Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 18, 1–15; http://www.math.u-szeged.hu/ejqtde/

POSITIVE SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENCE BOUNDARY VALUE PROBLEMS

Qingkai Kong and Min Wang

Department of Mathematics, Northern Illinois University DeKalb, IL 60115, USA. e-mail: kong@math.niu.edu, mwang@math.niu.edu

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract

We study a class of second order nonlinear difference boundary value problems with separated boundary conditions. A series of criteria are obtained for the existence of one, two, arbitrary number, and even an infinite number of positive solutions. A theorem for the nonexistence of positive solutions is also derived. Several examples are given to demonstrate the applications. Our results improve and supplement several results in the literature.

Key words and phrases: Second order difference boundary value problem, separated boundary condition, positive solutions, Green’s function, fixed point index theory. AMS (MOS) Subject Classifications: 39A12, 39A70

1 Introduction

In this paper, we study the existence of positive solutions of the second order boundary value problem (BVP) consisting of the difference equation

−Δ^{2}u(k) = w(k)f(k, u(k + 1)), k ∈ ℕ(0, K),  (1.1)

and the separated boundary condition (BC)

\begin{matrix}a_{11}u(0) − a_{12}Δu(0) = 0, \\ a_{21}u(K + 1) + a_{22}Δu(K + 1) = 0, \end{matrix}  (1.2)

where ℕ(0, K) = {0, 1, …, K}.

Second order difference BVPs have drawn attention in research in recent years. Results have been obtained for the existence of positive solutions for different types of BVPs, see Ma and Raffoul [12] and Pang and Ge [13] for multi-point BVPs, Tian and Ge [14] for BVPs on the half-line, and Yu, Zhu, and Guo [15] for BVPs with parameters. As regard to BVP (1.1), (1.2), Aykut and Guseinov [3] derived conditions for the existence of one positive solution. However, to the best of the knowledge of the authors, very little is known on the existence of multiple positive solutions and on the nonexistence of positive solutions.

In this paper, we will establish a series of criteria for BVP (1.1), (1.2) to have one, two, any arbitrary number, and even a countably infinite number of positive solutions. Criteria for nonexistence of positive solutions are also derived. Several examples are given to demonstrate the applications. Our results improve the results in [3] even for the existence of one positive solution.

This paper is organized as follows: After this introduction, our main results are stated in Section 2. Several examples are given in Section 3. All the proofs are given in Section 4.

2 Main Results

Throughout this paper, we assume without further mention that

(H1) a_{ij} ≥ 0 for i, j = 1, 2 such that a_{22} > 0, a_{11} + a_{12} > 0, and

b := a_{11}a_{21}(K + 1) + a_{11}a_{22} + a_{12}a_{21} > 0;  (2.1)

(H2) w(k) ≥ 0 on ℕ(0, K) such that ∑_{k=0}^{K} w(k) > 0;

(H3) for fixed k ∈ ℕ(0, K), f(k, ·) : [0, ∞) → [0, ∞) is continuous.

Define G : ℕ(0, K + 2) × ℕ(0, K) → ℝ as

G(k, l) = \frac{1}{b} \{ \begin{matrix}[a_{12} + a_{11}(l + 1)][a_{22} + a_{21}(K + 1 − k)], & 0 ≤ l ≤ k − 1, \\ (a_{12} + a_{11}k)[a_{22} + a_{21}(K − l)], & k ≤ l ≤ K, \end{matrix}  (2.2)

where b is defined by (2.1). It is known that G(k, l) is the Green’s function of the BVP consisting of the equation

−Δ^{2}u(k) = 0, k ∈ ℕ(0, K),  (2.3)

and BC (1.2), see Agarwal [1] for the detail. Let

α = min \{\frac{a_{11} + a_{12}}{a_{11}(K + 1) + a_{12}} , \frac{a_{22}}{a_{11}K + a_{22}} \} ,  (2.4)

β = max_{k∈ℕ(1,K+1)}{∑_{l=0}^{K} G(k, l)w(l)} .  (2.5)

From Aykut and Guseinov [3], we know that 0 < α < 1 and

0 < αG(l + 1, l) ≤ G(k, l) ≤ G(l + 1, l) on ℕ(1, K + 1) × ℕ(0, K).  (2.6)

Difference Boundary Value Problems 3 Definition 2.1 A function u(k) : ℕ(0, K + 2) → ℝ is said to be a positive solution of BVP (1.1), (1.2) if u(k) satisfies BVP (1.1), (1.2) and u(k) > 0 on ℕ(1, K + 1). Define

∥u∥ = max_{k∈ℕ(1,K+1)}|u(k)|.

