Existence of One-Signed Solutions of Discrete Second-Order Periodic Boundary Value Problems

and Applied Analysis 3 We say that the linear boundary value problem 1.4 , 1.5 is nonresonant when its unique solution is the trivial one. If 1.4 , 1.5 is nonresonant, and let h : 1, T Z → R, by the virtue of the Fredholm’s alternative theorem, we can get that the discrete second-order periodic boundary value problem Δ2y t − 1 a t y t h t , t ∈ 1, T Z 2.2 y 0 y T , Δy 0 Δy T 2.3 has a unique solution y,


Introduction
Let R be the set of real numbers, Z be the integers set, T, a, b ∈ Z with T > 2, a > b, and a, b Z {a, a 1, . . ., b}.
In recent years, the existence and multiplicity of positive solutions of periodic boundary value problems for difference equations have been studied extensively, see 1-5 and the references therein.In 2003, Atici and Cabada 2 studied the existence of solutions of second-order difference equation boundary value problem where a, f satisfy H1 a : 1, T Z → −∞, 0 and a • / ≡ 0; H2 f : 1, T Z × R → R is continuous with respect to y ∈ R. The authors obtained the existence results of solutions of 1.1 under conditions H1 , H2 , and the used tool is upper and lower solutions techniques.
Naturally, whether there exists the Green function G t, s of the homogeneous linear boundary value problem corresponding to 1.1 if a n ≥ 0? Moreover, if the answer is positive, whether G t, s keeps its sign?To the knowledge of the authors, there are very few works on the case a n ≥ 0.
Recently, in 2003, Torres 6 investigated the existence of one-signed periodic solutions for second-order differential equation boundary value problem x t f t, x t , t ∈ 1, T , x 0 x T , x 0 x T , 1.2 by applying the fixed-point theorem in cones, and constructed Green's function of where a ∈ L p 0, T satisfies either H3 a ≤ 0, a • / ≡ 0 on 0, T ; H4 a ≥ 0, a • / ≡ 0 on 0, T and a p ≤ K 2p * for some 1 ≤ p ≤ ∞.Motivated by Torres 6 , in Section 2, the paper gives the new expression of Green's function of the linear boundary value problem where a ∈ Λ ∪ Λ − and and obtains the sign properties of Green's function of 1.4 , 1.5 .
In Section 3, we obtain the existence of one-signed periodic solutions of the discrete second-order nonlinear periodic boundary value problem For related results on the associated differential equations, see Torres 6 .

Preliminaries
Let be a Banach space endowed with the norm y max t∈ 0,T 1 Z |y t |.
We say that the linear boundary value problem 1.4 , 1.5 is nonresonant when its unique solution is the trivial one.If 1.4 , 1.5 is nonresonant, and let h : 1, T Z → R, by the virtue of the Fredholm's alternative theorem, we can get that the discrete second-order periodic boundary value problem Proof.Obviously, G is well defined on 0, T 1 Z × 1, T Z .We only need to prove that G has no generalized zero in any point.Suppose on the contrary that there exists t 0 , s 0 ∈ 0, T 1 Z × 1, T Z such that t 0 , s 0 is a generalized zero of G t, s .It is well known that for a given s 0 ∈ 1, T Z , G t, s 0 as a function of t is a solution of 1.4 in the intervals 0, s 0 − 1 Z and s 0 1, Consequently, y is a solution of 1.4 in the whole interval s 0 , s 0 T Z .Since y t 0 0, we have Δ 2 y t 0 − 1 −a t y t 0 0, that is, y t 0 − 1 y t 0 1 < 0.Moreover, y s 0 y s 0 T , so there at least exists another generalized zero t 1 ∈ s 0 1, s 0 T Z of y.Note that the distance between t 0 and t 1 is smaller than T , which is a contradiction.
Analogously, if t 0 ∈ 0, s 0 − 1 Z , we get a contradiction by the same reasoning with If t 0 s 0 , we can apply y as defined 2.7 .Since y t 0 y s 0 0 and y s 0 − T y s 0 , which contradicts with the hypothesis.

Case 2 G t
we can construct y defined as 2.6 .It is not difficult to verify that y is a solution of 1.4 in the whole interval s 0 , s 0 T Z .Also, we have that y t 0 − 1 y t 0 < 0, that is, t 0 is a generalized zero of y.Moreover, y s 0 y s 0 T , so there at least exists another generalized zero t 1 ∈ s 0 1, s 0 T Z of y.Note that the distance between t 0 and t 1 is smaller than T , which is a contradiction.
Similarly, if t 0 ∈ 1, s 0 − 1 Z , we can get a contradiction by the same reasoning as y defined 2.7 .
If t 0 s 0 , we can construct y defined by 2.7 .Since y t 0 − 1 y t 0 y s 0 − 1 y s 0 < 0 and y s 0 −T y s 0 , it is clear that there exists another generalized zero t 1 ∈ s 0 −T, s 0 Z of y.Note that the distance between t 0 and t 1 is smaller than T , this contradicts with the hypothesis.
To apply the above result, we are going to study the two following cases.
Proof.If a ∈ Λ − , by 7, Corollary 6.7 , it is easy to verify that 1.4 is disconjugate on 0, T 1 Z , and any nontrivial solution of 1.4 has at most one generalized zero on 0, T 1 Z .Hence, by Theorem 2.2, Green's function G t, s has constant sign.We claim that the sign is negative.In fact, y t T s 1 G t, s is the unique T -periodic solution of the equation

