Eigenvalue Characterization for a Class of Boundary Value Problems

Chuan Jen Chyan Department of Mathematics Tamkang University Taipei, Taiwan, 251

                       email: chuanjen@mail.tku.edu.tw

Johnny Henderson Department of Mathematics Auburn University Auburn, Alabama 36849-5310 USA

                       email: hendej2@mail.auburn.edu

Abstract

We consider the nth order ordinary differential equation (−1)^{n−k}y^{(n)} = λa(t)f(y), t ∈ [0, 1], n ≥ 3 together with boundary condition y^{(i)}(0) = 0, 0 ≤ i ≤ k − 1, and y^{(l)}(1) = 0, j ≤ l ≤ j + n − k − 1, for 1 ≤ j ≤ k − 1 fixed. Values of λ are characterized so that the boundary value problem has a positive solution.

1 Introduction

Let n ≥ 3, 2 ≤ k ≤ n − 1, and 1 ≤ j ≤ k − 1 be given. In this paper we shall consider the nth order

differential equation

(−1)^{n−k}y^{(n)} = λa(t)f(y), t ∈ [0, 1],  (1)

satisfying the boundary conditions

\begin{matrix}y^{(i)}(0) = 0, 0 ≤ i ≤ k − 1, \\ y^{(l)}(1) = 0, j ≤ l ≤ j + n − k − 1. \end{matrix}  (2)

Throughout, we assume the following hypotheses :

(H_{1}) a(t) is a continuous nonnegative function on [0, 1] and is not identically equal to zero on any

subinterval of [0, 1].

(H_{2}) f : R → [0, ∞) is continuous and nonnegative.

(H_{3}) The limits f_{0} = lim_{u→0^{+}} \frac{f(u)}{u} and f_{∞} = lim_{u→∞} \frac{f(u)}{u} exist in [0, ∞).

We shall determine values of λ for which the boundary value problem (1), (2) has a positive solution. By a positive solution y of (1), (2), we mean y ∈ C^{(n)}[0, 1] satisfies (1) on [0, 1] and fulfills (2), and y is nonnegative and is not identically zero on [0, 1]. We let

Sp(a) = {λ > 0 | (1), (2) has a postive solution}.

The motivation for the present work originates from many recent investigations. In the case n = 2 the boundary value problem (1), (2) describes a vast spectrum of scientific phenomena; we refer the reader to [1, 3, 5, 6, 14, 16]. It is noted that only positive solutions are meaningful in those models. Our results complement the work of many authors, see, e.g. [2, 4, 8, 9, 10, 11, 12, 13, 17, 18, 19]. In Section 2, we provide some definitions and background results, and state a fixed point theorem due to Krasnosel’skii [15]. Also, we present some properties of certain Green’s function where needed. By defining an appropriate Banach space and cone, in Section 3, we characterize the set Sp(a).

2 Background Notation and Definitions

We first present the definition of a cone in a Banach space and the Krasnosel’skii Fixed Point Theorem. Definition 2.1. Let B be a Banach space over R. A nonempty closed convex set P ⊂ B is said to be a cone provided the following are satisfied:

(a) If y ∈ P and α ≥ 0 , then αy ∈ P;

(b) If y ∈ P and −y ∈ P , then y = 0.

Theorem 2.1 Let B be a Banach space, and let P ⊂ 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

T : P ∩ (\overline{Ω_{2}} \ Ω_{1}) → P

be a completely continuous operator such that, either

(i) ∥T u∥ ≤ ∥u∥, u ∈ P ∩ ∂Ω_{1}, and ∥T u∥ ≥ ∥u∥, u ∈ P ∩ ∂Ω_{2} ;

EJQTDE, 1999 No. 12, p. 2 (ii) ∥T u∥ ≥ ∥u∥, u ∈ P ∩ ∂Ω_{1}, and ∥T u∥ ≤ ∥u∥, u ∈ P ∩ ∂Ω_{2}. Then T has a fixed point in P ∩ (\overline{Ω_{2}} \ Ω_{1}). To obtain a solution for (1) and (2), we require a mapping whose kernel G(t, s) is the Green’s function of the boundary value problem