The first theorem is our basic result on the existence of positive solutions of BVP (1.1), (1.2). Theorem 2.1 Assume there exist 0 < r_{∗} < r^{∗} [respectively, 0 < r^{∗} < r_{∗}] such that

f(k, x) ≤ β^{−1}r_{∗} for all (k, x) ∈ ℕ(0, K) × [αr_{∗}, r_{∗}]  (2.7)

and

f(k, x) ≥ β^{−1}r^{∗} for all (k, x) ∈ ℕ(0, K) × [αr^{∗}, r^{∗}].  (2.8)

Then BVP (1.1), (1.2) has at least one positive solution u with r_{∗} ≤ ∥u∥ ≤ r^{∗} [respectively, r^{∗} ≤ ∥u∥ ≤ r_{∗}].

In the sequel, we will use the following notation:

f_{0} = lim inf_{x→0} min_{k∈ℕ(0,K)}f(k, x)/x, f_{∞} = lim inf_{x→∞} min_{k∈ℕ(0,K)}f(k, x)/x;
f^{0} = lim sup_{x→0} max_{k∈ℕ(0,K)}f(k, x)/x, f^{∞} = lim sup_{x→∞} max_{k∈ℕ(0,K)}f(k, x)/x.

The next three theorems are derived from Theorem 2.1 using f_{0}, f_{∞}, f^{0}, and f^{∞}. Theorem 2.2 BVP (1.1), (1.2) has at least one positive solution if either

(a) f^{0} < β^{−1}, and f_{∞} > (αβ)^{−1}; or

(b) f_{0} > (αβ)^{−1} and f^{∞} < β^{−1}. Theorem 2.3 Assume there exists r_{∗} > 0 such that (2.7) holds.

(a) If f_{0} > (αβ)^{−1}, then BVP (1.1), (1.2) has at least one positive solution u with ∥u∥ ≤ r_{∗};

(b) if f_{∞} > (αβ)^{−1}, then BVP (1.1), (1.2) has at least one positive solution u with ∥u∥ ≥ r_{∗}.

Theorem 2.4 Assume there exists r^{∗} > 0 such that (2.8) holds.

(a) If f^{0} < β^{−1}, then BVP (1.1), (1.2) has at least one positive solution u with

∥u∥ ≤ r^{∗};

EJQTDE Spec. Ed. I, 2009 No. 18

(b) if f^{∞} < β^{−1}, then BVP (1.1), (1.2) has at least one positive solution u with ∥u∥ ≥ r^{∗}.

Combining Theorems 2.3 and 2.4 we obtain results on the existence of at least two positive solutions.

Theorem 2.5 Assume either

(a) f_{0} > (αβ)^{−1} and f_{∞} > (αβ)^{−1}, and there exists r > 0 such that

f(k, x) < β^{−1}r for all (k, x) ∈ ℕ(0, K) × [αr, r]; or  (2.9)

(b) f^{0} < β^{−1} and f^{∞} < β^{−1}, and there exists r > 0 such that

f(k, x) > β^{−1}r for all (k, x) ∈ ℕ(0, K) × [αr, r].  (2.10)

Then BVP (1.1), (1.2) has at least two positive solutions u_{1} and u_{2} with ∥u_{1}∥ < r < ∥u_{2}∥.

Note that in Theorem 2.5, the inequalities in (2.9) and (2.10) are strict and hence are different from those in (2.7) and (2.8) in Theorem 2.1. This is to guarantee that the two solutions u_{1} and u_{2} are different.

By applying Theorem 2.1 repeatedly, we can generalize the conclusions to obtain criteria for the existence of multiple positive solutions.

Theorem 2.6 Let {r_{i}}_{i=1}^{N} ⊂ ℝ such that 0 < r_{1} < r_{2} < r_{3} < ⋯ < r_{N}. Assume either

(a) f satisfies (2.9) with r = r_{i} when i is odd, and satisfies (2.10) with r = r_{i} when i is even; or

(b) f satisfies (2.9) with r = r_{i} when i is even, and satisfies (2.10) with r = r_{i} when i is odd.

Then BVP (1.1), (1.2) has at least N − 1 positive solutions u_{i} with r_{i} < ∥u_{i}∥ < r_{i+1}, i = 1, 2, …, N − 1.

Theorem 2.7 Let {r_{i}}_{i=1}^{∞} ⊂ ℝ such that 0 < r_{1} < r_{2} < r_{3} < · · ·. Assume either

(a) f satisfies (2.7) with r_{∗} = r_{i} when i is odd, and satisfies (2.8) with r^{∗} = r_{i} when i is even; or

(b) f satisfies (2.7) with r_{∗} = r_{i} when i is even, and satisfies (2.8) with r^{∗} = r_{i} when i is odd.

Then BVP (1.1), (1.2) has an infinite number of positive solutions.

The following is an immediate consequence of Theorem 2.7.

