let g_{i}(t, s) be the Green’s function of the boundary value problem

\{ \begin{matrix}y′′′(t) = 0, t ∈ [a, b], \\ y(a) = y′(t_{i}) = 0,  γ_{i}y(b) + δ_{i}y′′(b) = 0. \end{matrix}

It is known [6] that for t, s ∈ [a, b],

g_{i}(t, s) = \{ \begin{matrix}s ∈ [a, t_{i}] : & \{ \begin{matrix}v_{1}(t, s), & t ≤ s \\ v_{2}(t, s), & t ≥ s \end{matrix} \\ s ∈ [t_{i}, b] : & \{ \begin{matrix}v_{3}(t, s), & t ≤ s \\ v_{4}(t, s), & t ≥ s \end{matrix} \end{matrix}  (3.1)

where

v_{1}(t, s) = \frac{t − a}{2}(2s − t − a) + \frac{γ_{i}(t − a)}{2η_{i}}(s − a)^{2}(2t_{i} − a − t),

v_{2}(t, s) = \frac{(s − a)^{2}}{2η_{i}} [η_{i} + γ_{i}(t − a)(2t_{i} − a − t)],

v_{3}(t, s) = \frac{t − a}{2η_{i}}(2t_{i} − a − t)[2δ_{i} + γ_{i}(b − s)^{2}],

v_{4}(t, s) = \frac{t − a}{2η_{i}}(2t_{i} − a − t)[2δ_{i} + γ_{i}(b − s)^{2}] + \frac{(t − s)^{2}}{2} .

Moreover, g_{i}(t, s) has the following properties:

g_{i}(t, s) ≥ 0, t, s ∈ [a, b];  g_{i}(t, s) > 0, t, s ∈ (a, b]  (3.2)

g_{i}(t, s) ≤ g_{i}(t_{i}, s), t, s ∈ [a, b]  (3.3)

\{ \begin{matrix}for fixed h_{i} satisfying 0 < h_{i} < b − t_{i}, 1 ≤ i ≤ n, we have \\ g_{i}(t, s) ≥ M_{i}g_{i}(t_{i}, s), t ∈ [t_{i} − h_{i}, t_{i} + h_{i}], s ∈ [a, b], \\ where M_{i} = \frac{(t_{i} − a + h_{i})(t_{i} − a − h_{i})}{(t_{i} − a)^{2}} ∈ (0, 1). \end{matrix}  (3.4)

Throughout, we shall denote u = (u_{1}, u_{2}, ⋯, u_{n}). Let the Banach space B = (C[a, b])^{n} be equipped with the norm

∥u∥ = max_{1≤i≤n} sup_{t∈[a,b]} |u_{i}(t)| = max_{1≤i≤n} |u_{i}|_{0}

where we let |u_{i}|_{0} = sup_{t∈[a,b]} |u_{i}(t)|, 1 ≤ i ≤ n.

For clarity, we shall list the conditions used later as follows. In these conditions, the numbers θ_{i} ∈ {1, −1}, 1 ≤ i ≤ n are fixed, and the sets K, K̃ are defined by

K̃ = {u ∈ B | θ_{i}u_{i}(t) ≥ 0, t ∈ [a, b], 1 ≤ i ≤ n}

and

K = {u ∈ K̃ | θ_{j}u_{j}(t) > 0 for some j ∈ {1, 2, ⋯, n} and some t ∈ [a, b]} = K̃\{0}.

(H1) For each 1 ≤ i ≤ n, assume that f_{i} : [a, b] × IR^{n} → IR is a L^{1}-Carathéodory

function. (H2) For each 1 ≤ i ≤ n, assume that

θ_{i}f_{i}(t, u) ≥ 0, u ∈ K̃, a.e. t ∈ (a, b) and θ_{i}f_{i}(t, u) > 0, u ∈ K, a.e. t ∈ (a, b).

(H3) For each 1 ≤ i ≤ n, there exist continuous functions p_{i}, a_{i}, b_{i} with p_{i} : IR^{n} → [0, ∞) and a_{i}, b_{i} : (a, b) → [0, ∞) such that

a_{i}(t)p_{i}(u) ≤ θ_{i}f_{i}(t, u) ≤ b_{i}(t)p_{i}(u), u ∈ K̃, a.e. t ∈ (a, b).

(H4) For each 1 ≤ i ≤ n, the function a_{i} is not identically zero on any nondegenerate subinterval of (a, b), and there exists a number 0 < ρ_{i} ≤ 1 such that

a_{i}(t) ≥ ρ_{i}b_{i}(t), a.e. t ∈ (a, b).

(H5) For each 1 ≤ i, j ≤ n, if |u_{j}| ≤ |v_{j}|, then for a.e. t ∈ (a, b),

θ_{i}f_{i}(t, u_{1}, ⋯, u_{j−1}, u_{j}, u_{j+1}, ⋯, u_{n}) ≤ θ_{i}f_{i}(t, u_{1}, ⋯, u_{j−1}, v_{j}, u_{j+1}, ⋯, u_{n}).

(H6) For each 1 ≤ i, j ≤ n, if |u_{j}| ≤ |v_{j}|, then

p_{i}(u_{1}, ⋯, u_{j−1}, u_{j}, u_{j+1}, ⋯, u_{n}) ≤ p_{i}(u_{1}, ⋯, u_{j−1}, v_{j}, u_{j+1}, ⋯, u_{n}).

