Existence of Periodic Solutions of Discrete Ricker Delay Models

article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this article is to investigate the sufficient conditions for the existence of periodic solutions of a generalized Ricker delay model, N (n + 1) = N (n) exp{f (n, N (n − r(n)))} n ≥ n 0 , n 0 ∈ Z, which appears as a model for dynamics with single species in changing environments, by applying Horn's fixed point theorem which assume some kind of bounded solutions.


Introduction
For ordinary differential equations and functional differential equations, the existence of periodic solutions of systems has been studied by many authors.One of the most popular methods is to assume the boundedness conditions and that is to find the certain Liapunov functions/functionals [1-3, 5, 8, 12].In 1950, for a nonlinear differential equation, Massera [11] showed that for scalar T -periodic equation, the existence of a bounded solution on [0, ∞) implies the existence of a periodic solution of period T .However, for functional differential equations, the existence of bounded solutions does not necessarily imply the existence of a periodic solution even for scalar equations, see [1,12], But, Hale and Lopes [4] showed that for a functional differential T -periodic system with finite delay, there exists a periodic solution of period T , if the solutions are uniformly bounded and uniformly ultimately bounded.Recently, He, Zhang and Gopalsamy [6] have shown the existence of periodic and almost periodic solutions for a non-autonomous scalar delay differential equation of modeling single species dynamics in a temporally changing environment.Their results are to extend results in Gopalsamy [3] to a non-autonomous differential equation by using the ultimately boundedness.To the best of our knowledge, there is a few relevant results on the existence of periodic solutions for discrete Ricker models by means of our approach of discrete boundedness theorems.We emphasize that our results extend [6] as a delay discrete periodic case.But [6] has forgotten to add uniformly bounded condition in main theorems for periodic case of the finite delay equation.In this paper, we discuss the existence of periodic solutions for a generalized non-autonomous discrete Ricker type difference equation with finite delay.In what follows, we denote by R real Euclidean space, Z is the set of integers,

Preliminary lemma
In this paper, we shall consider a discrete non-autonomous Ricker delay difference equation of the form and in particular, we obtain sufficient conditions for equation (1) to have periodic solution when f is a periodic function.As example of the type (1), we provide the following example We set the following assumptions for equation (1).
We assume that the initial conditions associated with (1) are as follows: By (1) and (H 3 ), we can see that solutions of (1) satisfy To prove this, from (1), ( 2) and (H 2 ), we have By recursively using inequality (4), we have where N(n 0 ) = φ(n 0 ).Thus, we obtain Similarly, we get the first inequality in (3).Then solutions of (1) are defined for all n ≥ n 0 and furthermore that N(n) > 0 for n ≥ n 0 .We show the following key lemma in which say that the solution of (1) is uniformly bounded and uniformly ultimately bounded.
Lemma 1.If the assumptions (H 1 ), (H 2 ) and (H 3 ) hold, then there exists an n 1 ≥ n 0 such that any solution N(n) of (1) satisfies where α = max 0≤x≤ξ 2 F 2 (x).Furthermore, if we assume that (H 1 ), (H 2 ) and (H 3 ), and if then there exists an n 2 ≥ n 1 such that Proof.Let N(n) be any positive solution of (1) defined for all n ≥ n 0 .From (1) and (H 2 ), By (H 3 ), there exists a ξ 2 > 0 such that Since F 2 is continuous on [0, ξ 2 ], it must attain its maximum say α at some point say (10) leads to (11) there exist > 0 and We can drive from ( 10) and (12), By (13), on [n 4 , n], (n > n 4 ), we have Hence, we obtain lim n→∞ N(n) = ξ 2 from which (7) follows.Next, we suppose that N(n) is oscillates about ξ 2 that is, there exists a sequence {n k } satisfying which together with (11) implies Similar as (10) Thus, it lead to N(n) ≤ ξ 2 e αr L .Since N(α k ) is an arbitrary local maximum of N, we can conclude that there exists an n 2 > n 1 such that (7) holds.To prove (9), we note first that we have from ( 1) and (H 2 ), From (H 3 ), we can see that there exists a ξ 1 > 0 such that F 1 (ξ 1 ) = 0. Suppose that N(n) does not oscillate about ξ 1 , then similar to the above arguments, we can show that there exists an n 2 such that (9) holds.We have already show that there exists an n 1 such that (9) holds.Hence when N(n) oscillates about ξ 1 we have from ( 8) Now, by using ( 14), (15) and similar above arguments, we can show that there exists an n 2 ≥ n 1 such that (9) holds.This completes the proof.