Corollary 2.1 Let {r_{i}}_{i=1}^{∞} ⊂ ℝ such that 0 < r_{1} < r_{2} < r_{3} < · · ·. Let E_{1} = ∪_{i=1}^{∞}[αr_{2i−1}, r_{2i−1}] and E_{2} = ∪_{i=1}^{∞}[αr_{2i}, r_{2i}]. Assume

lim sup_{E_{1}∋x→∞} max_{k∈ℕ(0,K)} \frac{f(k, x)}{x} < β^{−1} and lim inf_{E_{2}∋x→∞} min_{k∈ℕ(0,K)} \frac{f(k, x)}{x} > (αβ)^{−1}.

Then BVP (1.1), (1.2) has an infinite number of positive solutions.

We observe that in the above theorems, if one of f_{0}, f_{∞}, f^{0}, f^{∞} is involved and it is between β^{−1} and (αβ)^{−1}, then the corresponding conclusions fail. Motivated by the ideas in [6, 10], we employ the first eigenvalue of a Sturm-Liouville problem (SLP) associated with BVP (1.1), (1.2) and the topological homotopy invariance method to improve the criteria given in theorems 2.2-2.5.

Consider the SLP consisting of the equation

−Δ^{2}u(k) = μw(k)u(k + 1), k ∈ ℕ(0, K),  (2.11)

and BC (1.2). Let μ_{0} be its fist eigenvalue with an associated eigenfunction v_{0}(k). It is known from assumption (H1) that μ_{0} > 0, v_{0}(k) > 0 on ℕ(1, K + 1), see [2, 8, 9, 11] for the detail. Now we have

Theorem 2.8 BVP (1.1), (1.2) has at least one positive solution if f satisfies one of the following conditions:

(a) f^{0} < μ_{0} and f_{∞} > μ_{0};

(b) f_{0} > μ_{0} and f^{∞} < μ_{0};

(c) f_{0} > μ_{0} and there exists r_{∗} such that (2.7) holds;

(d) f_{∞} > μ_{0} and there exists r_{∗} such that (2.7) holds;

(e) f^{0} < μ_{0} and there exists r^{∗} such that (2.8) holds;

(f) f^{∞} < μ_{0} and there exists r^{∗} such that (2.8) holds.

Theorem 2.9 BVP (1.1), (1.2) has at least two positive solutions if either

(a) f_{0} > μ_{0}, f_{∞} > μ_{0}, and there exists r such that (2.9) holds; or

(b) f^{0} < μ_{0}, f^{∞} < μ_{0}, and there exists r such that (2.10) holds.

Remark 2.1 We claim that

β^{−1} ≤ μ_{0} ≤ (αβ)^{−1},  (2.12)

and hence Theorems 2.8 and 2.9 improve the results of Theorems 2.2-2.5. In fact, for k ∈ ℕ(1, K + 1) we have

v_{0}(k) = μ_{0} ∑_{l=0}^{K} G(k, l)w(l)v_{0}(l + 1).

Let k_{1} ∈ ℕ(1, K + 1) with v_{0}(k_{1}) = ∥v_{0}∥. Then

∥v_{0}∥ = μ_{0} ∑_{l=0}^{K} G(k_{1}, l)w(l)v_{0}(l + 1) ≤ μ_{0}β∥v_{0}∥.  (2.13)

On the other hand, we observe that w(l)v_{0}(l + 1) ≢ 0 on ℕ(0, K). For otherwise, every μ ∈ ℝ is an eigenvalue of SLP (2.11), (1.2). From (2.6) we have that for any k ∈ ℕ(1, K + 1)

v_{0}(k) ≥ μ_{0} ∑_{l=0}^{K} αG(l + 1, l)w(l)v_{0}(l + 1)

≥ αμ_{0} ∑_{l=0}^{K} G(k_{1}, l)w(l)v_{0}(l + 1) = αv_{0}(k_{1}) = α∥v_{0}∥.

Let k_{2} ∈ ℕ(1, K + 1) satisfy

∑_{l=0}^{K} G(k_{2}, l)w(l) = β.  (2.14)

Then

∥v_{0}∥ ≥ μ_{0} ∑_{l=0}^{K} G(k_{2}, l)w(l)v_{0}(l + 1) ≥ μ_{0}αβ∥v_{0}∥.  (2.15)

The combination of (2.13) and (2.15) proves our claim.

Finally, we present a result on the nonexistence of positive solutions of BVP (1.1), (1.2).

Theorem 2.10 BVP (1.1), (1.2) has no positive solutions if

(a) f(k, x)/x < β^{−1} for all (k, x) ∈ ℕ(0, K) × (0, ∞), or

(b) f(k, x)/x > (αβ)^{−1} for all (k, x) ∈ ℕ(0, K) × (0, ∞).

3 Examples

In this section, we give several examples to demonstrate the applications of the criteria obtained in Section 2. For simplicity we choose w(k) ≡ 1 and f(k, x) ≡ f(x) in all the examples.

Example 3.1 Let f(x) = x^{p}.

If p > 1, then lim_{x→0+} f(x)/x = 0 and lim_{x→∞} f(x)/x = ∞. By Theorem 2.2 (a), BVP (1.1), (1.2) has at least one positive solution.