To begin the discussion, define the operator S : (C[a, b])^{n} → (C[a, b])^{n} by

Su(t) = (S_{1}u(t), S_{2}u(t), ⋯, S_{n}u(t)) , t ∈ [a, b],  (3.5)

where

S_{i}u(t) = λ ∫_{a}^{b} g_{i}(t, s)f_{i}(s, u(s))ds, t ∈ [a, b], 1 ≤ i ≤ n.  (3.6)

Clearly, a fixed point of the operator S is a solution of the system (1.1).

Next, with θ_{i} and h_{i} (0 < h_{i} < b − t_{i}, see (3.4)) fixed for each 1 ≤ i ≤ n, we define a cone in B = (C[a, b])^{n} as

C = {u ∈ B | for each 1 ≤ i ≤ n, θ_{i}u_{i}(t) ≥ 0 for t ∈ [a, b],
and min_{t∈[t_{i}−h_{i},t_{i}+h_{i}]} θ_{i}u_{i}(t) ≥ M_{i}ρ_{i}|u_{i}|_{0}},  (3.7)

where M_{i} and ρ_{i} are defined in (3.4) and (H4), respectively. Note that C ⊆ K̃. A fixed point of S obtained in C or K̃ will be a constant-sign solution of the system (1.1). For R > 0, we denote

C(R) = {u ∈ C | ∥u∥ ≤ R}.

Assuming that (H2) and (H3) hold and noting (3.2), it follows from (3.6) that for u ∈ K̃,

λ ∫_{a}^{b} g_{i}(t, s)a_{i}(s)p_{i}(u(s))ds ≤ θ_{i}S_{i}u(t) ≤ λ ∫_{a}^{b} g_{i}(t, s)b_{i}(s)p_{i}(u(s))ds,
t ∈ [a, b], 1 ≤ i ≤ n. (3.8)

Lemma 3.1. Let (H1) hold. Then, the operator S is continuous and completely continuous.

Proof. As in [17, Theorem 4.2.2], (H1) ensures that S is continuous and completely continuous. □

Lemma 3.2. Let (H2)–(H4) hold. Then, the operator S maps C into itself.

Proof. Let u ∈ C. From (3.8) we have for t ∈ [a, b] and 1 ≤ i ≤ n,

θ_{i}S_{i}u(t) ≥ λ ∫_{a}^{b} g_{i}(t, s)a_{i}(s)p_{i}(u(s))ds ≥ 0.  (3.9)

Next, using (3.8) and (3.3) gives for t ∈ [a, b] and 1 ≤ i ≤ n,

|S_{i}u(t)| = θ_{i}S_{i}u(t) ≤ λ ∫_{a}^{b} g_{i}(t, s)b_{i}(s)p_{i}(u(s))ds ≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)p_{i}(u(s))ds.

Hence, we have

|S_{i}u|_{0} ≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)p_{i}(u(s))ds, 1 ≤ i ≤ n.  (3.10)

Now, employing (3.8), (3.4), (H4) and (3.10) we find for t ∈ [t_{i} − h_{i}, t_{i} + h_{i}] and 1 ≤ i ≤ n,

θ_{i}S_{i}u(t) ≥ λ ∫_{a}^{b} g_{i}(t, s)a_{i}(s)p_{i}(u(s))ds
≥ λ∫_{a}^{b} M_{i}g_{i}(t_{i}, s)a_{i}(s)p_{i}(u(s))ds
≥ λ∫_{a}^{b} M_{i}g_{i}(t_{i}, s)ρ_{i}b_{i}(s)p_{i}(u(s))ds
≥ M_{i}ρ_{i}|S_{i}u|_{0}.

This leads to

min_{t∈[t_{i}−h_{i},t_{i}+h_{i}]} θ_{i}S_{i}u(t) ≥ M_{i}ρ_{i}|S_{i}u|_{0}, 1 ≤ i ≤ n.  (3.11)

Inequalities (3.9) and (3.11) imply that Su ∈ C. □

Our first main result provides conditions for E to contain an interval. Theorem 3.1. Let (H1)–(H4) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. Then, there exists c > 0 such that the interval (0, c] ⊆ E. Proof. Let R > 0 be given. Define

c = R \{[ max_{1≤k≤n} sup_{\begin{matrix}|u_{j}| ≤ R \\ 1 ≤ j ≤ n \end{matrix}} p_{k}(u_{1}, u_{2}, ⋯, u_{n})] ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds\}^{−1} .  (3.12)

Let λ ∈ (0, c]. We shall prove that S(C(R)) ⊆ C(R). To begin, let u ∈ C(R). By Lemma 3.2, we have Su ∈ C. Thus, it remains to show that ∥Su∥ ≤ R. Using (3.10) and (3.12), we get for t ∈ [a, b] and 1 ≤ i ≤ n,

|S_{i}u(t)| = θ_{i}S_{i}u(t)
≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)p_{i}(u(s))ds
≤ λ[ sup_{\begin{matrix}|u_{j}| ≤ R \\ 1 ≤ j ≤ n \end{matrix}} p_{i}(u_{1}, u_{2}, ⋯, u_{n})] ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds

≤ λ[ max_{1≤k≤n} sup_{\begin{matrix}|u_{j}| ≤ R \\ 1 ≤ j ≤ n \end{matrix}} p_{k}(u_{1}, u_{2}, ⋯, u_{n})] ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds

≤ c[ max_{1≤k≤n} sup_{\begin{matrix}|u_{j}| ≤ R \\ 1 ≤ j ≤ n \end{matrix}} p_{k}(u_{1}, u_{2}, ⋯, u_{n})] ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds = R.

It follows immediately that

∥Su∥ ≤ R.

