Positive periodic solutions for high-order differential equations with multiple delays in Banach spaces

where Lnu(t) = u(n)(t) + ∑n–1 i=0 aiu (i)(t) is the nth-order linear differential operator, ai ∈R (i = 0, 1, . . . ,n – 1) are constants, f :R× Em → E is a continuous function which is ω-periodic with respect to t, and τi > 0 (i = 1, 2, . . . ,m) are constants which denote the time delays. We first prove the existence of ω-periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω-periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones.


Introduction
In recent years, the existence of periodic solutions for differential equations has been studied by many authors. But in some practice models, only positive periodic solutions are more important. For second-order differential equations without delay, the existence of positive periodic solutions has been discussed extensively, see [1,5,[10][11][12] and the references therein. In [2], Chu and Zhou considered the periodic solutions for the third-order differential equation solutions. In [4], Feng studied the third-order differential equation where δ and are positive constants. By utilizing the Guo-Krasnoselskii fixed point theorem in cones, he established some existence and multiplicity results of positive 2πperiodic solutions. For the more general case, in [7], Li proved some existence theorems of positive 2π -periodic solutions for the high-order differential equation where L n u(t) = u (n) (t) + n-1 i=0 a i u (i) (t) is the nth-order linear differential operator, a i ∈ R (i = 0, 1, . . . , n -1) are constants. However, in these works, the authors did not consider the effect of the delay in the equation. Recently, Li [8] discussed the existence of positive ω-periodic solutions of the second-order differential equation with delays of the form where a ∈ C(R, (0, ∞)) is an ω-periodic function, f : R × [0, ∞) m → [0, ∞) is a continuous function that is ω-periodic with respect to t, and τ 1 , τ 2 , . . . , τ m are positive constants. The results obtained in [8] can deal with the case of second-order differential equations, but for high-order differential equations, for example, L n u(t) = 1 2 u 2 (tτ 1 ) + 1 4 u 2 (tτ 2 ) + 1 8 u 2 (tτ 3 ), t ∈ R, the results of [8] are not valid. Motivated by the papers mentioned above, we consider the existence of positive ωperiodic solutions for the nth-order nonlinear ordinary differential equations in Banach space E where f : R × E m → E is a continuous function that is ω-periodic with respect to t, and τ 1 , τ 2 , . . . , τ m are positive constants which denote the time delays. The main features and crucial technique of the present paper are summarized as follows: (i) In this paper, we discuss the effect of multiple delays in the high-order ordinary differential equation in abstract Banach spaces, which has seldom been studied before. (ii) Since the integral operator Q is not compact in abstract Banach spaces, the fixed theorems of completely continuous mapping are not valid for this problem. In order to overcome this difficulty, we provide a measure of non-compactness condition (R1) on nonlinear term f , which is much weaker than some existing results. And we prove that the operator Q is a condensing mapping, see Lemma 2.7. (iii) By utilizing the perturbation method, we obtain the existence of positive ω-periodic solution of the linear differential equation corresponding to Eq. (1.1).
Then the strong positivity estimation of the operator T is established by using the positivity of G n (t, s) and T n , see Lemma 2.3. (iv) In our main results Theorem 3.1 and Theorem 3.2, we provide some order conditions on nonlinearity f to guarantee the existence of positive ω-periodic solutions of Eq. (1.1), which are much easier to verify in application. The rest of this paper is organized as follows. In Sect. 2, we introduce some preliminaries and prove the existence of positive solutions of the corresponding linear problem. The main results of this paper are presented in Sect. 3. Some remarks are given to show the superiority of this work.

