POSITIVE SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENCE BOUNDARY VALUE PROBLEMS

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.

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.

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) 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 and BC (1.2), see Agarwal [1] for the detail.Let From Aykut and Guseinov [3], we know that 0 < α < 1 and The first theorem is our basic result on the existence of positive solutions of BVP (1.1), (1.2).
Then BVP (1.1), (1.2) has at least one positive solution In the sequel, we will use the following notation: The next three theorems are derived from Theorem 2.1 using f 0 , f ∞ , f , and there exists r > 0 such that (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.
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.
) 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.
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.
Finally, we present a result on the nonexistence of positive solutions of BVP (1.1), (1.2).
. Then (a) BVP (1.1), (1.2) has at least one positive solution when h = r(r 2) has at least two positive solutions u 1 and u 2 with u 1 < r < u 2 when 0 < h < r(r (c) BVP (1.1), (1.2) has no positive solutions when h > r(r 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 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 h > r(r Then the conclusion follows from Theorem 2.10 (b).
. Then 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 where r * = α −1 r.By Theorem 2.4 (a) or (b), There exists at least one positive solution.
When 0 < h < (r Then the conclusion follows from Theorem 2.10 (a).
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].
Then Γ has a fixed point in Ωr  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), For any u ∈ ∂Ω r * , u = r * and αr Then for any r * with αr * ≥ r 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 This shows that for sufficiently large i, Therefore, the conclusion follows from Theorem 2.7.
To prove Theorems 2.8 and 2.9, we need the following lemma.
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).