Thus, we have shown that S(C(R)) ⊆ C(R). Also, from Lemma 3.1 the operator S is continuous and completely continuous. Schauder’s fixed point theorem guarantees that S has a fixed point in C(R). Clearly, this fixed point is a constant-sign solution of (1.1) and therefore λ is an eigenvalue of (1.1). Since λ ∈ (0, c] is arbitrary, we have proved that the interval (0, c] ⊆ E. □

We shall next establish criteria for E to be an interval. For this, we require the following lemma.

Lemma 3.3. Let (H1), (H2) and (H5) hold. Suppose that λ^{∗} ∈ E. Then, for any λ ∈ (0, λ^{∗}), we have λ ∈ E, i.e., (0, λ^{∗}] ⊆ E.

Proof. Let u^{∗} = (u_{1}^{∗}, u_{2}^{∗}, ⋯, u_{n}^{∗}) be the eigenfunction corresponding to the eigenvalue λ^{∗}. Thus, we have

u_{i}^{∗}(t) = S_{i}u^{∗}(t) = λ^{∗} ∫_{a}^{b} g_{i}(t, s)f_{i}(s, u^{∗}(s))ds, t ∈ [a, b], 1 ≤ i ≤ n.  (3.13)

Define

K^{∗} = {u ∈ K̃ | for each 1 ≤ i ≤ n, θ_{i}u_{i}(t) ≤ θ_{i}u_{i}^{∗}(t), t ∈ [a, b]}.

For u ∈ K^{∗} and λ ∈ (0, λ^{∗}), applying (3.2), (H2) and (H5) yields

\begin{aligned}0 & ≤ & θ_{i}S_{i}u(t) = θ_{i} [λ ∫_{a}^{b} g_{i}(t, s)f_{i}(s, u(s))ds] \\ & ≤ & θ_{i} [λ^{∗} ∫_{a}^{b} g_{i}(t, s)f_{i}(s, u^{∗}(s))ds] = θ_{i}S_{i}u^{∗}(t), t ∈ [a, b], 1 ≤ i ≤ n \end{aligned}

where the last equality follows from (3.13). This immediately implies that the operator S defined by (3.5) maps K^{∗} into K^{∗}. Moreover, from Lemma 3.1 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 constant-sign solution of (1.1). Hence, λ is an eigenvalue, i.e., λ ∈ E. □

Theorem 3.2. Let (H1), (H2) and (H5) hold. If E ≠ ∅, then E is an interval.

Proof. Suppose E is not an interval. Then, there exist λ_{0}, λ′_{0} ∈ E (λ_{0} < λ′_{0}) and τ ∈ (λ_{0}, λ′_{0}) with τ ∉ E. However, this is not possible as Lemma 3.3 guarantees that τ ∈ E. Hence, E is an interval. □

Our next result provides upper and lower bounds for an eigenvalue λ.

Theorem 3.3. Fix h_{i} ∈ (0, b − t_{i}), 1 ≤ i ≤ n. Let (H2)–(H4) and (H6) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. Suppose that λ is an eigenvalue of (1.1) and u ∈ C is a corresponding eigenfunction. Let q_{i} = |u_{i}|_{0}, 1 ≤ i ≤ n. Then, for each 1 ≤ i ≤ n, we have

λ ≥ \frac{q_{i}}{p_{i}(q_{1}, q_{2}, ⋯, q_{n})} [∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds]^{−1}  (3.14)

and

λ ≤ \frac{q_{i}}{p_{i}(M_{1}ρ_{1}q_{1}, M_{2}ρ_{2}q_{2}, ⋯, M_{n}ρ_{n}q_{n})} [∫_{t_{i}−h_{i}}^{t_{i}+h_{i}} M_{i}g_{i}(t_{i}, s)a_{i}(s)ds]^{−1} .  (3.15)

Proof. First, we shall prove (3.14). For each 1 ≤ i ≤ n, let x_{i}^{∗} ∈ [a, b] be such that

q_{i} = |u_{i}|_{0} = θ_{i}u_{i}(x_{i}^{∗}), 1 ≤ i ≤ n.

Then, in view of (3.8), (3.3) and (H6), we find

q_{i} = θ_{i}u_{i}(x_{i}^{∗}) = θ_{i}S_{i}u(x_{i}^{∗})
= θ_{i}λ ∫_{a}^{b} g_{i}(x_{i}^{∗}, s)f_{i}(s, u(s))ds
≤ λ ∫_{a}^{b} g_{i}(x_{i}^{∗}, s)b_{i}(s)p_{i}(u(s))ds
≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)p_{i}(q_{1}, q_{2}, ⋯, q_{n})ds

from which (3.14) is immediate.

Next, to verify (3.15), we employ (3.8), (3.4), (H6) and the fact that

min_{t∈[t_{i}−h_{i},t_{i}+h_{i}]} θ_{i}u_{i}(t) ≥ M_{i}ρ_{i}|u_{i}|_{0} = M_{i}ρ_{i}q_{i}

to get

q_{i} = |u_{i}|_{0} ≥ θ_{i}u_{i}(t_{i})
= θ_{i}λ ∫_{a}^{b} g_{i}(t_{i}, s)f_{i}(s, u(s))ds
≥ λ∫_{a}^{b} g_{i}(t_{i}, s)a_{i}(s)p_{i}(u(s))ds
≥ λ∫_{t_{i}−h_{i}}^{t_{i}+h_{i}} M_{i}g_{i}(t_{i}, s)a_{i}(s)p_{i}(u(s))ds
≥ λ∫_{t_{i}−h_{i}}^{t_{i}+h_{i}} M_{i}g_{i}(t_{i}, s)a_{i}(s)p_{i}(M_{1}ρ_{1}q_{1}, M_{2}ρ_{2}q_{2}, ⋯, M_{n}ρ_{n}q_{n})ds

which reduces to (3.15). □

We are now ready to establish criteria for E to be a bounded/unbounded interval.