Preliminaries
Let I = [0, ω], C(I, R) be the Banach space of all continuous functions furnished with the norm u C = max t∈I |u(t)|. For ∀h ∈ C(I, R), we first consider the linear boundary value problem (LBVP) Denote by P n (λ) the characteristic polynomial of L n : P n (λ) = λ n + a n-1 λ n-1 + · · · + a 0 .
Let E be a Banach space whose positive cone K is normal. Denote by C ω (R, E) the Banach space of all E-valued ω-periodic continuous functions on R endowed with the norm u C = max t∈I u(t) . Let K C = C ω (R, K) be the normal cone of C ω (R, E).
is called a positive ω-periodic solution of Eq. (1.1) if u(t) > 0 for any t ∈ R and u(t) satisfies Eq. (1.1).

Lemma 2.2
Let assumption (P) hold and a 0 > 0. Then, for ∀h ∈ C ω (R, E), the linear equation has a unique ω-periodic solution given by where r n (t) ∈ C ∞ (I, R) is the unique solution of LBVP(2.1), and T n : is a positive and bounded linear operator, whose norm satisfies T n = 1 a 0 .
Proof From Lemma 2.1, if assumption (P) holds and a 0 > 0, LBVP(2.1) has a unique solution r n (t) > 0 for any t ∈ I. By Lemma 1 of [7], the linear periodic boundary value problem(LPBVP) has a unique solution u ∈ C n (I, E), which is given by the expression
is a positive operator. It remains to prove that T n is bounded and T n = 1 a 0 . On the one hand, for ∀h ∈ C ω (R, E), the inequality This means that T n is bounded. On the other hand, let h 0 (t) ≡ 1 for all t ∈ R. Then h 0 ∈ C ω (R, E) and h 0 C = 1. So, In order to prove the existence of positive ω-periodic solutions of Eq. (1.1), for ∀h ∈ C ω (R, E), we consider the linear differential equation with delay of the form where ρ ≥ 0 is a constant.
If r n (t) > 0 for t ∈ I, let m n = min t∈I r n (t) and M n = max t∈I r n (t). Then 0 < m n ≤ r n (t) ≤ M n . By Lemmas 2.1 and 2.2, we obtain the following lemma.

Lemma 2.3 Assume that (P) holds and
is a positive linear bounded operator satisfying the strong positivity estimate Proof By Lemma 2.2, the ω-periodic solution of Eq. (2.5) is expressed by Define an operator B by Then Hence, by (2.6) and (2.7), we have Since T n B ≤ T n B ≤ ρ a 0 < 1, by the perturbation theorem, (I + T n B) -1 exists and By direct calculation, we get Hence, by (2.8), we have which is an ω-periodic solution of (2.5). By (2.10) and (2.11), we have Therefore, is a positive operator and (Th)(t) ≥ γ (Th)(s) for any t, s ∈ R. By (2.9) and (2.11), we have we get is a positive operator. Moreover, for any h ∈ C ω (R, K), by (2.11), we have So, we have Consequently, for any t, s ∈ R, we have (2.14) Hence, (Th)(t) ≥ γ (Th)(s) for any t, s ∈ R.
Let E be a separable Banach space. Denote by β E (·) and β C (·) the Hausdorff measure of non-compactness(MNC) of the bounded set in E and C ω (R, E), respectively.

Lemma 2.4 Let D ⊂ C(I, E) be a bounded and equicontinuous subset. Then β E (D(t)) is continuous on I and
Now, we consider the existence of positive ω-periodic solutions for the high-order differential equation with delays of the form (1.1). By Lemma 2.3, we define an operator By the continuity of f , the operator Q : C ω (R, E) − → C ω (R, E) is continuous. The positive ω-periodic solution of the high-order differential equation (1.1) is equivalent to the positive fixed point of Q. It is noted that the integral operator Q is not compact in an abstract Banach space. In order to employ the topological degree theory of condensing mapping, it demands that the nonlinear term f satisfies some MNC conditions. Thus, we make the following assumption. Proof For any r > 0, let K r, for any t ∈ R and x i ∈ K r , i = 1, 2, . . . , m. Hence, for any u ∈ K r,C , by (2.15), we have By assumption (R1) and Lemma 2.6, we have Consequently, we have K) is a condensing mapping due to (2.16).
Remark 1 In Lemma 2.7, if the nonlinearity f satisfies linear growth condition, for example, f satisfies the following condition: (R2) There exist constants C i > 0 (i = 1, 2, . . . , m) and b > 0 such that for any t ∈ R and x i ∈ K , i = 1, 2, . . . , m, then (2.17) holds for where γ = m n M n . Then we can obtain the following lemma.
Proof For any t, s ∈ R and u ∈ C ω (R, K), by (2.15), we have It follows from the above two inequalities that (I + T n B)(Qu)(t) ≥ γ (I + T n B)(Qu)(s), ∀t, s ∈ R.