(−1)^{n−k}y^{(n)} = 0,  (3)

\begin{matrix}y^{(i)}(0) = 0, 0 ≤ i ≤ k − 1, \\ y^{(l)}(1) = 0, j ≤ l ≤ j + n − k − 1. \end{matrix}

Wong and Agarwal [20] have found that if y satisfies

(−1)^{n−p}y^{(n)} ≥ 0,  (4)
\begin{matrix}y^{(i)}(0) = 0, 0 ≤ i ≤ p − 1, \\ y^{(l)}(1) = 0, 0 ≤ l ≤ n − p − 1, \end{matrix}  (5)

then, for δ ∈ (0, \frac{1}{2}) and t ∈ [δ, 1 − δ],

y(t) ≥ min{b(p) min{c(p), c(n − p − 1)}, b(p − 1) min{c(p − 1), c(n − p)}}∥y∥  (6)

where the functions b and c are defined as

b(x) = \frac{(n − 1)^{n−1}}{x^{x}(n − x − 1)^{n−x−1}} , c(x) = δ^{x}(1 − δ)^{n−x−1}.

Aided by this, we have the following lemma. Lemma 2.2 Let n ≥ 3. Assume u ∈ C^{(n)}[0, 1], (−1)^{n−k}u^{(n)}(t) ≥ 0, 0 ≤ t ≤ 1 and u satifies (2). Then for 0 ≤ t ≤ 1,

u^{(j)}(t) ≥ 0

and for t ∈ [δ, 1 − δ]

u^{(j)}(t) ≥ σ_{1}|u^{(j)}|_{∞}

where

σ_{1} = min{b(k − j) min{c(k − j), c(n − k − 1)}, b(k − j − 1) min{c(k − j − 1), c(n − k)}}. EJQTDE, 1999 No. 12, p. 3 Proof: First, u^{(j)} ∈ C^{(n−j)}[0, 1]. Also u^{(j)} satisfies

(−1)^{n−k}y^{(n−j)}(t) ≥ 0.

Let the boundary condition (2) be partitioned into two parts:

\begin{matrix}y^{(i)}(0) = 0, j ≤ i ≤ k − 1 \\ y^{(l)}(1) = 0, j ≤ l ≤ j + n − k − 1 \end{matrix}  (7)

and

y^{(i)}(1) = 0, 0 ≤ i ≤ j − 1.  (8)

Now u satisfies (7), so u^{(j)} satisfies (k − j, n − k) homogeneous conjugate boundary conditions. The conclusion then follows from inequality (6).

Lemma 2.3 Let n ≥ 3. Assume u ∈ C^{(n)}[0, 1], (−1)^{n−k}y^{(n)}(t) ≥ 0, 0 ≤ t ≤ 1, and u satifies (2). Then for 0 ≤ t ≤ 1,

u(t) ≥ 0

and for t ∈ [\frac{1}{2}, 1 − δ],

u(t) ≥ σ_{2}|u^{(j)}|_{∞}

where σ_{2} = ^{\frac{σ_{1}( \frac{1}{2} −δ)^{j}}{j!}} and |u^{(j)}|_{∞} = max_{t∈[0,1]}|u^{(j)}(t)|.

Proof: Since u satisfies (2), u satisfies (8) as well. Thus for 0 ≤ t ≤ 1,

u(t) = ∫_{0}^{t} \frac{(t − s)^{j−1}}{(j − 1)!} u^{(j)}(s)ds.

Aided by Lemma 2.2

u(t) = ∫_{δ}^{t} \frac{(t − s)^{j−1}}{(j − 1)!} u^{(j)}(s)ds + ∫_{0}^{δ} \frac{(t − s)^{j−1}}{(j − 1)!} u^{(j)}(s)ds.
≥ ∫_{δ}^{t} \frac{(t − s)^{j−1}}{(j − 1)!} u^{(j)}(s)ds
≥ \frac{σ_{1}(t − δ)^{j}}{j!} |u^{(j)}|_{∞}.

Consequently, for t ∈ [\frac{1}{2}, 1 − δ],

u(t) ≥ \frac{σ_{1}(\frac{1}{2} − δ)^{j}}{j!} |u^{(j)}|_{∞}.