Theorem 3.4. Let (H1)–(H6) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. For each 1 ≤ i ≤ n, define

F_{i}^{B} = \{p : IR^{n} → [0, ∞) | \frac{|u_{i}|}{p(u_{1}, u_{2}, ⋯, u_{n})} is bounded for u ∈ IR^{n}\} ,

F_{i}^{0} = \{p : IR^{n} → [0, ∞) | lim_{min_{1≤j≤n} |u_{j}|→∞} \frac{|u_{i}|}{p(u_{1}, u_{2}, ⋯, u_{n})} = 0\} ,

F_{i}^{∞} = \{p : IR^{n} → [0, ∞) | lim_{min_{1≤j≤n} |u_{j}|→∞} \frac{|u_{i}|}{p(u_{1}, u_{2}, ⋯, u_{n})} = ∞\} .

(a) If p_{i} ∈ F_{i}^{B} for each 1 ≤ i ≤ n, then E = (0, c) or (0, c] for some c ∈ (0, ∞).

(b) If p_{i} ∈ F_{i}^{0} for each 1 ≤ i ≤ n, then E = (0, c] for some c ∈ (0, ∞).

(c) If p_{i} ∈ F_{i}^{∞} for each 1 ≤ i ≤ n, then E = (0, ∞).

Proof. (a) This is immediate from (3.15) and Theorem 3.2.

(b) Since F_{i}^{0} ⊆ F_{i}^{B}, 1 ≤ i ≤ n, it follows from Case (a) that E = (0, c) or (0, c] for some c ∈ (0, ∞). In particular,

c = sup E.

182 PATRICIA J. Y. WONG

Let {λ_{m}}_{m=1}^{∞} be a monotonically increasing sequence in E which converges to c, and let

{u^{m} = (u_{1}^{m}, u_{2}^{m}, ⋯, u_{n}^{m})}_{m=1}^{∞} ∈ K̃

be a corresponding sequence of eigenfunctions. Further, let q_{i}^{m} = |u_{i}^{m}|_{0}, 1 ≤ i ≤ n. Then, (3.15) together with p_{i} ∈ F_{i}^{0} implies that no subsequence of {q_{i}^{m}}_{m=1}^{∞} can diverge to infinity. Thus, there exists R_{i} > 0, 1 ≤ i ≤ n such that q_{i}^{m} ≤ R_{i}, 1 ≤ i ≤ n for all m. So u_{i}^{m} is uniformly bounded for each 1 ≤ i ≤ n. This together with Su^{m} = u^{m} (note Lemma 3.1) implies that for each 1 ≤ i ≤ n there is a subsequence of {u_{i}^{m}}_{m=1}^{∞}, relabeled as the original sequence, which converges uniformly to some u_{i} ∈ K̃_{i}, where

K̃_{i} = {y ∈ C[a, b] | θ_{i}y(t) ≥ 0, t ∈ [a, b]}.

Clearly, we have

u_{i}^{m}(t) = λ_{m} ∫_{a}^{b} g_{i}(t, s)f_{i}(s, u_{1}^{m}(s), u_{2}^{m}(s), ⋯, u_{n}^{m}(s))ds, t ∈ [a, b], 1 ≤ i ≤ n. (3.16)

Since u_{i}^{m} converges to u_{i} and λ_{m} converges to c, letting m → ∞ in (3.16) yields

u_{i}(t) = c ∫_{a}^{b} g_{i}(t, s)p_{i}(s, u_{1}(s), u_{2}(s), ⋯, u_{n}(s))ds, t ∈ [a, b], 1 ≤ i ≤ n.

Hence, c is an eigenvalue with corresponding eigenfunction u = (u_{1}, u_{2}, ⋯, u_{n}), i.e., c = sup E ∈ E. This completes the proof for Case (b).

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

λ max_{1≤i≤n} ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds ≤ \frac{1}{ϵ}.  (3.17)

By definition, if p_{i} ∈ F_{i}^{∞}, 1 ≤ i ≤ n, then there exists R = R(ϵ) > 0 such that the following holds for each 1 ≤ i ≤ n:

p_{i}(u_{1}, u_{2}, ⋯, u_{n}) < ϵ|u_{i}|, |u_{j}| ≥ R, 1 ≤ j ≤ n.  (3.18)

We shall prove that S(C(R)) ⊆ C(R). To begin, let u ∈ C(R). By Lemma 3.2, we have Su ∈ C. Thus, it remains to show that ∥Su∥ ≤ R. Using (3.10), (H6), (3.18) and (3.17), we find for t ∈ [a, b] and 1 ≤ i ≤ n,

|S_{i}u(t)| = θ_{i}S_{i}u(t)
≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)p_{i}(u(s))ds
≤ λp_{i}(R, R, ⋯, R) ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds
≤ λ(ϵR) ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds ≤ R.