Existence of positive ω-periodic solutions
Let E be a separable Banach space and K ⊂ E be a positive cone of E. For any positive constants R and r, let for any t ∈ R and x i ∈ Ξ , i = 1, 2, . . . , m.
Proof Let Ξ be the closed convex cone of C ω (R, K) defined by (2.20). Define an operator Q : C ω (R, K) → C ω (R, K) by (2.15). We show that Q has a fixed point in Ξ ∩ Ω r,R for r > 0 small enough and R > 0 sufficiently large.
Let r ∈ (0, δ), where δ is the positive constant in assumption (H1). We prove that Q satisfies the conditions of Lemma 2.9 in Ξ ∩ Ω r , namely, In fact, if there exist u 0 ∈ Ξ ∩ ∂Ω r and 0 < λ 0 ≤ 1 such that then by the definition of Q and Lemma 2.3, u 0 satisfies the delayed differential equation i.e., Since u 0 ∈ ∂Ω r , by the definition of Ω r , we have It follows from (H1) that Hence, by (3.2), we have Integrating both sides of this inequality from 0 to ω and using the periodicity of u 0 , we have By the definition of cone Ξ , we have Consequently, Since m i=1 c i ≤ a 0 γ 2 , it follows that u 0 (s) ≤ 0 for s ∈ R, which is a contradiction to u 0 ∈ ∂Ω r . Hence, for any u ∈ Ξ ∩ ∂Ω r and 0 < λ ≤ 1, we have λQu = u.
By Lemma 2.9, we have Let e ∈ C(R, K) with e(t) ≡ 1 for any t ∈ R. Then e ∈ Ξ \ {θ }. We show that Q satisfies the conditions of Lemma 2.10 in Ξ ∩ ∂Ω R , that is, for R > 0 sufficiently large. In fact, if there exist u 1 ∈ Ξ ∩ ∂Ω R and μ 1 ≥ 0 such that Then, by the definition of Q and Lemma 2.3, u 1 satisfies the delayed differential equation Since u 1 ∈ Ξ ∩ ∂Ω R , by condition (H2), we have Integrating both sides of this inequality from 0 to ω and using the periodicity of u 1 , we have (H4) There exist positive constants c 1 , c 2 , . . . , c m satisfying m i=1 c i < a 0 and h 1 ∈ C ω (R, K) such that for any t ∈ R and x i ∈ Ξ , i = 1, 2, . . . , m.
Proof For any 0 < r < R < +∞, choose Ξ , Ω r , Ω R , and Ω r,R as in the proof of Theorem 3.1. Define an operator Q by (2.15), then by (R1), Q : C ω (R, K) − → C ω (R, K) is a condensing mapping. We will show that the operator Q has at least one fixed point in Ξ ∩ Ω r,R .
Hence, from the above inequality, we have Consequently, we have Let R > max{R, r}. Then all the conditions of Lemma 2.9 are satisfied. By Lemma 2.9, we have i(Q, Ξ ∩ ∂Ω R , Ξ ) = 1.

Conclusion
In the present work, we establish some sufficient conditions on nonlinear term f to guarantee the existence of positive ω-periodic solutions of Eq.