If 0 < p < 1, then lim_{x→0+} f(x)/x = ∞ and lim_{x→∞} f(x)/x = 0. By Theorem 2.2 (b), BVP (1.1), (1.2) has at least one positive solution.

Example 3.2 Let f(x) = h(x^{p_{1}} + x^{p_{2}}), where 0 < p_{1} < 1 < p_{2} < ∞. Let r =

(\frac{1−p_{1}}{p_{2}−1})^{1/(p_{2}−p_{1})} . Then

(a) BVP (1.1), (1.2) has at least one positive solution when h = r(r^{p_{1}} + r^{p_{2}})^{−1}β^{−1};

(b) BVP (1.1), (1.2) has at least two positive solutions u_{1} and u_{2} with ∥u_{1}∥ < r < ∥u_{2}∥ when 0 < h < r(r^{p_{1}} + r^{p_{2}})^{−1}β^{−1};

(c) BVP (1.1), (1.2) has no positive solutions when h > r(r^{p_{1}} + r^{p_{2}})^{−1}(αβ)^{−1}.

In fact, it is clear that lim_{x→0+} f(x)/x = lim_{x→∞} f(x)/x = ∞, f(x) is strictly increasing, and r is the minimum point of f(x)/x on (0, ∞).

When h = r(r^{p_{1}} + r^{p_{2}})^{−1}β^{−1}, we have f(x) ≤ f(r) = β^{−1}r for all x ∈ [αr, r]. Then from Theorem 2.3 (a), BVP (1.1), (1.2) has a positive solution u_{1} with ∥u_{1}∥ ≤ r. Similarly from Theorem 2.3 (b), BVP (1.1), (1.2) has a positive solution u_{2} with ∥u_{2}∥ ≥ r. However, u_{1} and u_{2} may be the same for the case when ∥u_{1}∥ = ∥u_{2}∥ = r.

When 0 < h < r(r^{p_{1}} + r^{p_{2}})^{−1}β^{−1}, by a similar argument and from Theorem 2.5 (a), we obtain the conclusion.

When h > r(r^{p_{1}} + r^{p_{2}})^{−1}(αβ)^{−1}, f(x)/x ≥ f(r)/r > (αβ)^{−1} on (0, ∞). Then the conclusion follows from Theorem 2.10 (b).

Example 3.3 Let

f(x) = \{ \begin{matrix}h/(x^{−p_{1}} + x^{−p_{2}}), & x > 0, \\ 0, & x = 0, \end{matrix}

where h > 0, p_{1} > 1 and 0 ≤ p_{2} < 1. Let r = (\frac{p_{1}−1}{1−p_{2}})^{1/(p_{1}−p_{2})}. Then

(a) BVP (1.1), (1.2) has at least one positive solution when h = (r^{1−p_{1}}+r^{1−p2})(αβ)^{−1};

(b) BVP (1.1), (1.2) has at least two positive solutions u_{1} and u_{2} with ∥u_{1}∥ < r < ∥u_{2}∥ when h > (r^{1−p_{1}} + r^{1−p_{2}})(αβ)^{−1};

(c) BVP (1.1), (1.2) has no positive solutions when 0 < h < (r^{1−p_{1}} + r^{1−p_{2}})β^{−1}.

In fact, it is clear that lim_{x→0+} f(x)/x = lim_{x→∞} f(x)/x = 0, f(x) is strictly increasing and r is the maximum point of f(x)/x on (0, ∞).

When h = (r^{1−p_{1}} + r^{1−p_{2}})(αβ)^{−1}, f(r) = (αβ)^{−1}r. Then f(x) ≥ f(r) = (αβ)^{−1}r on [r, α^{−1}r], i.e., f(x) ≥ β^{−1}r^{∗} on [αr^{∗}, r^{∗}], where r^{∗} = α^{−1}r. By Theorem 2.4 (a) or (b), There exists at least one positive solution.

When h > (r^{1−p_{1}} + r^{1−p_{2}})(αβ)^{−1}, by a similar argument and from Theorem 2.5 (b), we obtain the conclusion.

When 0 < h < (r^{1−p_{1}} + r^{1−p_{2}})β^{−1}, f(x)/x ≤ f(r)/r < β^{−1} on (0, ∞). Then the conclusion follows from Theorem 2.10 (a).

Example 3.4 Let

f(x) = \{ \begin{matrix}(α^{−1} + 1)β^{−1}x(sin(h ln x) + 1)/2, & x > 0, \\ 0, & x = 0, \end{matrix}

where 0 < h < (π − 2 sin^{−1} δ)/ ln(α^{−1}) with δ = (α^{−1} − 1)/(α^{−1} + 1). We claim that BVP (1.1), (1.2) has an infinite number of positive solutions.

To show this, for j = 2i + 1, i ∈ ℕ, let

ξ_{j} = exp(h^{−1}(sin^{−1} δ + (j − 1)π)), η_{j} = exp(h^{−1}(jπ − sin^{−1} δ)).