EIGENVALUE CHARACTERIZATION 183

It follows that ∥Su∥ ≤ R and hence S(C(R)) ⊆ C(R). From Lemma 3.1, the operator S is continuous and completely continuous. Schauder’s fixed point theorem guarantees that S has a fixed point in C(R). Clearly, this fixed point is a constant-sign solution of (1.1) and therefore λ is an eigenvalue of (1.1). Since λ > 0 is arbitrary, we have proved that E = (0, ∞). □

4. SUBINTERVALS OF E

In this section we shall derive explicit subintervals of E. The main tool used is Theorem 2.1. For each p_{i}, 1 ≤ i ≤ n introduced in (H3), we define

\overline{p}_{0,i} = lim sup_{max_{1≤j≤n} |u_{j}|→0} \frac{p_{i}(u_{1}, u_{2}, ⋯, u_{n})}{|u_{i}|},

\underline{p}_{0,i} = lim inf_{max_{1≤j≤n} |u_{j}|→0} \frac{p_{i}(u_{1}, u_{2}, ⋯, u_{n})}{|u_{i}|},

\overline{p}_{∞,i} = lim sup_{min_{1≤j≤n} |u_{j}|→∞} \frac{p_{i}(u_{1}, u_{2}, ⋯, u_{n})}{|u_{i}|},

\underline{p}_{∞,i} = lim inf_{min_{1≤j≤n} |u_{j}|→∞} \frac{p_{i}(u_{1}, u_{2}, ⋯, u_{n})}{|u_{i}|}.

Theorem 4.1. Fix h_{i} ∈ (0, b − t_{i}), 1 ≤ i ≤ n. Let (H1)–(H4) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. If λ satisfies

σ_{1,i} < λ < σ_{2,i}, 1 ≤ i ≤ n  (4.1)

where

σ_{1,i} = [\underline{p}_{∞,i} M_{i}ρ_{i} ∫_{t_{i}−h_{i}}^{t_{i}+h_{i}} M_{i}g_{i}(t_{i}, s)a_{i}(s)ds]^{−1}

and

σ_{2,i} = [\overline{p}_{0,i} ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds]^{−1} ,

then λ ∈ E.

Proof. Let λ satisfy (4.1) and let ϵ_{i} > 0, 1 ≤ i ≤ n be such that

[(\underline{p}_{∞,i} − ϵ_{i})M_{i}ρ_{i} ∫_{t_{i}−h_{i}}^{t_{i}+h_{i}} M_{i}g_{i}(t_{i}, s)a_{i}(s)ds]^{−1} ≤ λ ≤ [(\overline{p}_{0,i} + ϵ_{i}) ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds]^{−1} ,

1 ≤ i ≤ n. (4.2)

First, we choose w > 0 so that

p_{i}(u) ≤ (\overline{p}_{0,i} + ϵ_{i})|u_{i}|, 0 < |u_{i}| ≤ w, 1 ≤ i ≤ n.  (4.3)

Let u ∈ C be such that ∥u∥ = w. Then, applying (3.10), (4.3) and (4.2) successively, we find for t ∈ [a, b] and 1 ≤ i ≤ n,

|S_{i}u(t)| = θ_{i}S_{i}u(t) ≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)p_{i}(u(s))ds
≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)(\overline{p}_{0,i} + ϵ_{i})|u_{i}(s)|ds
≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)(\overline{p}_{0,i} + ϵ_{i})∥u∥ds ≤ ∥u∥.

Hence,

∥Su∥ ≤ ∥u∥.  (4.4)

If we set Ω_{1} = {u ∈ B | ∥u∥ < w}, then (4.4) holds for u ∈ C ∩ ∂Ω_{1}.

Next, pick T > w > 0 such that

p_{i}(u) ≥ (\underline{p}_{∞,i} − ϵ_{i})|u_{i}|, |u_{i}| ≥ T, 1 ≤ i ≤ n.  (4.5)

Let u ∈ C be such that

∥u∥ = T′ ≡ max_{1≤j≤n} \frac{T}{M_{j}ρ_{j}} (> w).

Suppose ∥u∥ = |u_{z}|_{0} for some z ∈ {1, 2, ⋯, n}. Then, for s ∈ [t_{z} − h_{z}, t_{z} + h_{z}] we have

|u_{z}(s)| ≥ M_{z}ρ_{z}|u_{z}|_{0} = M_{z}ρ_{z}∥u∥ ≥ M_{z}ρ_{z} · \frac{T}{M_{z}ρ_{z}} = T,

which, in view of (4.5), yields

p_{z}(u(s)) ≥ (\underline{p}_{∞,z} − ϵ_{z})|u_{z}(s)|, s ∈ [t_{z} − h_{z}, t_{z} + h_{z}].  (4.6)

Using (3.8), (3.4), (4.6) and (4.2), we find

|S_{z}u(t_{z})| = θ_{z}S_{z}u(t_{z}) ≥ λ ∫_{a}^{b} g_{z}(t_{z}, s)a_{z}(s)p_{z}(u(s))ds
≥ λ ∫_{a}^{b} M_{z}g_{z}(t_{z}, s)a_{z}(s)p_{z}(u(s))ds
≥ λ ∫_{t_{z}−h_{z}}^{t_{z}+h_{z}} M_{z}g_{z}(t_{z}, s)a_{z}(s)p_{z}(u(s))ds
≥ λ ∫_{t_{z}−h_{z}}^{t_{z}+h_{z}} M_{z}g_{z}(t_{z}, s)a_{z}(s)(\underline{p}_{∞,z} − ϵ_{z})|u_{z}(s)|ds
≥ λ ∫_{t_{z}−h_{z}}^{t_{z}+h_{z}} M_{z}g_{z}(t_{z}, s)a_{z}(s)(\underline{p}_{∞,z} − ϵ_{z})M_{z}ρ_{z}|u_{z}|_{0} ds
= λ ∫_{t_{z}−h_{z}}^{t_{z}+h_{z}} M_{z}g_{z}(t_{z}, s)a_{z}(s)(\underline{p}_{∞,z} − ϵ_{z})M_{z}ρ_{z}∥u∥ds
≥ ∥u∥.

