Periodic Boundary Value Problems for First Order Difference Equations

In this paper, existence criteria for single and multiple positive solutions of periodic boundary value problems for first order difference equations of the form � 4 x(k) + f(k,x(k + 1)) = 0, k 2 [0,T], x(0) = x(T + 1), are established by using the fixed point theorem in cones. An example is also given to illustrate the main results.


Introduction
Due to the wide application in many fields such as science, economics, neural network, ecology, cybernetics, etc., the theory of nonlinear difference equations has been widely studied since 70's of last century, see, for example, [1,2,19,20].At the same time, Boundary value problems (BVPs) of difference equations have received much attention from many authors, see [3-12, 14-18, 21, 22, 24-29] and the references therein.However, to the best our knowledge, few papers can be found in the literature for periodic boundary value problems (PBVPs) of difference equations [9,10,22,24,29].
In this paper, we are concerned with the existence of single and multiple positive solutions of PBVP for first order difference equation where T is a fixed positive integer, △ denotes the forward difference operator with stepsize 1, and In [22], by using a fixed point theorem, Sun considered the existence of one positive solution of the PBVP (1.1) when the following condition holds: (A) There exists a positive number M > 1 such that In [24], Wang obtained the existence of multiple positive solutions of PBVP (1.1) by using the Leggett-Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression when condition (A) holds.
Motivated by the results mentioned above, in this paper, we shall obtain existence criteria for single and multiple positive solutions to the PBVP (1.1) by means of a fixed point theorem in cones.It is worth noticing that our hypotheses on nonlinearity f in this paper are weaker than condition (A) of [22,24].This paper's ideas come from [23].
Theorem 1.1 ( [13]).Let X be a Banach space and K is a cone in be a continuous and completely continuous operator such that (i) Φx ≤ x for x ∈ K ∩ ∂Ω 1 ; (ii) there exists e ∈ K\{0} such that x = Φx + λe for x ∈ K ∩ ∂Ω 2 and λ > 0.

Preliminaries
Let C = {x : [0, T ] → R}.For σ ∈ C, we first consider the following linear PBVP : where It is easy to see that the following lemma holds.
EJQTDE, 2012 No. 52, p. 2 Lemma 2.1.Suppose M > 1.Then for any σ ∈ C, PBVP (2.1) has a unique solution: In addition, if we choose σ(k) ≡ 1, then we know where δ = M −(T +1) < 1, one may readily verify that K is a cone in E. Now for u ∈ K, we consider the following PBVP: It follows from Lemma 2.1 that PBVP (2.3) has a unique solution: Define an operator Φ : It is obviously that fixed points of Φ are solutions of PBVP (1.1) and Φ : K → E is continuous and completely continuous.

Main results
In this section, by defining an appropriate cones, we impose the conditions on f which allow us to apply the fixed point theorem in cones to establish the existence criteria for single and multiple positive solutions of the PBVP (1.1).Theorem 3.1.Suppose that there exist a positive number M > 1 and 0 < α < β such that EJQTDE, 2012 No. 52, p. 3 Then the PBVP (1.1) has at least one positive solution if one of the following two conditions holds (i) Proof.Define the open sets Firstly, we claim that Φ : Secondly, we prove the result provided conditions (i) holds.By the first inequality of (i), we have Let e ≡ 1, then e ∈ K.We assert that x = Φx + λe for x ∈ K ∩ ∂Ω 1 and λ > 0. (3.1) If not, there would exist x 0 ∈ K ∩ ∂Ω 1 and λ 0 > 0 such that x 0 = Φx 0 + λ 0 e.

EJQTDE, 2012
No. 52, p. 4 Since This implies that µ ≥ µ + λ 0 , and this is a contradiction.Therefore (3.1) holds.On the other hand, by using the second inequality of (i), we have In fact, for any In a similar way, we can prove the result by Theorem 1.1 if condition (ii) holds.
Theorem 3.2.Suppose that there exist a positive number M > 1 and 0 Then the PBVP (1.1) has at least two positive solutions if one of the following two conditions holds EJQTDE, 2012 No. 52, p. 5 Proof.We only prove the result when condition (i) holds.In a similar way we can obtain the result if condition (ii) holds.
Define Ω 1 , Ω 2 as in Theorem 3.1 and define Similar to the proof of Theorem 3.1, we can prove that x = Φx + λe for x ∈ K ∩ ∂Ω 2 and λ > 0, ( where e ≡ 1 ∈ K, and Φx < x for x ∈ K ∩ ∂Ω 3 . (3.5) Thus we can obtain the existence of two positive solutions x 1 and x 2 by using Theorem 1.1 and Remark 1.1, respectively.It is easy to see that α ≤ x 1 < ρ < x 2 ≤ β.Theorem 3.3.Suppose that there exist a positive number M > 1 and 0 < α Then the PBVP (1.1) has at least n multiple positive solutions Then the PBVP (1.1) has at least 2n − 1 multiple positive solutions.