The nonnegativity of u follows. It is noted that Eloe [7] proved that G^{(j)}(t, s) = \frac{∂^{j}}{∂t^{j}} G(t, s) is the Green’s function of y^{(n−j)} = 0 subject to the boundary conditions

\begin{matrix}y^{(i)}(0) = 0, 0 ≤ i ≤ k − j − 1, \\ y^{(l)}(1) = 0, 0 ≤ l ≤ n − k − 1. \end{matrix}  (9)

The proof follows from the four properties of the Green’s function. Consequently we have the following result, whose conclusion follows from Lemma 2.2.

Lemma 2.4 For each s ∈ (0, 1), and t ∈ [δ, 1 − δ]

(−1)^{n−k}G^{(j)}(t, s) ≥ σ_{1}|G^{(j)}(·, s)|_{∞}

where |G^{(j)}(·, s)|_{∞} = max_{0≤t≤1}|G^{(j)}(t, s)|.

3 Main Results

We are now in a position to give some chacterization of Sp(a). Define a Banach space, B, by

B = {u ∈ C^{(j)}[0, 1]|u satisfies (8)}

with norm ∥u∥ = max_{0≤t≤1}|u^{(j)}(t)|.

Let σ = σ_{2} = \frac{σ_{1}( \frac{1}{2} −δ)^{j}}{j!} . Define a cone, P_{σ} ⊂ B, by

P_{σ} = {u ∈ B| u^{(j)}(t) ≥ 0 on[0, 1], and min_{t∈[δ,1−δ]} u(t) ≥ σ∥u∥}.

Let

T u(t) = (−1)^{n−k} ∫_{0}^{1}G(t, s)a(s)f(u(s)) ds, 0 ≤ t ≤ 1, u ∈ B.

To obtain a solution of (1), (2), we shall seek a fixed point of the operator λT in the cone P_{σ}. In order to apply the Krasnosel’skii Fixed Point Theorem, for λ > 0, we need the following.

Lemma 3.1 For λ > 0, λT : P_{σ} → P_{σ} and is a completely continuous operator.

Proof: Let u ∈ P_{σ}. It sufffices to verify this lemma when λ = 1. By properties of (−1)^{n−k}G^{(j)}(t, s), it is clear that (T u)^{(j)}(t) ≥ 0 and (T u)^{(j)}(t) is continuous on [0, 1]. Furthermore, for any 0 ≤ τ ≤ 1

min_{t∈[δ,1−δ]}(T u)^{(j)}(t) ≥ ∫_{0}^{1} min_{t∈[δ,1−δ]}(−1)^{n−k}G^{(j)}(t, s)a(s)f(u(s))ds
≥ σ ∫_{0}^{1}(−1)^{n−k}G^{(j)}(τ, s)a(s)f(u(s))ds
≥ σ ∫_{0}^{1} |G^{(j)}(·, s)|_{∞}a(s)f(u(s))ds
≥ σ∥T u∥.

Also, the standard arguments yield that λT is completely continuous.

Theorem 3.2 Assume (H_{1}), (H_{2}), and (H_{3}) with f_{0} < f_{∞} < ∞. Assume there exists a value of λ such that

λf_{0} ∫_{0}^{1} ∥G(·, s)∥a(s)ds < 1,  (10)

and

λσ^{2}f_{∞} ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds > 1.  (11)

Then the BVP (1),(2) has a positive solution in the cone P_{σ}.

Proof: For each λ > 0 satisfying both of the conditions (10) and (11), let ϵ(λ) > 0 be sufficiently small such that

λ(f_{0} + ϵ) ∫_{0}^{1} ∥G(·, s)∥a(s)ds ≤ 1,  (12)

and

λσ^{2}(f_{∞} − ϵ) ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds ≥ 1.  (13)

Consider f_{0} first. There exists H_{1}(ϵ) > 0 such that f(u) ≤ (f_{0} + ϵ)u, for all 0 < u ≤ H_{1}. Let

Ω_{1} = {u ∈ B|∥u∥ < H_{1}}.