Then

η_{j}/ξ_{j} = exp(h^{−1}(π − 2 sin^{−1} δ)) > exp(ln(α^{−1})) = α^{−1},

hence ξ_{j} < αη_{j}. Note that for x ∈ [αη_{j}, η_{j}] ⊂ [ξ_{j}, η_{j}], sin(h ln x) ≥ sin(sin^{−1} δ) = δ. Therefore for x ∈ [αη_{j}, η_{j}],

f(x) ≥ (α^{−1} + 1)β^{−1}αη_{j}(δ + 1)/2 = β^{−1}η_{j},

i.e., (2.8) holds with r_{∗} = η_{j}.

For j = 2i, i ∈ ℕ, let

ξ_{j} = exp(h^{−1}((j − 1)π − sin^{−1} δ)), η_{j} = exp(h^{−1}(jπ + sin^{−1} δ)).

Then

η_{j}/ξ_{j} = exp(h^{−1}(π + 2 sin^{−1} δ)) > exp(ln(α^{−1})) = α^{−1},

hence ξ_{j} < αη_{j}. Note that for x ∈ [αη_{j}, η_{j}] ⊂ [ξ_{j}, η_{j}], sin(h ln x) ≤ −δ. Therefore for x ∈ [αη_{j}, η_{j}],

f(x) ≤ (α^{−1} + 1)β^{−1}η_{j}(−δ + 1)/2 = β^{−1}η_{j},

i.e., (2.7) holds with r^{∗} = η_{j}.

Therefore by Theorem 2.7, BVP (1.1), (1.2) has an infinite number of positive solutions.

4 Proofs

Let X be a Banach space, Ω ⊂ X a cone in X, and Γ : X → X a completely continuous operator. For r > 0, define

Ω_{r} = {u ∈ Ω|∥u∥ < r} and ∂Ω_{r} = {u ∈ Ω|∥u∥ = r}.

Let 𝔦(Γ, Ω_{r}, Ω) be the fixed point index of Γ on Ω_{r} with respect to Ω. We will use the following well-known lemmas on fixed-point indices to prove our main results. For the detail of the fixed point index theory, see [5, 7].

Lemma 4.1 Assume Γu ≠ u for u ∈ ∂Ω_{r}. Then

(a) If ∥Γu∥ ≥ ∥u∥ for u ∈ ∂Ω_{r}, then 𝔦(Γ, Ω_{r}, Ω) = 0.

(b) If ∥Γu∥ ≤ ∥u∥ for u ∈ ∂Ω_{r}, then 𝔦(Γ, Ω_{r}, Ω) = 1.

Lemma 4.2 Let 0 < r_{1} < r_{2} satisfy

𝔦(Γ, Ω_{r_{1}}, Ω) = 0 and 𝔦(Γ, Ω_{r_{2}}, Ω) = 1;

or

𝔦(Γ, Ω_{r_{1}}, Ω) = 1 and 𝔦(Γ, Ω_{r_{2}}, Ω) = 0.

Then Γ has a fixed point in Ω̄_{r_{2}} \ Ω_{r_{1}}.

Define D = {u : ℕ(1, K + 1) → ℝ}, and let ∥u∥ = max_{k∈ℕ(1,K+1)} |u(k)| for u ∈ D. It is easy to see (D, ∥ · ∥) is a Banach space. Let α be defined by (2.4). Define a cone Ω in D by

Ω = {u ∈ D | u(k) ≥ 0, k ∈ ℕ(1, K + 1) and min_{k∈ℕ(1,K+1)}u(k) ≥ α∥u∥}  (4.1)

and an operator Γ : D → D by

Γu = ∑_{l=0}^{K} G(k, l)w(l)f(l, u(l + 1)), k ∈ ℕ(1, K + 1).  (4.2)

Lemma 4.3 Γ(Ω) ⊂ Ω and Γ is completely continuous.

Proof: For any u ∈ Ω, Γu ≥ 0 on ℕ(1, K + 1). By (2.6),

min_{k∈ℕ(1,K+1)}(Γu)(k) = min_{k∈ℕ(1,K+1)} ∑_{l=0}^{K} G(k, l)w(l)f(l, u(l + 1))

≥ α ∑_{l=0}^{K} G(l + 1, l)w(l)f(l, u(l + 1))

≥ α max_{k∈ℕ(1,K+1)} ∑_{l=0}^{K} G(k, l)w(l)f(l, u(l + 1)) = α∥Γu∥.

Therefore, Γ(Ω) ⊂ Ω. The complete continuity of Γ can be shown by a standard argument using the Arzela-Arscoli Theorem. We omit the details.

Proof of Theorem 2.1. We observe that BVP (1.1), (1.2) has a positive solution u(k), k ∈ ℕ(0, K + 2), if and only if the operator Γ defined by (4.2) has a positive fixed point u(k), k ∈ ℕ(1, K + 1). In fact, the fixed point of Γ can be extended to ℕ(0, K + 2) so that BC (1.2) is satisfied. Therefore, it is enough to show that Γ has a positive fixed point.