Periodic delay models
To construct the discrete type existence theorem of the periodic solution, we first consider the scalar general functional difference equation (16) In (16), g(n, φ) is continuous for φ (second term) defined on Z ×BS to R and it takes bounded sets into bounded sets.Moreover, it satisfies a local Lipschitz condition in φ and there is a T with g(n + T, φ) = g(n, φ) whenever φ is also T -periodic.A solution is denoted by x(n 0 , φ) with value at being x(n, n 0 , φ) and with x(n 0 , n 0 , φ) = φ.The solution of ( 16) is unique for the initial function φ (cf.[1,2]).
Lemma 2. Suppose that x(n + T ) is a solution of (16) whenever x(n) is a solution of (16).Then Equation ( 16) has a T -periodic solution if and only if there is a n 0 ∈ Z and φ : n 0 , φ) and x(n + T, n 0 , φ) are both solutions with the same initial function and so, by uniqueness, they are equal.The proof is complete.
We consider the following fixed point theorems by Schauder [cf.1] and Horn [7].

Theorem A (Schauder).
A continuous mapping Q of a compact convex nonempty subset Y in the Banach space X into itself has at least one fixedpoint.

Theorem B (Horn).
Let S 0 ⊂ S 1 ⊂ S 2 be convex subsets of the Banach space X, with S 0 and S 2 compact and S 1 open relative to S 2 .Let Q : S 2 → X be a continuous function such that for some integer q > 0, (a) Then Q has a fixed point in S 0 .
In order to show the existence of periodic solution for equation (16), we now define the uniformly bounded and uniformly ultimately bounded of solutions for (16).Definition 1.The solutions of (16) are uniformly bounded if for any K > 0, there exists a Definition 2. The solutions of (16) are uniform ultimately bounded for bound B, if there exists a B > 0 and if corresponding to any K > 0, there exists a T * = T * (K) > 0 such that |φ| ≤ K implies that |x(n, n 0 , φ)| < B for all n ≥ n 0 + T * .
The Definition 1 and 2 can employ to our finite delay difference equation (1) and so, Lemma 1 shows that the solution N(n) of equation ( 1) is uniformly bounded and uniformly ultimately bounded under assumptions (H 1 ), (H 2 ) and (H 3 ).We show the following theorems by improving, as discrete case, the proof of theorems in [1].
Theorem 1. Suppose that x(n + T ) is a solution of (16) whenever x(n) is a solution of ( 16).If solutions of equation ( 16) are uniformly ultimately bounded for bound B, then it has a mT -periodic solution for some positive integer m.