For all u ∈ ∂Ω_{1} ∩ P_{σ}, 0 ≤ u(s) ≤ ∥u∥, and

∥λT u∥ ≤ λ ∫_{0}^{1} ∥G(·, s)∥a(s)f(u(s))ds
≤ λ ∫_{0}^{1} ∥G(·, s)∥a(s)(f_{0} + ϵ)u(s)ds
≤ λ(f_{0} + ϵ) ∫_{0}^{1} ∥G(·, s)∥a(s)ds · ∥u∥.

Hence, (12) implies that

∥λT u∥ ≤ ∥u∥.

On the other hand, consider f_{∞}. There exists H̄_{2}(ϵ) > 0 such that f(u) ≥ (f_{∞} − ϵ)u , for all u ≥ H̄_{2}. Let

H_{2} = max\{2H_{1}, \frac{1}{σ} H̄_{2}\},

Ω_{2} = {u ∈ B | ∥u∥ < H_{2}}.

For all u ∈ ∂Ω_{2} ∩ P_{σ}, u(s) ≥ σ∥u∥, \frac{1}{2} ≤ s ≤ 1 − δ, and

∥λT u∥ ≥ min_{t∈[δ,1−δ]}λT u(t)
≥ ∫_{0}^{1} min_{t∈[δ,1−δ]}(−1)^{n−k}G(t, s)a(s)f(u(s))ds
≥ λ ∫_{0}^{1} σ∥G(·, s)∥a(s)f(u(s))ds
≥ λσ ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)f(u(s))ds
≥ λσ ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)(f_{∞} − ϵ)u(s)ds
≥ λσ ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)(f_{∞} − ϵ)σ∥u∥ds
≥ λσ^{2}(f_{∞} − ϵ) ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds∥u∥.

Hence, (13) implies that

∥λT u∥ ≥ ∥u∥.

Finally, we apply part (i) of Krasnosel’skii’s Fixed Point Theorem and obtain a fixed point u_{1} of λT in P_{σ} ∩ (\overline{Ω_{2}}\Ω_{1}). Note that for \frac{1}{2} ≤ t ≤ 1 − δ,

u_{1}(t) ≥ σ∥u_{1}∥ ≥ σH_{1} > 0.

EJQTDE, 1999 No. 12, p. 7 Hence, u_{1} is a nontrivial solution of (1),(2). Successive applications of Rolle’s theorem imply that u_{1} does not vanish on (0, 1) and so u_{1} is a positive solution.

This completes the proof.

Corollary 3.3 Assume all the conditions for Theorem 3.2 hold. Then

(i) For f_{0} = 0 and f_{∞} = ∞ (superlinear), Sp(a) = (0, ∞).

(ii) For f_{0} = 0 and f_{∞} < ∞, ((σ^{2}f_{∞} ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds)^{−1}, ∞) ⊆ Sp(a).

(iii) For f_{0} > 0 and f_{∞} = ∞, (0, (f_{0} ∫_{0}^{1} ∥G(·, s)∥a(s)ds)^{−1}) ⊆ Sp(a).

(iv) For 0 < f_{0} < f_{∞} < ∞,

((σ^{2}f_{∞} ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds)^{−1}, (f_{0} ∫_{0}^{1} ∥G(·, s)∥a(s)ds)^{−1}) ⊆ Sp(a).

Theorem 3.4 Assume (H_{1}), (H_{2}) ,and (H_{3}) with f_{∞} < f_{0} < ∞. Assume there exists a value of λ such that

λσ^{2}f_{0} ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds > 1.  (14)

In addition, if f is not bounded, assume also that

λf_{∞} ∫_{0}^{1} ∥G(·, s)∥a(s)ds < 1.  (15)

Then the BVP (1),(2) has a positive solution in the cone P_{σ}.

Proof: For each λ > 0 satisfying the condition (14), let ϵ(λ) > 0 be sufficiently small such that

λσ^{2}(f_{0} − ϵ) ∫ 1−δ_{\frac{1}{2}} 1∥G(·, s)∥a(s)ds ≥ 1.  (16)

Consider f_{0} ∈ R^{+} first. There exists H_{1}(ϵ) > 0 such that f(u) ≥ (f_{0} − ϵ)u, for 0 < u ≤ H_{1}. Let

Ω_{1} = {u ∈ B | ∥u∥ < H_{1}}.

