Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

: This paper deals with the existence of ω -periodic solutions for n th-order ordinary differential equation involving ﬁxed delay in Banach space E . L n u ( t ) = f ( t , u ( t ) , u ( t − τ )) , t ∈ R , where = , a ∈ R , i = 0,1, · · · , n − 1, are constants, f ( t , x , y ) : R × E × E −→ E is continuous and ω -periodic with respect to t , τ > 0. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.


Introduction
The properties of periodic solutions of differential equations are significant problems in application science. A great number of works have focused on the existence of periodic solutions of differential equations, but they mainly studied the self-adjoint equations. For the case of non-self-adjoint differential equations, the researches are seldom because of their complex spectral structure. Since the nth-order differential equations are typical non-self-adjoint differential equations, it is very important both in theory and practice to prove the existence theorems of periodic solutions for nth-order ordinary differential equations. Recently, there are many beautiful results are obtained, for instance, see Cabada [1][2][3], Li [4][5][6], Liu [7] and V. Seda [8] and the references therein. The higher-order differential equation and its application in optimization and control theory were also studied, see [9][10][11] and the references therein. In some publications, the maximum principle is essential in the proof of main results. In [4], by using the obtained maximum principle, Li extended the results of Cabada in [1][2][3] and proved some existence results for the nth-order periodic boundary value problem of ordinary differential equations. Later, Li in [5] discussed the existence as well as the uniqueness of solutions for the nth-order periodic boundary value problem under spectral conditions. The maximum principle was also used in [6] to deal with the periodic boundary value problem of nth-order ordinary differential equation L n u(t) = f (t, u(t)), 0 ≤ t ≤ ω, u (i) (0) = u (i) (ω), i = 0, 1, 2, · · · , n − 1, where L n u(t) := u (n) (t) + n−1 ∑ i=0 a i u (i) (t), a i ∈ R, i = 0, 1, · · · , n − 1, are constants, f : [0, ω] × R −→ R is a continuous mapping. By using the obtained maximum principle, the author proved some existence and uniqueness theorems. In [7], Liu investigated the existence results of periodic solutions for the two special cases of nth-order delay differential equation by applying the coincidence degree theory, but the above mentioned literatures did not consider the periodic solutions for the general delayed differential equations in abstract spaces.
In the present work, we consider the existence as well as the uniqueness of ω-periodic solutions for nth-order ordinary differential equation involving delay in Banach space E where f (t, x, y) : R × E × E −→ E is a continuous mapping and it is ω-periodic with respect to t and τ > 0. Firstly, we establish the maximum principle to the corresponding linear delayed equation where h : R → E is an ω-periodic continuous function and b ≥ 0 is a constant. Then, by applying the obtained maximum principle, some existence and uniqueness theorems are proved by applying the fixed point approach and monotone iterative technique. The next Table 1 describes several symbols which will be later used within the body of the manuscript.
N the set of natural number C the complex plane

Preliminaries
Let J := [0, ω] and C ω (R, R) be the set of all continuous and ω-periodic functions. Then C ω (R, R) is a Banach space equipped with norm u C := max t∈[0,ω] |u(t)| and C(J, R) is also the Banach space. In general, C n (J, R) is the Banach space of nth-order continuous and differentiable functions.
For all h ∈ C(J, R), we know that the linear periodic boundary value problem(LPBVP) possesses a unique solution u ∈ C n (J, R): Let P n (λ) be the characteristic polynomial of L n defined by P n (λ) = λ n + a n−1 λ n−1 + . . . + a 0 .
And let N (P n (λ)) be the set of null points of P n (λ) in C. For the LBVP (3), we assume the following hypothesis.
Proof of Lemma 1. Denote by U(t) := (u(t), u (t), . . . u (n−1) (t)) T and B := (0, 0, . . . 0, 1) T . Then the LBVP (3) equivalents to the linear system where A is defined by If we take U(0) as the initial value, the first equation of (5) has a unique solution expressed by U(t) = e tA U(0). This implies that the linear system (5) has a unique solution The first component of V 0 (t) is denoted by r n (t), then it follows from (6) that r n (t) ∈ C ∞ (I, R) and it is a unique solution of the LBVP (3). Lemma 2. If the Hypothesis 1 (H1) holds, then for each h ∈ C ω (R, R) and t ∈ R, the linear equation L n u(t) = h(t) possesses a unique solution u := T n h ∈ C n ω (R, R), and T n : C ω (R, R) −→ C ω (R, R) is a bounded linear operator satisfying T n = 1 |a 0 | when a 0 = 0.