2.10
where If a ≥ 0, then the solutions of 1.4 are oscillating, that is, there are infinite zeros, and to get the required distance between generalized zeros, a should satisfy Λ .
Proof.We claim that the distance between two consecutive generalized zeros of a nontrivial solution y of 1.4 is strictly greater than T .In fact, it is not hard to verify that Δ 2 y t − 1 a y t 0 is disconjugate on 0, T 1 Z under assumption a < 4 sin 2 π/2T .Since a t ≤ a , t ∈ 1, T Z , by Sturm comparison theorem 7, Theorem 6.19 , 1.4 is disconjugate on 0, T 1 Z , that is, any nontrivial solution of 1.4 has at most one generalized zero on 0, T 1 .
Hence, by Theorem 2.2, G t, s has constant sign on 0, T 1 Z × 1, T Z , and the positive sign of G is determined as the proof process of Corollary 2.3.

2.12
Obviously, Green's function G t, s 0 for t s and G t, s > 0 for t / s.If a • ≡ a and a 4 sin 2 π/T , then θ 2π/T, and it is not difficult to verify that
Next, we provide a way to get the expression of G t, s .Let u be the unique solution of the initial value problem and v be the unique solution of the initial value problem

2.16
Lemma 2.7.Let a ∈ Λ − ∪ Λ .Then Green's function G t, s of 1.4 , 1.5 is explicitly given by 2.17 Proof.Suppose that Green's function of 1.4 , 1.5 is of the form where α s , β s can be determined by imposing the boundary conditions.From the basis theory of Green's function, we know that

2.21
Note that α • has the same sign with a • .In fact, by the comparison theorem 7, Theorem 6.6 , it is easy to prove that u, v ≥ 0 on 0, T Z .If a t ≥ 0, then

2.22
Thus Δu T < Δu 0 1.Similarly, we can get that Δv 0 > Δv T −1.Since v 0 u T , we have where G t, s is defined by 2.17 .
Proof.We check that y satisfies 2.2 .In fact,

2.25
On the other hand, it is easy to verify that y 0 y T , y 1 y T 1 .

2.26
As a direct application, we can compute the maximum and the minimum of the Green's function when a • ≡ a 0 , it follows that where λ 1 is defined in Remark 2.4.Similarly, when 0 < a • ≡ a < 4 sin 2 π/2T , we can get where θ is defined in Remark 2.6.

Main Results
In this section, we consider the existence of one-signed solutions of 1.7 .The following wellknown fixed-point theorem in cones is crucial to our arguments.
Then A has a fixed point in K ∩ Ω 2 \ Ω 1 . Theorem by the similar method, we can prove α • < 0. Let a ∈ Λ − ∪ Λ .Then the periodic boundary value problem 2.2 , 2.3 has the unique solution

Theorem 3.1 see
8 .Let E be a Banach space and K ⊂ E be a cone.Suppose Ω 1 and Ω 2 are bounded open subsets of E with Clearly, if y ∈ P ∩ Ω 2 \ Ω 1 , then m/M r ≤ y t ≤ M/m R, for all t ∈ 0, T 1 Z .From Lemma 2.8, we define the operator A : E → E by Thus A P ∩ Ω 2 \ Ω 1 ⊂ P .Moreover, E is a finite space, it is easy to prove that A : P ∩ Ω 2 \ Ω 1 → P is a completely continuous operator.Clearly, y is the solution of problem 1.7 if and only if y is the fixed point of the operator A.We only prove i .ii can be obtained by the similar method.If y ∈ ∂Ω 1 ∩ P , then y r and m/M r ≤ y t ≤ r for all t ∈ 0, T 1 Z .Therefore, from i , Applying the sign properties of G t, s when a ∈ Λ − and the similar argument to prove Theorem 3.2 with obvious changes, we can prove the following.Assume that there exist a ∈ Λ − and 0 < r < R such that Define the operator A as 3.7 , and the proof is analogous to that of Theorem 3.2 and is omitted.Assume that there exist a ∈ Λ − and 0 < r < R such that Similar to the proof of Theorem 3.2, we can prove the following.Corollary 3.3.Assume that there exist a ∈ Λ and 0 < r < R such that