EJQTDE, 1999 No. 12, p. 8 For all u ∈ ∂Ω_{1} ∩ P_{σ}, u(s) ≥ σ∥u∥, \frac{1}{2} ≤ s ≤ 1 − δ, and so

∥λT u∥ ≥ min_{t∈[δ,1−δ]} λT u(t)
≥ λ ∫_{0}^{1} min_{t∈[δ,1−δ]}(−1)^{n−k}G(t, s)a(s)f(u(s))ds
≥ λ ∫_{0}^{1} σ∥G(·, s)∥a(s)f(u(s))ds
≥ λσ ∫_{0}^{1} ∥G(·, s)∥a(s)(f_{0} − ϵ)u(s)ds
≥ λσ(f_{0} − ϵ) ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)u(s)ds
≥ λσ(f_{0} − ϵ) ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)σ∥u∥ds
≥ λσ^{2}(f_{0} − ϵ) ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds∥u∥.

Hence, (16) implies that

∥λT u∥ ≥ ∥u∥.

On the other hand, consider f_{∞} ∈ R^{+}. Given f_{0} > f_{∞}, there are two subcases for us to consider: Case 1: f is bounded. Let λ > 0 satisfying condition (14) be given throughout this case. Let

N > 0 be large enough so that

f(u) ≤ N, for all u ≥ 0,

and

λN ∫_{0}^{1} ∥G(·, s)∥a(s)ds > H_{1}.

Let

H_{2} = λN ∫_{0}^{1} ∥G(·, s)∥a(s)ds,

and

Ω_{2} = {u ∈ B | ∥u∥ < H_{2}}.

Then, for all u ∈ ∂Ω_{2} ∩ P_{σ},

∥λT u∥ ≤ λ ∫_{0}^{1} ∥G(·, s)∥a(s)f(u(s))ds
≤ λN ∫_{0}^{1} ∥G(·, s)∥a(s)ds
= ∥u∥.

EJQTDE, 1999 No. 12, p. 9 Coupled with condition (14), we apply part (ii) of Krasnosel’skii’s Fixed Point Theorem and obtain a fixed point of λT in P_{σ} ∩ (\overline{Ω_{2}}\Ω_{1}).

Case 2: f is not bounded. Assume now that λ > 0 also satisfies the condition (15). Without loss of generality, we let the preceding ϵ also satisfy

λ(f_{∞} + ϵ) ∫_{0}^{1} ∥G(·, s)∥a(s)ds ≤ 1.  (17)

There exists H̄_{2} > 0 such that for all u ≥ H̄_{2}, f(u) ≤ (f_{∞} +ϵ)u. Since f is continuous at u = 0, it is unbounded on (0, ∞) as u approaches +∞. Let

H_{2} > max{2H_{1}, H̄_{2}}

be such that

f(u) ≤ f(H_{2})

for all 0 ≤ u ≤ H_{2}. Let

Ω_{2} = {u ∈ B | ∥u∥ < H_{2}}.

For all u ∈ ∂Ω_{2} ∩ P_{σ}, 0 ≤ s ≤ 1,

f(u(s)) ≤ f(H_{2})
≤ (f_{∞} + ϵ)H_{2},

and so,

∥λT u∥ ≤ λ ∫_{0}^{1} ∥G(·, s)∥a(s)f(u(s))ds
≤ λ ∫_{0}^{1} ∥G(·, s)∥a(s)(f_{∞} + ϵ)H_{2}ds
≤ λ(f_{∞} + ϵ) ∫_{0}^{1} ∥G(·, s)∥a(s)ds · ∥u∥.

Hence,(17) implies that

∥λT u∥ ≤ ∥u∥.

Finally, we apply part (ii) of Krasnosel’skii’s Fixed Point Theorem and obtain a fixed point u_{1} of λT in P_{σ} ∩ \overline{Ω_{2}}\Ω_{1}.

By an argument similar to that in the proof of Theorem 3.2 there is a positive solution, u_{1}, of (1), (2).

Corollary 3.5 (Case 1) Assume all the conditions for Theorem 3.4 hold and in addition that f is bounded. Then

(i) For f_{0} = 0, Sp(a) = (0, ∞).

(ii) For f_{0} < ∞, ((σ^{2}f_{0} ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds)^{−1}, ∞) ⊆ Sp(a).

