Multiple Positive Periodic Solutions for Two Kinds of Higher

Hindawi Publishing Corporation Journal of Mathematics Volume 2014, Article ID 214093, 13 pages http://dx.doi.org/10.1155/2014/214093

Research Article Multiple Positive Periodic Solutions for Two Kinds of Higher-Dimension Impulsive Differential Equations with Multiple Delays and Two Parameters Zhenguo Luo1,2 1 2

Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Correspondence should be addressed to Zhenguo Luo; [email protected] Received 8 September 2013; Revised 13 February 2014; Accepted 13 February 2014; Published 6 April 2014 Academic Editor: Nan-Jing Huang Copyright © 2014 Zhenguo Luo. This is an open access 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. By applying the fixed point theorem, we derive some new criteria for the existence of multiple positive periodic solutions for two kinds of 𝑛-dimension periodic impulsive functional differential equations with multiple delays and two parameters: 𝑥𝑖󸀠 (𝑡) = 𝑎𝑖 (𝑡)𝑥𝑖 (𝑡) − 𝜆𝑏𝑖 (𝑡)𝑓𝑖 (𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏1 (𝑡)), . . . , 𝑥(𝑡 − 𝜏𝑛 (𝑡)))), a.e., 𝑡 > 0, 𝑡 ≠ 𝑡𝑘 , 𝑘 ∈ 𝑍+ , 𝑥𝑖 (𝑡𝑘+ ) − 𝑥𝑖 (𝑡𝑘− ) = 𝜇𝑐𝑖𝑘 𝑥𝑖 (𝑡𝑘 ), 𝑖 = 1, 2, . . . , 𝑛, 𝑘 ∈ 𝑍+ , and 𝑥𝑖󸀠 (𝑡) = −𝑎𝑖 (𝑡)𝑥𝑖 (𝑡) + 𝜆𝑏𝑖 (𝑡)𝑓𝑖 (𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏1 (𝑡)), . . . , 𝑥(𝑡 − 𝜏𝑛 (𝑡)))), a.e., 𝑡 > 0, 𝑡 ≠ 𝑡𝑘 , 𝑘 ∈ 𝑍+ , 𝑥𝑖 (𝑡𝑘+ ) − 𝑥𝑖 (𝑡𝑘− ) = 𝜇𝑐𝑖𝑘 𝑥𝑖 (𝑡𝑘 ), 𝑖 = 1, 2, . . . , 𝑛, 𝑘 ∈ 𝑍+ . As an application, we study some special cases of the previous systems, which have been studied extensively in the literature.

1. Introduction Let 𝑅 = (−∞, +∞), 𝑅+ = [0, +∞), 𝑅− = (−∞, 0], 𝑅𝑛 = {(𝑥1 , . . . , 𝑥𝑛 )𝑇 : 𝑥𝑖 ≥ 0, 1 ≤ 𝑖 ≤ 𝑛}, 𝐽 ⊂ 𝑅, and 𝑍+ = {1, 2, 3, . . .}, respectively. Denote by 𝑃𝐶(𝐽, 𝑅𝑛 ) the set of operators 𝜑 : 𝐽 → 𝑅𝑛 which are continuous for 𝑡 ∈ 𝐽, 𝑡 ≠ 𝑡𝑘 and have discontinuities of the first kind at the points 𝑡𝑘 ∈ 𝐽 (𝑘 ∈ 𝑍+ ) but are continuous from the left at these points. For each 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 )𝑇 ∈ 𝑅𝑛 , the norm of 𝑥 is defined as |𝑥| = ∑𝑛𝑖=1 |𝑥𝑖 |. Let 𝐵𝐶(𝑅, 𝑅+𝑛 ) denote the Banach space of bounded continuous functions 𝜓 : 𝑅 → 𝑅+𝑛 with the norm ‖𝜓‖ = sup𝜉∈𝑅 ∑𝑛𝑖=1 |𝜓𝑖 (𝜉)|, where 𝜓 = (𝜓1 , 𝜓2 , . . . , 𝜓𝑛 )𝑇 . The matrix 𝐴 > 𝐵 (𝐴 ≤ 𝐵) means that each pair of corresponding elements of 𝐴 and 𝐵 satisfies the inequality “ > ” (“ ≤ ”). In particular, 𝐴 is called a positive matrix if 𝐴 > 0. Impulsive differential equations are suitable for the mathematical simulation of evolutionary process whose states are subject to sudden changes at certain moments. Equations of this kind are found in almost every domain of applied sciences, and numerous examples are given in [1–4]. In recent years, the existence theory of positive periodic solutions of delay differential equations with impulsive effects or without