Therefore, |S_{z}u|_{0} ≥ ∥u∥ and this leads to

∥Su∥ ≥ ∥u∥.  (4.7)

If we set Ω_{2} = {u ∈ B | ∥u∥ < T′}, then (4.7) holds for u ∈ C ∩ ∂Ω_{2}.

Now that we have obtained (4.4) and (4.7), it follows from Theorem 2.1 and Lemma 3.1 that S has a fixed point u ∈ C ∩ (\overline{Ω}_{2}\Ω_{1}) such that w ≤ ∥u∥ ≤ T′. Since this u is a constant-sign solution of (1.1), the conclusion of the theorem follows immediately. □

The following corollary is immediate from Theorem 4.1.

Corollary 4.1. Fix h_{i} ∈ (0, b − t_{i}), 1 ≤ i ≤ n. Let (H1)–(H4) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. Then,

(σ_{1,i}, σ_{2,i}) ⊆ E, 1 ≤ i ≤ n

where σ_{1,i} and σ_{2,i} are defined in Theorem 4.1.

Corollary 4.2. Let (H1)–(H5) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. Then,

(0, max_{1≤i≤n} σ_{2,i}) ⊆ E,

where σ_{2,i} is defined in Theorem 4.1.

Proof. This is immediate from Corollary 4.1 and Lemma 3.3. □

186 PATRICIA J. Y. WONG

Theorem 4.2. Fix h_{i} ∈ (0, b − t_{i}), 1 ≤ i ≤ n. Let (H1)–(H4) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. If λ satisfies

σ_{3,i} < λ < σ_{4,i}, 1 ≤ i ≤ n  (4.8)

where

σ_{3,i} = [\underline{p}_{0,i} M_{i}ρ_{i} ∫_{t_{i}−h_{i}}^{t_{i}+h_{i}} M_{i}g_{i}(t_{i}, s)a_{i}(s)ds]^{−1}

and

σ_{4,i} = [\overline{p}_{∞,i} ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds]^{−1} ,

then λ ∈ E.

Proof. Let λ satisfy (4.8) and let ϵ_{i} > 0, 1 ≤ i ≤ n, be such that

[(\underline{p}_{0,i} − ϵ_{i})M_{i}ρ_{i} ∫_{t_{i}−h_{i}}^{t_{i}+h_{i}} M_{i}g_{i}(t_{i}, s)a_{i}(s)ds]^{−1} ≤ λ ≤ [(\overline{p}_{∞,i} + ϵ_{i}) ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds]^{−1} ,

1 ≤ i ≤ n. (4.9)

First, pick \overline{w} > 0 such that
p_{i}(u) ≥ (\underline{p}_{0,i} − ϵ_{i})|u_{i}|, 0 < |u_{i}| ≤ \overline{w}, 1 ≤ i ≤ n.  (4.10)

Let u ∈ C be such that ∥u∥ = \overline{w}. Suppose ∥u∥ = |u_{z}|_{0} for some z ∈ \{1, 2, ⋯, n\}. Employing (3.8), (3.4), (4.10) and (4.9) successively, we get

|S_{z}u(t_{z})| = θ_{z}S_{z}u(t_{z}) ≥ λ ∫_{a}^{b} g_{z}(t_{z}, s)a_{z}(s)p_{z}(u(s))ds
≥ λ ∫_{a}^{b} M_{z}g_{z}(t_{z}, s)a_{z}(s)p_{z}(u(s))ds
≥ λ ∫_{a}^{b} M_{z}g_{z}(t_{z}, s)a_{z}(s)(\underline{p}_{0,z} − ϵ_{z})|u_{z}(s)|ds
≥ λ ∫_{t_{z}−h_{z}}^{t_{z}+h_{z}} M_{z}g_{z}(t_{z}, s)a_{z}(s)(\underline{p}_{0,z} − ϵ_{z})|u_{z}(s)|ds
≥ λ ∫_{t_{z}−h_{z}}^{t_{z}+h_{z}} M_{z}g_{z}(t_{z}, s)a_{z}(s)(\underline{p}_{0,z} − ϵ_{z})M_{z}ρ_{z}|u_{z}|_{0} ds
= λ ∫_{t_{z}−h_{z}}^{t_{z}+h_{z}} M_{z}g_{z}(t_{z}, s)a_{z}(s)(\underline{p}_{0,z} − ϵ_{z})M_{z}ρ_{z}∥u∥ds ≥ ∥u∥.

Therefore, |Su_{z}|_{0} ≥ ∥u∥ and inequality (4.7) follows immediately. By setting Ω_{1} = \{u ∈ B | ∥u∥ < \overline{w}\}, we see that (4.7) holds for u ∈ C ∩ ∂Ω_{1}.

Next, choose \overline{T} > \overline{w} > 0 such that

p_{i}(u) ≤ (\overline{p}_{∞,i} + ϵ_{i})|u_{i}|, |u_{i}| ≥ \overline{T} , 1 ≤ i ≤ n.  (4.11)

For each p_{i}, 1 ≤ i ≤ n, we shall consider two cases, namely, p_{i} is bounded and p_{i} is unbounded. Let N_{b} and N_{u} be subsets of {1, 2, ⋯, n} such that

p_{i} is bounded for i ∈ N_{b},
p_{i} is unbounded for i ∈ N_{u},
N_{b} ∩ N_{u} = ∅ and N_{b} ∪ N_{u} = {1, 2, ⋯, n}.