Proof of Lemma 2.
If h ∈ C ω (R, R), since the ω-periodic solution of (7) is equivalent to the solution of the LPBVP (2), by Lemma 1, the linear Equation (7) possesses a unique ω-periodic solution Let (E, · , ≤) be an ordered and separable Banach space, K := {x ∈ E : x ≥ θ} be a positive cone of E, where θ denotes the zero element of E. Then K is a normal cone with the constant N. Denote by C ω (R, E) the set of E-valued continuous and ω-periodic functions. Then C ω (R, E) is a Banach space whose norm is defined by u C := max Then K C is also a normal cone with the same constant of cone K, and C ω (R, E) is an ordered Banach space. Generally, C n ω (R, E) is the Banach space of all ω-periodic and nth-order continuous differentiable functions for n ∈ N. Now, for any h ∈ C ω (R, E), we consider the linear delayed differential equation(LDDE) where b ≥ 0 and τ > 0 are constants.
Hence, by Lemma 2, when r n (t) > 0 and the Hypothesis 1 (H1) holds, the operator T n : is a positive operator. Let m n := min t∈I r n (t) and M n := max t∈I r n (t). It is clear that 0 < m n ≤ r n (t) ≤ M n . By Lemma 2, we obtain the following lemma.

is a linear bounded and positive operator.
Proof of Lemma 3. By Lemma 2, it is easy to see that the LDDE (8) possesses a solution Obviously, is a linear operator and B 1 ≤ b. Then (9) and (10) yield This implies that Since T n B 1 ≤ T n · B 1 ≤ b a 0 < 1, the perturbation theorem yields that (I + T n B 1 ) −1 exists and which implies Hence, by (11), we conclude that Consequently, u(t) is an ω-periodic solution of the LDDE (8). It follows from (13) that Next, we prove that T : is a positive operator when r n (t) > 0. By (12), for any h ∈ C ω (R, K), we have Form the above equality, it remains to prove the positivity of (I − T n B 1 )T n . Since Consequently, the operator T : In Lemma 3, the condition r n (t) > 0 is essential. We now introduce a condition to guarantee r n (t) > 0 for all t ∈ R: for more detail.
Hence, from Lemmas 3 and 4, the following lemma is easy to obtain.
Let β E (·) and β C (·) denote the Kuratowski's measure of non-compactness(MNC) of bounded subsets in E and C ω (R, E), respectively. For every bounded subset For more detail of the MNC, we refer to [12,13] and the references therein. The following lemmas can be found in [12,14], which are more useful in our arguments.
By Lemma 3, we present the definition of ω-periodic solution of Equation (1) as follows.

Definition 1.
A function u ∈ C ω (R, E) is called an ω-periodic solution of Equation (1) if it satisfies the integral equation (10).
To end this section, we introduce the definitions of lower and upper ω-periodic solutions of Equation (1).
then it is called the lower ω-periodic solution of Equation (1). If we inverse the inequality in (14), then it is called the upper ω-periodic solution of Equation (1).

The Method of Upper and Lower Solutions and the Monotone Iterative Technique
In this section, by utilizing the Sadovskii's fixed point theorem, we first consider the existence of ω-periodic solutions of Equation (1) between the lower and upper ω-periodic solutions. Then the monotone iterative technique is applied to study the existence as well as the uniqueness of ω-periodic solutions of Equation (1). At last, A sufficient condition is established for the existence of lower and upper ω-periodic solutions of the Equation (1).
At first, we make the following assumptions: .
for any countable subsets D i ⊂ E, i = 1, 2. Proof of Theorem 1. Since Equation (1) can be rewritten as by Lemma 3 and Definition 1, we define Q : Let D = [v 0 , w 0 ]. It is obvious that D ⊂ C ω (R, E) is nonempty bounded, convex and closed. We will apply the approach of fixed point to discuss the existence of fixed points of Q in D. These fixed points are the ω-periodic solutions of Equation (1) between v 0 and w 0 due to Lemma 3 and Definition 1.
First of all, we prove Q(D) ⊂ D. Let u ∈ D. Then for all t ∈ R. By the Hypothesis 3 (H3), we have For any u ∈ D, let v = Qu, then Secondly, we prove the equi-continuity of Q(D). For any u ∈ D and 0 ≤ t 1 ≤ t 2 ≤ ω, since r n ∈ C ∞ (I, R), by the definition of G n (t, s), we have Together this fact with the definition of Q, we obtain that Therefore, the set Q(D) is equi-continuous. It remains to prove that Q : D −→ D is a condensing mapping. By Lemma 7, since Q(D) is bounded, there is a countable subset D 0 ⊂ D such that β E (Q(D)(t)) ≤ 2β E (Q(D 0 )(t)).

Hence, Lemma 8 and the Hypothesis 4 (H4) yield
By the equi-continuity and boundedness of Q(D), we have Since 8L 1 a 0 −b < 1, it follows that Q : D −→ D is a condensing operator. Therefore, the Sadovskii's fixed point theorem guarantees that there is at least one fixed point of Q in D. So, the Equation (1) possesses at least one ω-periodic solution in D.