impulsive effects has been an object of active research, and we refer the reader to [5–17]. Recently, in [5], Jiang and Wei studied the following nonimpulsive delay differential equation: 𝑥󸀠 (𝑡) = −𝑎 (𝑡) 𝑥 (𝑡) + 𝑓 (𝑡, 𝑥 (𝑡 − 𝜏0 (𝑡)) , 𝑥 (𝑡 − 𝜏1 (𝑡))) , . . . , 𝑥 (𝑡 − 𝜏𝑛 (𝑡)) . (1) They obtained sufficient conditions for the existence of the positive periodic solutions of (1). Motivated by [5], in [6], Zhao et al. investigated the following impulsive delay differential equation: 𝑥󸀠 (𝑡) = −𝑎 (𝑡) 𝑥 (𝑡) + 𝑓 (𝑡, 𝑥 (𝑡 − 𝜏0 (𝑡)) , 𝑥 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑥 (𝑡 − 𝜏𝑛 (𝑡))) a.e., 𝑡 > 0, 𝑡 ≠ 𝑡𝑘 ; 𝑥 (𝑡𝑘+ ) − 𝑥 (𝑡𝑘 ) = 𝑏𝑘 𝑥 (𝑡𝑘 ) ,

𝑘 = 1, 2, . . . . (2)

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Journal of Mathematics

They derived some sufficient conditions for the existence of the positive periodic solutions of (2). In [7], Huo et al. considered the following impulsive delay differential equation: 𝑥󸀠 (𝑡) + 𝛼 (𝑡) 𝑥 (𝑡) = 𝑝 (𝑡) 𝑓 (𝑡, 𝑥 (𝑡 − 𝜎 (𝑡))) a.e., 𝑡 > 0, 𝑡 ≠ 𝑡𝑘 ; 𝑥 (𝑡𝑘+ ) − 𝑥 (𝑡𝑘 ) = 𝑏𝑘 𝑥 (𝑡𝑘 ) ,

(3)

𝑘 = 1, 2, . . . .

They got sufficient conditions for the existence and attractivity of the positive periodic solutions of (3). Motivated by [5– 7], in [8], Zhang et al. studied the following impulsive delay differential equation: 𝑥󸀠 (𝑡) = −𝑎 (𝑡) 𝑥 (𝑡) + 𝑝 (𝑡) 𝑓 (𝑡, 𝑥 (𝑡 − 𝜏0 (𝑡)) , 𝑥 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑥 (𝑡 − 𝜏𝑛 (𝑡))) 𝑥 (𝑡𝑘+ ) − 𝑥 (𝑡𝑘 ) = 𝑏𝑘 𝑥 (𝑡𝑘 ) ,

a.e., 𝑡 > 0, 𝑡 ≠ 𝑡𝑘 ;

𝑘 = 1, 2, . . . . (4)

They obtained some sufficient conditions for the existence of the positive periodic solutions of (4). However, to this day, only a little work has been done on the existence of positive periodic solutions to the high-dimension impulsive differential equations based on the theory in cones. Motivated by this, in this paper, we mainly consider the following two classes of impulsive functional differential equations with two parameters: 𝑥󸀠 (𝑡) = 𝐴 (𝑡) 𝑥 (𝑡) − 𝜆𝐵 (𝑡) 𝑓 (𝑡, 𝑢 (𝑡)) Δ𝑥 (𝑡𝑘 ) = 𝜇𝐶𝑘 𝑥 (𝑡𝑘 ) ,

a.e. , 𝑡 > 0, 𝑡 ≠ 𝑡𝑘 , 𝑘 ∈ 𝑍+ , (5)

𝑥󸀠 (𝑡) = −𝐴 (𝑡) 𝑥 (𝑡) + 𝜆𝐵 (𝑡) 𝑓 (𝑡, 𝑢 (𝑡)) Δ𝑥 (𝑡𝑘 ) = 𝜇𝐶𝑘 𝑥 (𝑡𝑘 ) ,

a.e., 𝑡 > 0, 𝑡 ≠ 𝑡𝑘 ,

𝑘 ∈ 𝑍+ , (6)