Proof. The space BS
be the solution of ( 16) with initial time n 0 = 0.For the B > 0, we can find an integer mT > 0 such that |φ| ≤ B implies that |x(n, 0, φ)| < B for n ≥ mT −h.Now, for some L > 0, |x(n+1, 0, φ)| ≤ L holds on [mT, mT +2h].Thus, we can find an integer m > 0 such that mT > mT + 2h, and let Also, we have Since m > 0 and η m > 0 are arbitrary small, m + η m + L|u − v| → L|u − v| as m → ∞.Thus, Hence, φ ∈ S and S is compact.Define Q : S → S by Qφ = Qφ(n) = x(n + mT, 0, φ) for −h ≤ n ≤ 0. We can see that Q is continuous, because g is locally Lipschitz and hence, Q has a fixed point by Schauder's fixed point theorem.Thus, x(n+ mT, 0, φ) and x(n, 0, φ) are both solutions with the same initial function φ and hence, by the uniqueness they are equal this.
Theorem 2. Suppose that x(n + T ) is a solution of (16) whenever x(n) is a solution of ( 16).If solutions of equation ( 16) are uniformly bounded and uniformly ultimately bounded for bound B, then it has a T -periodic solution which is bounded by B.
Proof.Let x(n) = x(n, 0, φ) be the solution defined on Z + with initial time n 0 = 0. Since solution x(n) of ( 16) is uniformly bounded, for B > 0 of uniformly ultimately bounded for bound B, there is a Then, the S i , (i = 0, 1, 2) are convex, and moreover S 0 and S 2 are compact by Ascoli's theorem.Furthermore, by the uniqueness theorem.Next, and x(n + T, 0, Qφ) is a solution with initial function Qφ.Hence, by uniqueness.Now in (17) let n be replaced by n + T so that by (18).In general, for each integer k > 0, x(n + kT, 0, φ) = x(n, 0, Q k φ).By construction of S 1 and S 2 we have Q j S 1 ⊂ S 2 for 1 ≤ j ≤ m.By choice of m we have Q j S 1 ⊂ S 0 for j ≥ m.Also, Q j S 0 ⊂ S 1 for all j.We can see that, by Horn's fixed point theorem, Q has a fixed point φ ∈ S 0 .Then, by Lemma 2, we conclude that x(n, 0, φ) is T -periodic solution of ( 16).This completes the proof.
Now, in our main theorem, we show that the existence of a periodic solution for the equation (1).Theorem 3.Under the assumptions (H 1 ), (H 2 ) and (H 3 ), if f (n, φ) is continuous in its second term, satisfies a local Lipschity condition in φ and periodic in n with period T , then equation ( 1) has a positive periodic solution of period T .
Proof.We can assert a consequence of the uniformly bounded and uniformly ultimately bounded of positive solutions of (1) by Lemma 1 and the periodicity of function f in n.Our result immediately follows from Theorem 2.

Examples
We first consider the following periodic delay model of the form where a, b, c and r are defined on [n 0 , ∞) with a(n) > 0, b(n) ∈ R, c(n) > 0 and r(n) > 0, and for n ≥ n 0 .The original differential model of ( 19) is considered by Ladas and Qian [9].Let We note that F 1 (y) ≤ f (n, y) ≤ F 2 (y) for n ≥ n 0 , y ∈ [0, ∞).It is easy to see that all the assumptions (H 1 ), (H 2 ) and (H 3 ) are satisfied.We denote by y * and y * the unique positive solution of the equations F 1 (y) = 0 and F 2 (y) = 0, respectively.Then, any positive solution N(n) of ( 19) satisfies eventually for all large n where If a(n) ≡ a, b(n) ≡ b, c(n) ≡ c and r(n) ≡ r for n ≥ n 0 and y 0 is the unique positive solution of the equation According to the above result and Theorem 3, we can obtain the following corollary.
Corollary 1. Suppose that a(n), b(n) and c(n) are periodic functions with period T .If we assume (20) and (21), then equation ( 19) has a periodic solution of period T , and furthermore, all solutions of (19) satisfy in which m 1 and M 1 are defined by (22).Remark 1.As we have seen the above result (by using boundedness theorems 2 and 3), we do not need any stability result to establish the existence of periodic solutions of equations ( 1), ( 16) and (19).It is also noticed that we do not need any restriction on the delay term r (see cf. [10]).Finding conditions for the bounded of solutions for equations ( 1) and ( 19) are of very useful and interest in applications.
We next consider the delay difference equation in which η 2 is defined (24).
is the set of nonnegative integers and | • | will denote the Euclidean norm in R. For any discrete interval I ⊂ Z := (−∞, ∞), we denote by BS(I) the set of all bounded functions mapping I into R, and set |φ| I = sup{|φ(s)| : s ∈ I}.
Then, S is convex.We next show S is compact sets.If {φ m } is any sequence in S, then it is uniformly bounded and equi-continuous.By Ascoli's theorem it has a subsequence converging uniformly to a function φ.But |φ m (n)| ≤ B for any fixed n, so |φ| ≤ B.Moreover, if we denote the subsequence by {φ m } again, then for fixed u and v there exist m > 0 and