\underline{Case 1} Suppose that p_{i} is bounded (i ∈ N_{b}). Then, there exists some R_{i} > 0 such that

p_{i}(u) ≤ R_{i}, u ∈ IR^{n}, i ∈ N_{b}.  (4.12)

We define

T′ = max_{i∈N_{b}} σ_{4,i}R_{i} ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)ds.

Let u ∈ C be such that ∥u∥ ≥ T′. Applying (3.10), (4.12) and (4.8) gives for i ∈ N_{b} and t ∈ [a, b],

|S_{i}u(t)| = θ_{i}S_{i}u(t) ≤ λ ∫_{a}^{b} g_{i}(t, s)b_{i}(s)p_{i}(u(s))ds
≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)R_{i} ds
< σ_{4,i} ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)R_{i} ds ≤ T′ ≤ ∥u∥.

It follows that for u ∈ C with ∥u∥ ≥ T′,

max_{i∈N_{b}} |S_{i}u|_{0} ≤ ∥u∥.  (4.13)

\underline{Case 2} Suppose that p_{i} is unbounded (i ∈ N_{u}). Then, there exists

T′′ > max\{\overline{T} , T′\} (> \overline{w})

such that

p_{i}(u) ≤ max_{\begin{matrix}η_{j} ∈ \{−1, 1\} \\ 1 ≤ j ≤ n \end{matrix}} p_{i}(η_{1}T′′, η_{2}T′′, ⋯, η_{n}T′′), |u_{j}| ≤ T′′, 1 ≤ j ≤ n.  (4.14)

Let u ∈ C be such that ∥u∥ = T′′. Then, successive use of (3.10), (4.14), (4.11), (3.3) and (4.9) provides for i ∈ N_{u} and t ∈ [a, b],

|S_{i}u(t)| = θ_{i}S_{i}u(t) ≤ λ ∫_{a}^{b} g_{i}(t, s)b_{i}(s)p_{i}(u(s))ds
≤ λ ∫_{a}^{b} g_{i}(t, s)b_{i}(s) max_{\begin{matrix}η_{j} ∈ \{−1, 1\} \\ 1 ≤ j ≤ n \end{matrix}} p_{i}(η_{1}T′′, η_{2}T′′, ⋯, η_{n}T′′)ds

≤ λ ∫_{a}^{b} g_{i}(t, s)b_{i}(s)(\overline{p}_{∞,i} + ϵ_{i})T′′ds
≤ λ ∫_{a}^{b} g_{i}(t_{i}, s)b_{i}(s)(\overline{p}_{∞,i} + ϵ_{i})∥u∥ds ≤ ∥u∥.

Therefore, we have for u ∈ C with ∥u∥ = T′′,

max_{i∈N_{u}} |S_{i}u|_{0} ≤ ∥u∥.  (4.15)

Combining (4.13) and (4.15), we obtain for u ∈ C with ∥u∥ = T′′,

max_{i∈N_{b}∪N_{u}} |S_{i}u|_{0} = max_{1≤i≤n} |S_{i}u|_{0} = ∥Su∥ ≤ ∥u∥.

Hence, by setting Ω_{2} = {u ∈ B | ∥u∥ < T′′}, we see that (4.4) holds for u ∈ C ∩ ∂Ω_{2}.

Having obtained (4.7) and (4.4), an application of Theorem 2.1 and Lemma 3.1 leads to the existence of a fixed point u of S in C ∩ (\overline{Ω}_{2}\Ω_{1}) such that \overline{w} ≤ ∥y∥ ≤ T′′. This u is a constant-sign solution of (1.1) and the conclusion of the theorem follows immediately. □

Theorem 4.2 leads to the following corollary.

Corollary 4.3. Fix h_{i} ∈ (0, b − t_{i}), 1 ≤ i ≤ n. Let (H1)–(H4) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. Then,

(σ_{3,i}, σ_{4,i}) ⊆ E, 1 ≤ i ≤ n,

where σ_{3,i} and σ_{4,i} are defined in Theorem 4.2.

Corollary 4.4. Let (H1)–(H5) hold and let g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[a, b], 1 ≤ i ≤ n. Then,

(0, max_{1≤i≤n} σ_{4,i}) ⊆ E,

where σ_{4,i} is defined in Theorem 4.2.

Proof. This is immediate from Corollary 4.3 and Lemma 3.3. □

EIGENVALUE CHARACTERIZATION 189

Remark 4.1. For a fixed i ∈ \{1, 2, ⋯, n\}, if p_{i} is superlinear (i.e., \overline{p}_{0,i} = 0 and \underline{p}_{∞,i} = ∞) or sublinear (i.e., \underline{p}_{0,i} = ∞ and \overline{p}_{∞,i} = 0), then we conclude from Corollaries 4.1 and 4.3 that E = (0, ∞), i.e., (1.1) has a constant-sign solution for any λ > 0. We remark that superlinearity and sublinearity conditions have also been discussed for various boundary value problems in the literature for the single equation case (n = 1); see for example [2–5, 11, 12, 20] and the references cited therein.