For any u ∈ ∂Ω_{r_{∗}}, ∥u∥ = r_{∗} and αr_{∗} ≤ u(k) ≤ r_{∗} on ℕ(1, K + 1). Without loss of generality, we assume Γu ≠ u. For otherwise, Γu = u implies u is a positive fixed point. From (2.7), f(k, u(k + 1)) ≤ β^{−1}r_{∗} on ℕ(0, K). For any k ∈ ℕ(1, K + 1)

(Γu)(k) = ∑_{l=0}^{K} G(k, l)w(l)f(l, u(l + 1))

≤ β^{−1}r_{∗} ∑_{l=0}^{K} G(k, l)w(l) ≤ β^{−1}r_{∗}β = r_{∗} = ∥u∥.

Thus ∥Γu∥ ≤ ∥u∥. By Lemma 4.1 (b), 𝔦(Γ, Ω_{r_{∗}}, Ω) = 1.

For any u ∈ ∂Ω_{r^{∗}}, ∥u∥ = r^{∗} and αr^{∗} ≤ u(k) ≤ r^{∗} on ℕ(1, K + 1). From (2.8), f(k, u(k + 1)) ≥ β^{−1}r^{∗} on ℕ(0, K). Let k_{2} ∈ ℕ(1, K + 1) be defined by (2.14). Then

(Γu)(k_{2}) = ∑_{l=0}^{K} G(k_{2}, l)w(l)f(l, u(l + 1))

≥ β^{−1}r^{∗} ∑_{l=0}^{K} G(k_{2}, l)w(l) = ββ^{−1}r^{∗} = ∥u∥.

Thus ∥Γu∥ ≥ ∥u∥. By Lemma 4.1 (a), 𝔦(Γ, Ω_{r^{∗}}, Ω) = 0.

If r_{∗} < r^{∗}, then by Lemma 4.2, Γ has a fixed point u ∈ Ω̄_{r^{∗}} \ Ω_{r_{∗}}. Similarly, if r_{∗} > r^{∗}, then Γ has a fixed point u ∈ Ω̄_{r_{∗}} \ Ω_{r^{∗}}. In each case, u is a positive function with min{r_{∗}, r^{∗}} ≤ ∥u∥ ≤ max{r_{∗}, r^{∗}}.

Proof of Theorem 2.2. (a) If f^{0} < β^{−1}, there exists r_{∗} > 0 such that

f(k, x) < β^{−1}x ≤ β^{−1}r_{∗}, (k, x) ∈ ℕ(0, K) × [0, r_{∗}].

If f_{∞} > (αβ)^{−1}, there exists r̂ > r_{∗} such that

f(k, x) > (αβ)^{−1}x, (k, x) ∈ ℕ(0, K) × [r̂, ∞).

Then for any r^{∗} with αr^{∗} ≥ r̂

f(k, x) > (αβ)^{−1}x ≥ β^{−1}r^{∗} for all (k, x) ∈ ℕ(0, K) × [αr^{∗}, r^{∗}].

Then the conclusion follows from Theorem 2.1.

(b) The proof is similar to Part (a) and hence is omitted.

The proofs of Theorems 2.3 and 2.4 are in the same way and hence are omitted.

Proof of Theorem 2.5. (a) If there exists r > 0 such that (2.9) holds, then there exist r_{1} and r_{2} such that r_{1} < r < r_{2} and f(k, x) < β^{−1}x for all (k, x) ∈ ℕ(0, K) × [αr_{i}, r_{i}], i = 1, 2. By Theorem 2.3 (a) and (b), BVP (1.1), (1.2) has two positive solutions u_{1} and u_{2} satisfying ∥u_{1}∥ ≤ r_{1} and ∥u_{2}∥ ≥ r_{2}.

Similarly, case (b) follows from Theorem 2.4.

The proofs of Theorems 2.6 and 2.7 are in the same way and are hence omitted.

Proof of Corollary 2.1. From the assumption we see that for sufficiently large i

\frac{f(k, x)}{x} < β^{−1} for all (k, x) ∈ ℕ(0, K) × [αr_{2i−1}, r_{2i−1}]

and

\frac{f(k, x)}{x} > (αβ)^{−1} for all (k, x) ∈ ℕ(0, K) × [αr_{2i}, r_{2i}].

This shows that for sufficiently large i,

f(k, x) < β^{−1}x ≤ β^{−1}r_{2i−1}

for all (k, x) ∈ ℕ(0, K) × [αr_{2i−1}, r_{2i−1}] and

f(k, x) > (αβ)^{−1}x ≥ (αβ)^{−1}αr_{2i} = β^{−1}r_{2i}

for all (k, x) ∈ ℕ(0, K) × [αr_{2i}, r_{2i}].