(𝑃1 ) 𝑎𝑖 , 𝑏𝑖 , 𝜏𝑖 : 𝑅+ → 𝑅+ are locally summable 𝜔-periodic functions; that is, 𝑎𝑖 (𝑡 + 𝜔) = 𝑎𝑖 (𝑡), 𝑏𝑖 (𝑡 + 𝜔) = 𝑏𝑖 (𝑡), and 𝜏𝑖 (𝑡 + 𝜔) = 𝜏𝑖 (𝑡) for all 𝑡 ≥ 0, 𝜆 > 0, and 𝜇 > 0 are two parameters; (𝑃2 ) 𝑓 = (𝑓1 , . . . , 𝑓𝑛 )𝑇 ∈ 𝑅 × 𝐵𝐶(𝑅, 𝑅+𝑛 ) and for all (𝑡, 𝑢1 , . . . , 𝑢𝑛 ) ∈ 𝑅 × 𝐵𝐶(𝑅, 𝑅𝑛 ), 𝑓𝑖 (𝑡 + 𝜔, 𝑢1 , . . . , 𝑢𝑛 ) = 𝑓𝑖 (𝑡, 𝑢1 , . . . , 𝑢𝑛 ) such that 𝑓𝑖 (𝑡, 𝑢1 , . . . , 𝑢𝑛 ) ≢ 0, 𝑖 = 1, 2, . . . , 𝑛; (𝑃3 ) {𝑡𝑘 }, 𝑘 ∈ 𝑍+ satisfies 0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑘 < ⋅ ⋅ ⋅ and lim𝑘 → +∞ 𝑡𝑘 = +∞. 𝐶𝑘 : 𝑅+𝑛 → 𝑅 (𝑘 ∈ 𝑍+ ) satisfy Caratheodory conditions and are 𝜔-periodic functions in 𝑡. Moreover, there exists a positive constant 𝑞 such that 𝑡𝑘+𝑞 = 𝑡𝑘 + 𝜔, 𝑘 ∈ 𝑍+ . Without loss of generality, we can assume that 𝑡𝑘 ≠ 0 and [0, 𝜔] ∩ {𝑡𝑘 , 𝑘 ∈ 𝑍+ } = {𝑡1 , 𝑡2 , . . . , 𝑡𝑞 }; (𝑃4 ) {𝑐𝑖𝑘 } is a real sequence such that 𝜇𝑐𝑖𝑘 > −1, 𝑖 = 1, 2, . . . , 𝑛, 𝑘 ∈ 𝑍+ and 𝑐𝑖 (𝑡) := ∏0 0 is a constant. We assume that there exists an integer 𝑞 > 0 such that 𝑡𝑘+𝑞 = 𝑡𝑘 + 𝜔, 𝑐𝑖(𝑘+𝑞) = 𝑐𝑖𝑘 (𝑖 = 1, 2, . . . , 𝑛), where 0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑞 < 𝜔. Throughout the paper, we make the following assumptions:

(b) for each 𝑘 ∈ 𝑍+ , 𝑥𝑖 (𝑡𝑘+ ) and 𝑥𝑖 (𝑡𝑘− ) exist, and 𝑥𝑖 (𝑡𝑘− ) = 𝑥𝑖 (𝑡𝑘 );

𝜃 ∈ [−𝜏, 0] ,

𝜙𝑖 ∈ 𝐶 ([−𝜏, 0) , [0, +∞)) ,

(7)

𝑖 = 1, 2 . . . , 𝑛, where

Under the previous hypotheses (𝑃1 )–(𝑃4 ), we consider the neutral nonimpulsive system:

𝑢 (𝑡) = (𝑥 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑥 (𝑡 − 𝜏𝑛 (𝑡))) = (𝑢1 (𝑡) , . . . , 𝑢𝑛 (𝑡)) ,

(c) 𝑥𝑖 (𝑡) satisfies the first equation of (5) and (6) for almost everywhere (for short a.e.) in [0, ∞] \ {𝑡𝑘 } and satisfies 𝑥𝑖 (𝑡𝑘+ ) = (1 + 𝑐𝑖𝑘 )𝑥𝑖 (𝑡𝑘 ) for 𝑡 = 𝑡𝑘 , 𝑘 ∈ 𝑍+ = {1, 2, . . .}.