If we replace the conditions Hypothesis 3 (H3) and Hypothesis 4 (H4) in Theorem 1 by
Hypothesis 6 (H6). There is a constant L 2 ∈ (0, a 0 −b 4 ) such that Proof of Theorem 2. We first prove that Q has properties: where the operator Q is defined as in (15).
By the countability and boundedness of {v n }, we conclude from Lemma 8 and Hypothesis 6 (H6) that Furthermore, {v n } is equi-continuous, by Lemma 6, we get Hence β({v n }) = 0 due to 4L 2 a 0 −b < 1. Similarly, we obtain β C ({w n }) = 0. Hence, the sets {v n } and {w n } have convergent subsequences due to their relative compactness in C ω (R, E). Since the cone K C is normal and {v n }, {w n } are monotone, we assume that {v n } and {w n } are convergent. That is, there exist u and u belong to C ω (R, E) such that Putting n → ∞ in (16), we get This means that u and u are all the fixed points of Q. Consequently, u and u are ω-periodic solutions of Equation (1).
Therefore, u and u are minimal and maximal ω-periodic solutions of Equation (1).
The MNC conditions are necessary in Theorems 1 and 2, but they are not easy to verify in application. The next theorem establishes sufficient conditions to guarantee the existence as well as the uniqueness of ω-periodic solution of Equation (1), where the nonlinearity f is not asked to satisfy the MNC condition. Hypothesis 7 (H7). there is a constant L 3 satisfying max{2b − a 0 , 0} < L 3 < b such that , then there is a unique ω-periodic solution of Equation (1) between v 0 and w 0 .

Proof of Theorem 3. Define a mapping Φ by
is a continuous mapping. By Lemma 3, for any h ∈ C ω (R, E), the linear equation has a unique ω-periodic solution, which is given by Then Q : C ω (R, E) → C ω (R, E) is a continuous operator. It follows from (19) that the fixed point of operator Q is the ω-periodic solution of Equation (1).
From the proof of Theorem 2, the operator Q satisfies the properties: Let {v n } and {w n } be two sequences defined by (16). The properties (i) and (ii) yield that (17) holds. Then for t ∈ R, we have The normality of cone K yields Therefore, there exists a unique function u belongs to ∩ ∞ n=1 [v n , w n ] such that v n → u, w n → u as n → ∞. Since Qv n−1 = v n ≤ u ≤ w n ≤ Qw n , taking n → ∞ we get u = Q u. This implies that Equation (1) possesses unique ω-periodic solution.
In Theorems 1-3, we always suppose that Equation (1) possesses lower and upper ω-periodic solutions v 0 and w 0 satisfying v 0 ≤ w 0 , but it is still a problem whether Equation (1) possesses lower and upper ω-periodic solutions. Next, we will prove that Equation (1)   Hypothesis 8 (H8). There exist L * ∈ [0, m n a 0 M n ) and h ∈ C ω (R, K) such that for any u, v ∈ E and t ∈ R.
Proof of Theorem 4. By Lemma 4, if the condition Hypothesis 2 (H2) holds and a 0 > 0, the LBVP (3) possesses a unique solution r n (t) > 0 for t ∈ R. By the definition of G n (t, s) and Lemma 2, we know that T n : C ω (R, E) → C ω (R, E) is a positive linear bounded operator with T n = 1 a 0 . If the condition Hypothesis 8 (H8) holds, we consider the linear differential equation Let Then B 2 : C ω (R, E) → C ω (R, E) is positive and linear bounded, and B 2 ≤ L * . Lemma 2 yields that Equation (20) possesses a unique ω-periodic solution Since T n B 2 ≤ L * a 0 < m n M n ≤ 1, we get that (I − T n B 2 ) −1 exists and (I − T n B 2 ) −1 = ∞ ∑ i=0 (T n B 2 ) i is a positive linear operator. Hence, from (21), u(t) is given by and u(t) ≥ 0 for any t ∈ R owing to h(t) ∈ K. Let v 0 = − u and w 0 = u, by the Hypothesis 8 (H8), we have and L n w 0 (t) = L n ( u(t)) = L * ( u(t − τ)) + h(t) ≥ f (t, u(t), u(t − τ)) = f (t, w 0 (t), w 0 (t − τ)).
Hence, the Equation (1) possesses lower and upper ω-periodic solutions v 0 and w 0 satisfying v 0 ≤ w 0 . Example 1. Consider the following fourth-order ordinary differential equation in Banach space E u (4) (t) + u (t) + u (t) + u (t) + u(t) = F(t, u(t), u(t − τ)), t ∈ R, where u ∈ C ω (R, E) and F(t, u, v) : R × E × E → E is continuous and ω-periodic with respect to t. We suppose that the following conditions hold.
Thus, by Theorem 4, the fourth-order ordinary differential Equation (22) possesses lower and upper ω-periodic solutions v 0 and w 0 satisfying v 0 ≤ w 0 .

Conclusions
In this work, the maximum principle of the linear problem involving delay term is first established. Then the approach of upper and lower solutions and the monotone iterative technique are applied to consider the existence as well as the uniqueness of ωperiodic solutions for the nth-order ordinary differential Equation (1) by using the obtained maximum principle. The existence of lower and upper ω-periodic solutions of Equation (1) is also discussed in this paper. The results extend and improve some existing works.
Author Contributions: All authors contributed equally in writing this paper. All authors read and approved the final manuscript.