Corollary 3.6 (Case 2) Assume all the conditions for Theorem 3.4 hold.Then

(i) For f_{0} = ∞ and f_{∞} = 0 (Sublinear), Sp(a) = (0, ∞).

(ii) For f_{0} = ∞ and f_{∞} > 0, (0, (f_{∞} ∫_{0}^{1} ∥G(·, s)∥a(s)ds)^{−1}) ⊆ Sp(a).

(iii) For 0 < f_{0} < ∞ and f_{∞} = 0, ((σ^{2}f_{0} ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds)^{−1}, ∞) ⊆ Sp(a).

(iv) For 0 < f_{∞} < f_{0} < ∞,

((σ^{2}f_{0} ∫ 1−δ_{\frac{1}{2}} ∥G(·, s)∥a(s)ds)^{−1}, (f_{∞} ∫_{0}^{1} ∥G(·, s)∥a(s)ds)^{−1}) ⊆ Sp(a).

References

[1] D. Aronson, M. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Analysis 6(1982), 1001-1002.

[2] N. P. Cac, A. M. Fink and J. A. Gatica, Nonnegative solutions of quasilinear elliptic boundary value problems with nonnegative coefficients, J. Math. Anal. Appl. 206(1997), 1-9.

[3] Y. S. Choi and G. S. Ludford, An unexpected stablility result of near-extinction diffusion flame for non-unity Lewis numbers, . J. Mech. Appl. Math. 42, part 1(1989), 143-158.

[4] C. J. Chyan and J. Henderson, Positive solutions for singular higher order nonlinear equations, Diff. Eqs. Dyn. Sys. 2(1994), 153-160.

[5] D. S. Cohen, Multiple stable solutions of nonlinear boundary value problems arising in chemical reactor theory, SIAM J. Appl. Math. 20(1971), 1-13.

[6] E. N. Dancer, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Differential Equations 37(1980), 404-437.

[7] P. W. Eloe, Sign properties of Green’s functions for two classes of boundary value problems, Canad. Math. Bull. 30(1987), 28-35.

[8] P. W. Eloe and J. Henderson, Positive solutions for higher order ordinary differential equations, Electronic J. Differential Equations 3(1995), 1-8.

[9] P. W. Eloe and J. Henderson, Positive solutions for (n-1, 1) boundary value problems, Nonlinear Analysis 28(1997), 1669-1680.

[10] P. W. Eloe, J. Henderson and P. J. Y. Wong, Positive solutions for two-point boundary value problems, In Dynamic Systems and Applications, Vol.2, (eds. G. S. Ladde and M. Sambandham), 135-144, Dynamic, Atlanta, GA, 1996.

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

[12] A. M. Fink, J. A. Gatica and G. E. Hernandez, Eigenvalues of generalized Gel’fand models, Nonlinear Analysis 20(1993), 1453-1468.

[13] A. M. Fink and J. A. Gatica, Positive solutions of second order systems of boundary value problems, J. Math. Anal. Appl. 180(1993), 93-108.

[14] I. M. Gel’fand, Some problems in the theory of quasilinear equations, Uspehi Mat. Nauka 14(1959), 87-158.; English translation, Translations Amer. Math. Soc. 29(1963), 295-381.

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

[16] S. Parter, Solutions of differential equations arising in chemical reactor processes, SIAM J. Appl. Math. 26(1974), 687-716.

[17] P. J. Y. Wong and R. P. Agarwal, On the existence of positive solutions of higher order difference equations, Topol. Methods Nonlinear Anal. 10(1997), 339-351.

[18] P. J. Y. Wong and R. P. Agarwal, On the eigenvalues of boundary value problems for higher order difference equations, Rocky Mountain J. Math. 28(1998), 767-791.

EJQTDE, 1999 No. 12, p. 12 [19] P. J. Y. Wong and R. P. Agarwal, Eigenvalues of boundary value problems for higher order

differential equations, Mathematical Problems in Engineering 2(1996), 401-434.

[20] P. J. Y. Wong and R. P. Agarwal, Extension of continuous and discrete inequalities due to Eloe and Henderson, Nonlinear Anal. 34(1998), 479-487.