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𝜏 = max sup 𝜏𝑖 (𝑡) , 1≤𝑖≤𝑛

and 𝐴(𝑡) = diag[𝑎1 (𝑡), 𝑎2 (𝑡), . . . , 𝑎𝑛 (𝑡)], 𝐵(𝑡) = diag[𝑏1 (𝑡), 𝑏2 (𝑡), . . . , 𝑏𝑛 (𝑡)], 𝑎𝑖 , 𝑏𝑖 ∈ 𝐶(𝑅, 𝑅+ ) (𝑖 = 1, 2 . . . , 𝑛) are 𝜔periodic; that is, 𝑎𝑖 (𝑡 + 𝜔) = 𝑎𝑖 (𝑡), 𝑏𝑖 (𝑡 + 𝜔) = 𝑏𝑖 (𝑡),

𝑑𝑦 = 𝐴 (𝑡) 𝑦 (𝑡) − 𝜆𝐵 (𝑡) 𝑓 (𝑡, V (𝑡)) , 𝑑𝑡

a.e., 𝑡 ≥ 0,

𝑑𝑦 = −𝐴 (𝑡) 𝑦 (𝑡) + 𝜆𝐵 (𝑡) 𝑓 (𝑡, V (𝑡)) , 𝑑𝑡

a.e., 𝑡 ≥ 0,

(9)

(10)


3 = ∏ (1 + 𝜇𝑐𝑖𝑘 )

with initial conditions: 𝑦𝑖 (𝜉) = 𝜑𝑖 (𝜉) , 𝜑𝑖 (0) > 0,

0 󵄩󵄩𝑦󵄩󵄩 ,

󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 0 󵄩󵄩 󵄩󵄩 󵄩𝑦󵄩 𝜎𝑅 𝑅 > 󵄩 󵄩𝐿 ≤ = , 𝜆𝛼𝜎𝑏 𝜆𝛼𝜎𝑏𝐿 𝜆𝛼𝑏𝐿

for any 𝑦 ∈ 𝐸 ∩ 𝜕Ω𝑅 .

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Theorem 15. In addition to (𝑃1 )–(𝑃4 ), if (𝐻5 ) and (𝐻8 ) hold, then system (5) has at least one positive 𝜔-periodic solution.

Proof. By (𝐻7 ), for any 𝜖 = 𝜃2 − 1/𝜆𝛼𝜎𝑏𝐿 > 0, there exists a sufficiently small 𝑅 > 0 such that 𝜔󵄨 󵄨 ∫0 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑦󵄩󵄩

1 > 𝜃2 + 𝜖 = , 𝜆𝛼𝜎𝑏𝐿

𝜔󵄨 󵄨 ∫0 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑦󵄩󵄩

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󵄩 󵄩 for 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑟;

𝜔

󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 0 󵄩󵄩 󵄩󵄩 󵄩𝑦󵄩 𝜎𝑅 𝑅 > 󵄩 󵄩𝐿 ≥ = , 𝐿 𝜆𝛼𝜎𝑏 𝜆𝛼𝜎𝑏 𝜆𝛼𝑏𝐿

𝜔

which implies that (𝐻10 ) is satisfied.

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which implies that (𝐻9 ) is satisfied. On the other hand, by (𝐻6 ), for any 𝜖 = 1/𝜆𝛽𝑏𝑀 − 𝛾1 > 0, there exists a sufficiently large 𝑟∗ > 0 such that 𝜔󵄨 󵄨 ∫0 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑦󵄩󵄩

1 < 𝛾1 + 𝜖 = , 𝜆𝛽𝑏𝑀

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󵄩 󵄩 for 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≥ 𝑟∗ .

In the following, we consider two cases to prove (𝐻6 ) to be 𝜔 satisfied: ∫0 |𝑓(𝑡, 𝑦(𝑡 − 𝜏1 (𝑡)), . . . , 𝑦(𝑡 − 𝜏𝑛 (𝑡)))|𝑑𝑡 are bounded 𝜔

󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 0 󵄩 󵄩 for 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑟,

󵄩 󵄩 for 𝜎𝑅 ≤ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑅,

and unbounded. The bounded case is clear. If ∫0 |𝑓(𝑡, 𝑦(𝑡 − 𝜏1 (𝑡)), . . . , 𝑦(𝑡 − 𝜏𝑛 (𝑡)))|𝑑𝑡 are unbounded, then there exist 𝑦∗ ∈ 𝑅+𝑛 , 𝑟 = ‖𝑦∗ ‖ ≥ 𝑟∗ and 𝑡0 ∈ [0, 𝜔] such that

that is

󵄩󵄩 󵄩󵄩 󵄩𝑦󵄩 𝑟 < 󵄩 󵄩𝑀 ≤ , 𝜆𝛽𝑏 𝜆𝛽𝑏𝑀

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󵄩 󵄩 for 0 < 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑅;

that is

Proof. By (𝐻5 ), for any 𝜖 = 1/𝜆𝛽𝑏𝑀 − 𝜃1 > 0, there exists a sufficiently small 𝑟 > 0 such that

1 , 𝜆𝛽𝑏𝑀

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which implies that (𝐻9 ) is satisfied. Therefore, by Theorem 14, we complete the proof.