5. EXAMPLE

To illustrate the results obtained in the previous sections, we consider the system of boundary value problems

\{ \begin{matrix}u′′′_{1}(t) = λ \frac{(|u_{1}(t)| + 1)^{2}}{|u_{2}(t)| + 1} [t(t − 2) (t − \frac{517}{360}) + 1]^{−1} , t ∈ [0, 1] \\ u′′′_{2}(t) = λ \frac{3(|u_{2}(t)| + 1)^{2}}{|u_{1}(t)| + 1} [3t(t − 2) (t − \frac{517}{360}) + 3]^{−1} , t ∈ [0, 1] \\ u_{i}(0) = u′_{i}(0.55) = 0,  u_{i}(1) + 0.5u′′_{i}(1) = 0,  i = 1, 2. \end{matrix}  (5.1)

In this example, n = 2, [a, b] = [0, 1], t_{i} = 0.55, γ_{i} = 1, δ_{i} = 0.5, i = 1, 2,

f_{1}(t, u_{1}(t), u_{2}(t)) = \frac{(|u_{1}(t)| + 1)^{2}}{|u_{2}(t)| + 1} [t(t − 2) (t − \frac{517}{360}) + 1]^{−1}

and

f_{2}(t, u_{1}(t), u_{2}(t)) = \frac{3(|u_{2}(t)| + 1)^{2}}{|u_{1}(t)| + 1} [3t(t − 2) (t − \frac{517}{360}) + 3]^{−1} .

Clearly, (H1) is satisfied. Fix θ_{1} = θ_{2} = 1. Then, (H2) is fulfilled. Next, choose

p_{1}(u_{1}, u_{2}) = \frac{(|u_{1}| + 1)^{2}}{|u_{2}| + 1} ,

p_{2}(u_{1}, u_{2}) = \frac{3(|u_{2}| + 1)^{2}}{|u_{1}| + 1} ,

a_{1}(t) = b_{1}(t) = [t(t − 2) (t − \frac{517}{360}) + 1]^{−1} ,

a_{2}(t) = b_{2}(t) = [3t(t − 2) (t − \frac{517}{360}) + 3]^{−1} .

Then, (H3) and (H4) (with ρ_{1} = ρ_{2} = 1) are satisfied. Moreover, we have g_{i}(t_{i}, s)b_{i}(s) ∈ L^{1}[0, 1], i = 1, 2.

By Theorem 3.1, there exists c > 0 such that

(0, c] ⊆ E.  (5.2)

On the other hand, it is easy to see that

\overline{p}_{0,1} = \underline{p}_{0,1} = ∞, \overline{p}_{∞,1} = \underline{p}_{∞,1} = 1, \overline{p}_{0,2} = \underline{p}_{0,2} = ∞ and \overline{p}_{∞,2} = \underline{p}_{∞,2} = 3.

Using expression (3.1), by direct computation we get

σ_{3,1} = σ_{3,2} = 0 and σ_{4,1} = σ_{4,2} = 13.7382.  (5.3)

It follows from Corollary 4.3 that

(0, 13.7382) ⊆ E.  (5.4)

In view of (5.4), we see that (5.2) holds for any positive c less than 13.7382. Moreover, as an example, when λ = 6 ∈ (0, 13.7382) ⊆ E, the system (5.1) has a positive solution u = (u_{1}, u_{2}) given by

u_{1}(t) = u_{2}(t) = t(t − 2) (t − \frac{517}{360}) .  (5.5)

REFERENCES

[1] R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer, Dordrecht, 1998.

[2] R. P. Agarwal, M. Bohner and P. J. Y. Wong, Positive solutions and eigenvalues of conjugate boundary value problems. Proc. Edinburgh Math. Soc. (Series 2) 42 (1999), 349-374.

[3] R. P. Agarwal, J. Henderson and P. J. Y. Wong, On superlinear and sublinear (n, p) boundary value problems for higher order difference equations. Nonlinear World 4 (1997), 101-115.

[4] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer, Dordrecht, 1999.

[5] R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer, Dordrecht, 1997.

[6] D. Anderson, Green’s function for a third-order generalized right focal problem, J. Math. Anal. Appl. 288 (2003), 1-14.

[7] D. Anderson and J. Davis, Multiple solutions and eigenvalues for third order right focal boundary value problems, J. Math. Anal. Appl. 267 (2002), 135-157.

[8] J. V. Baxley and P. T. Carroll, Nonlinear boundary value problems with multiple positive solutions, Discrete Contin. Dyn. Syst., Vol suppl. (2003), 83-90.

[9] J. V. Baxley and C. R. Houmand, Nonlinear higher order boundary value problems with multiple positive solutions, J. Math. Anal. Appl. 286 (2003), 682-691.

[10] J. M. Davis, J. Henderson, K. R. Prasad and W. Yin, Eigenvalue intervals for nonlinear right focal problems, Applicable Analysis 74 (2000), 215-231.

[11] P. W. Eloe and J. Henderson, Positive solutions and nonlinear (k, n − k) conjugate eigenvalue problems, Diff. Eqns. Dyn. Sys. 6(1998), 309-317.

[12] L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), 743-748.

[13] J. R. Graef and J. Henderson, Double solutions of boundary value problems for 2mth-order differential equations and difference equations, Comput. Math. Appl. 45 (2003), 873-885.

[14] J. R. Graef, C. Qian and B. Yang, A three point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl. 287 (2003), 217-233.

[15] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.

[16] W. Lian, F. Wong and C. Yeh, On the existence of positive solutions of nonlinear second order differential equations, Proc. Amer. Math. Soc. 124 (1996), 1117-1126.

[17] D. O’Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer, Dordrecht, 1998.

[18] P. J. Y. Wong, Constant-sign solutions for a system of third-order generalized right focal problems, Preprint.

[19] P. J. Y. Wong, Two-point right focal eigenvalue problems for difference equations, Dynamic Systems and Applications 7 (1998), 345-364.

[20] P. J. Y. Wong, Positive solutions of difference equations with two-point right focal boundary conditions, J. Math. Anal. Appl. 224 (1998), 34-58.

[21] P. J. Y. Wong and R. P. Agarwal, Existence of multiple positive solutions of discrete two-point right focal boundary value problems, J. Difference Equ. Appl. 5 (1999), 517-540.