Therefore, the conclusion follows from Theorem 2.7.

To prove Theorems 2.8 and 2.9, we need the following lemma.

Lemma 4.4 (a) Assume f_{0} > μ_{0}. Then 𝔦(Γ, Ω_{r}, Ω) = 0 for all sufficiently small r > 0.

(b) Assume f_{∞} > μ_{0}. Then 𝔦(Γ, Ω_{r}, Ω) = 0 for all sufficiently large r > 0.

(c) Assume f^{0} < μ_{0}. Then 𝔦(Γ, Ω_{r}, Ω) = 1 for all sufficiently small r > 0.

(d) Assume f^{∞} < μ_{0}. Then 𝔦(Γ, Ω_{r}, Ω) = 1 for all sufficiently large r > 0.

Proof. In this proof, we will use the following integration by parts formulas given in [1, 4]

∑_{k=a}^{b} f(k + 1)Δg(k) = f(k)g(k)|_{a}^{b+1} − ∑_{k=a}^{b} Δf(k)g(k)  (4.3)

and

∑_{k=a}^{b} f(k)Δg(k) = f(k)g(k)|_{a}^{b+1} − ∑_{k=a}^{b} Δf(k)g(k + 1).  (4.4)

(a) Let 0 < p < 1 be fixed and define Γ_{1} : D → D by

(Γ_{1}u)(k) = ∑_{l=0}^{K} G(k, l)w(l)u^{p}(l + 1).

Similar to the proof of Lemma 4.3, we can show that Γ_{1} is compact and Γ_{1}Ω ⊂ Ω. Define r_{1} = (α^{p+1} ∑_{l=0}^{K} G(l + 1, l)w(l))^{1/(1−p)}. Then for 0 < r ≤ r_{1} and u ∈ ∂Ω_{r}, u^{p}(l) ≥ (αr)^{p} on ℕ(1, K + 1) and hence

∥Γ_{1}u∥ = max_{k∈ℕ(1,K+1)}(Γ_{1}u)(k) = max_{k∈ℕ(1,K+1)} ∑_{l=0}^{K} G(k, l)w(l)u^{p}(l + 1)

≥ α ∑_{l=0}^{K} G(l + 1, l)w(l)(αr)^{p} = α(αr)^{p} ∑_{l=0}^{K} G(l + 1, l)w(l)

= α(αr_{1})^{p}(r/r_{1})^{p} ∑_{l=0}^{K} G(l + 1, l)w(l) ≥ r_{1}r/r_{1} = r = ∥u∥.  (4.5)

By Lemma 4.1 (a), 𝔦(Γ_{1}, Ω_{r}, Ω) = 0.

Define a homotopy operator H : [0, 1] × Ω → Ω by

H(s, u) = (1 − s)Γu + sΓ_{1}u.

Then H(s, ·) is compact for 0 ≤ s ≤ 1. Since f_{0} > μ_{0}, we can choose ε > 0 and 0 < r_{2} ≤ r_{1} such that for (k, x) ∈ ℕ(0, K) × [0, r_{2}]

f(k, x) ≥ (μ_{0} + ε)x and x^{p} ≥ (μ_{0} + ε)x.

Let 0 < r ≤ r_{2}. We now show that H(s, u) ≠ u for all 0 ≤ s ≤ 1 and u ∈ K ∩ ∂Ω_{r}. Assume the contrary, i.e., there exists s_{1} ∈ [0, 1] and u_{1} ∈ ∂Ω_{r} with H(s_{1}, u_{1}) = u_{1}. Then u_{1} satisfies

−Δ^{2}u_{1}(k) = (1 − s_{1})w(k)f(k, u_{1}(k + 1)) + s_{1}w(k)u_{1}^{p}(w + 1)

and BC (1.2). Hence

∑_{l=0}^{K} [−Δ^{2}u_{1}(l)v_{0}(l + 1)]

= ∑_{l=0}^{K} w(l) [(1 − s_{1})f(l, u_{1}(l + 1))v_{0}(l + 1) + s_{1}u_{1}^{p}(l + 1)v_{0}(l + 1)] , (4.6)

where v_{0} is the eigenfunction of SLP (2.11), (1.2) associated to the eigenvalue μ_{0}. Using the integration by parts formulas (4.3) and (4.4) and BC (1.2), we have

∑_{l=0}^{K}[−Δ^{2}u_{1}(l)v_{0}(l + 1)] = μ_{0} ∑_{l=0}^{K} w(l)u_{1}(l + 1)v_{0}(l + 1).  (4.7)

Combining (4.6) and (4.7) we obtain

μ_{0} ∑_{l=0}^{K} w(l)u_{1}(l + 1)v_{0}(l + 1)

= ∑_{l=0}^{K} w(l)[(1 − s_{1})f(l, u_{1}(l + 1))v_{0}(l + 1) + s_{1}u_{1}^{p}(l + 1)v_{0}(l + 1)]