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In conclusion, under the assumptions (𝐻8 ) and (𝐻10 ), 𝜓 satisfies the conditions in Lemma 4, and then 𝜓 has a fixed point 𝑥1 ∈ 𝐸 ∩ (Ω𝑅 \ Ω𝑟 ). By Lemma 4, the system (5) has at least one positive 𝜔-periodic solution 𝑥1 satisfying 𝑟 < ‖𝑥1 ‖ < 𝑅, where 𝑟 and 𝑅 are defined in (𝐻9 ) and (𝐻10 ), respectively. The proof of Theorem 14 is complete.

< 𝜃1 + 𝜖 =

󵄩 󵄩 for 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑅,

Theorem 16. In addition to (𝑃1 )–(𝑃4 ), if (𝐻6 ) and (𝐻7 ) hold, then system (5) has at least one positive 𝜔-periodic solution.

󵄨 󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜓𝑦󵄩󵄩 = ∑ sup 󵄨󵄨󵄨(𝜓𝑖 𝑦) (𝑡)󵄨󵄨󵄨

> 𝜆𝑏𝐿 𝛼

󵄩 󵄩 for 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≥ 𝜎𝑅;

that is

𝑟 󵄩 󵄩 = 𝑟 = 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 , 𝜆𝑏𝑀𝛽

which yields 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜓𝑦󵄩󵄩 < 󵄩󵄩𝑦󵄩󵄩 ,

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𝜔

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󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 0 𝜔

(76) 󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦∗ (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦∗ (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡, 0 󵄩 󵄩 󵄩 󵄩 for any 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑦∗ 󵄩󵄩󵄩 = 𝑟.

10


Since 𝑟 = ‖𝑦‖ ≥ ‖𝑦∗ ‖ ≥ 𝑟∗ , then we get

In conclusion, under the assumptions (𝐻1 ) and (𝐻6 ), 𝜓 satisfies the conditions in Lemma 4, and then 𝜓 has a fixed point in 𝐸∩(Ω𝑟2 \Ω𝑟1 ). By Lemma 4, the system (5) has at least one positive 𝜔-periodic solution. The proof of Theorem 19 is complete.

𝜔

󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 0 𝜔

󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦∗ (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦∗ (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 0 󵄩󵄩 ∗ 󵄩󵄩 󵄩𝑦 󵄩󵄩 𝑟 < 󵄩 𝑀 = , 𝜆𝛽𝑏 𝜆𝛽𝑏𝑀

(77) Similar to Theorem 19, we can get the following consequences.

󵄩 󵄩 for any 0 < 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑟,

which implies that the condition (𝐻6 ) holds. By Theorem 14, we complete the proof. Theorem 17. In addition to (𝑃1 )–(𝑃4 ), if (𝐻5 ), (𝐻6 ), and (𝐻9 ) hold, then system (5) has at two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑅 < ‖𝑥2 ‖, where 𝑅 is defined in (𝐻9 ). Proof. By (𝐻5 ) and the proof of Theorem 15, there exists a sufficiently small 𝑟1 ∈ (0, 𝑟) such that 𝜔

󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 0 󵄩󵄩 󵄩󵄩 󵄩𝑦󵄩 𝑟1 < 󵄩 󵄩𝑀 ≤ , 𝜆𝛽𝑏 𝜆𝛽𝑏𝑀

󵄩 󵄩 for 0 < 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑟1 .

(78)

On the other hand, from (𝐻6 ) and the proof of Theorem 16, there exists a sufficiently large 𝑟2 ∈ (𝑟, ∞) such that