≥ (μ_{0} + ε) ∑_{l=0}^{K} w(l)[(1 − s_{1})u_{1}(l + 1)v_{0}(l + 1) + s_{1}u_{1}(l + 1)v_{0}(l + 1)]

= (μ_{0} + ε) ∑_{l=0}^{K} w(l)u_{1}(l + 1)v_{0}(l + 1),

which is a contradiction since u_{1} and v_{0} are positive on ℕ(1, K + 1) and u_{1}(0) ≥ 0 which is implied by (H1). Hence by (4.5)

𝔦(Γ, Ω_{r}, Ω) = 𝔦(H(0, ·), Ω_{r}, Ω) = 𝔦(H(1, ·), Ω_{r}, Ω) = 𝔦(Γ_{1}, Ω_{r}, Ω) = 0.

The proofs of Parts (b), (c) and (d) are similar to Part (a) and hence are omitted.

Proof of Theorem 2.8. We only prove case (a). The rest can be proved similarly.

By Lemma 4.4, f^{0} < μ_{0} implies 𝔦(Γ, Ω_{r}, Ω) = 1 when r > 0 small enough. f_{∞} > μ_{0} implies 𝔦(Γ, Ω_{R}, Ω) = 0 when R > r large enough. Therefore, by Lemma 4.2, Γ has a positive fixed point and hence BVP (1.1), (1.2) has a positive solution.

The proof of Theorem 2.9 is similar and hence is omitted.

Proof of Theorem 2.10. (a) Assume BVP (1.1), (1.2) has a positive solution u with ∥u∥ = r for some r > 0. Then u is a fixed point of the operator Γ defined by (4.2).

From the assumption, f(k, u(k + 1)) < β^{−1}u(k) ≤ β^{−1}r on ℕ(0, K). Thus for any k ∈ ℕ(1, K + 1)

u(k) = (Γu)(k) = ∑_{l=0}^{K} G(k, l)w(l)f(l, u(l + 1))

< β^{−1}r ∑_{l=0}^{K} G(k, l)w(l) ≤ r

which contradicts that ∥u∥ = r. Therefore, BVP (1.1), (1.2) has no positive solutions.

(b) Assume BVP (1.1), (1.2) has a positive solution u with ∥u∥ = r. Similar to the proof of Lemma 4.3 we can show that Γu ∈ Ω and hence αr ≤ u(k) ≤ r on ℕ(1, K + 1). From the assumption, f(k, u(k + 1)) > (αβ)^{−1}u(k) ≥ β^{−1}r on ℕ(0, K). Let k_{2} ∈ ℕ(1, K + 1) be defined as in (2.14). Then

u(k_{2}) = Γu(k_{2}) = ∑_{l=0}^{K} G(k_{2}, l)w(l)f(l, u(l + 1))

> β^{−1}r ∑_{l=0}^{K} G(k_{2}, l)w(l) = r

which contradicts that ∥u∥ = r. Therefore, BVP (1.1), (1.2) has no positive solutions.

References

[1] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second edition, Marcel Dekker, Inc, 2000.

[2] R. P. Agarwal, M. Bohner and P. J. Y. Wong, Sturm-Liouville eigenvalue problems on time scales. Appl. Math. Comput. 99 (1999), 153–166.

[3] N. Aykut and G. Guseinov, On positive solutions of boundary value problems for nonlinear second order difference equations, Turkish J. Math. 27 (2003), 481–507.

[4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.

[5] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.

[6] L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Computer Modelling 32 (2000) 529–539.

[7] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, 1988.

[8] J. Ji and B. Yang, Eigenvalue comparisons for boundary value problems for second order difference equations, J. Math. Anal. Appl. 320 (2) (2006), 964–972.

[9] J. Ji and B. Yang, Eigenvalue comparisons for a class of boundary value problems of second order difference equations, Linear Algebra and its Applications 420 (2007), 218–227.

[10] Q. Kong, Existence and nonexistence of solutions of second-order nonlinear boundary value problems, Nonlinear Anal. 66 (2007), 2635–2651.

[11] Q. Kong, Sturm-Liouville problems on time scales with separated boundary conditions, Result. Math. 52 (2008), 111–121.

[12] R. Ma and T. Raffoul, Positive solutions of three-point nonlinear discrete second order boundary value problem, J. Difference Eqns. Appl. 10 (2) (2004), 129–138.

[13] H. Pang and W. Ge, Positive solution of second-order multi-point boundary value problems for finite difference equation with a p-Laplacian, J. Appl. Math. Comput. 26 (2008), 133–150.

[14] Y. Tian and W. Ge, Multiple positive solutions of boundary value problems for second-order discrete equations on the half-line, J. Difference Eqns. Appl. 12 (2) (2006), 191–208.

[15] J. Yu, B. Zhu and Z. Guo, Positive solutions for multiparameter semipositone discrete boundary value problems via variational method, Advances in Difference Equations 2008 (2008), Article ID 840458.