Theorem 20. In addition to (𝑃1 )–(𝑃4 ), if (𝐻2 ) and (𝐻5 ) hold, then system (5) has at least one positive 𝜔-periodic solution. Theorem 21. In addition to (𝑃1 )–(𝑃4 ), if (𝐻3 ) and (𝐻8 ) hold, then system (5) has at least one positive 𝜔-periodic solution. Theorem 22. In addition to (𝑃1 )–(𝑃4 ), if (𝐻4 ) and (𝐻7 ) hold, then system (5) has at least one positive 𝜔-periodic solution. Theorem 23. In addition to (𝑃1 )–(𝑃4 ), if (𝐻1 ), (𝐻8 ), and (𝐻10 ) hold, then system (5) has two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑟 < ‖𝑥2 ‖, where 𝑟 is defined in (𝐻10 ). Proof. Let Ω𝑟 = {𝑦 ∈ 𝑋 : ‖𝑦‖ < 𝑟}. By (𝐻1 ) and the proof of Theorem 10, there exists a sufficiently small 𝑟1 ∈ (0, 𝑟) such that 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 𝐿 󵄩󵄩𝜓𝑦󵄩󵄩 ≥ 𝜆𝑏 𝛽𝜂 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,

for any 𝑦 ∈ 𝐸 ∩ 𝜕Ω𝑟1 .

𝜔

𝑟 󵄨 󵄨 , ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 < 𝜆𝛽𝑏𝑀 0 󵄩 󵄩 for 0 < 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≤ 𝑟2 .

(82)

Likewise, by (𝐻8 ) and the proof of Theorem 15, there exists a sufficiently large 𝑟2 ∈ (0, 𝑟) such that (79)

Therefore, from the proof of Theorem 14, there exist two positive solutions 𝑦1 and 𝑦2 satisfying 𝑟 < ‖𝑦1 ‖ < 𝑟 < ‖𝑦2 ‖ < 𝑟2 , where 𝑟 is defined in (𝐻9 ); the proof of Theorem 17 is complete. Theorem 18. In addition to (𝑃1 )–(𝑃4 ), if (𝐻7 ), (𝐻8 ), and (𝐻10 ) hold, then system (5) has at least two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑟 < ‖𝑥2 ‖, where R is defined in (𝐻10 ). Proof. The proof is similar to that of Theorem 17, and we omit the details here. Theorem 19. In addition to (𝑃1 )–(𝑃4 ), if (𝐻1 ) and (𝐻6 ) hold, then system (5) has at least one positive 𝜔-periodic solution. Proof. Let Ω𝑟 = {𝑦 ∈ 𝑋 : ‖𝑦‖ < 𝑟}. By (𝐻1 ) and the proof of Theorem 10, there exists a sufficiently small 𝑟1 ∈ (0, 𝑟) such that 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 𝐿 󵄩󵄩𝜓𝑦󵄩󵄩 ≥ 𝜆𝑏 𝛽𝜂 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 , for any 𝑦 ∈ 𝐸 ∩ 𝜕Ω𝑟1 . (80) Likewise, by (𝐻6 ) and the proof of Theorem 16, there exists a sufficiently large 𝑟2 ∈ (𝑟, ∞) such that 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜓𝑦󵄩󵄩 < 󵄩󵄩𝑦󵄩󵄩 , for any 𝑦 ∈ 𝐸 ∩ 𝜕Ω𝑟2 . (81)

󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 𝐿 󵄩󵄩𝜓𝑦󵄩󵄩 ≥ 𝜆𝑏 𝛽𝜂 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,

for any 𝑦 ∈ 𝐸 ∩ 𝜕Ω𝑟2 .

(83)

Incorporating (𝐻10 ) and the proof of Theorem 14, we know that there exist two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 𝑟1 < ‖𝑥1 ‖ < 𝑟 < ‖𝑥2 ‖ < 𝑟2 , where 𝑟 is defined in (𝐻10 ). The proof of Theorem 23 is complete. Similar to Theorem 23, one immediately has the following consequences. Theorem 24. In addition to (𝑃1 )–(𝑃4 ), if (𝐻2 ), (𝐻7 ), and (𝐻10 ) hold, then system (5) has two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑟 < ‖𝑥2 ‖, where 𝑟 is defined in (𝐻10 ). Theorem 25. In addition to (𝑃1 )–(𝑃4 ), if (𝐻3 ), (𝐻6 ), and (𝐻9 ) hold, then system (5) has two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑅 < ‖𝑥2 ‖, where R is defined in (𝐻9 ). Theorem 26. In addition to (𝑃1 )–(𝑃4 ), if (𝐻4 ), (𝐻5 ), and (𝐻9 ) hold, then system (5) has two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑅 < ‖𝑥2 ‖, where R is defined in (𝐻9 ).


11

3. Existence of Periodic Solutions of System (6) Now, we are at the position to study the existence of positive periodic solutions of system (6). By carrying out similar arguments as in Section 2, it is not difficult to derive sufficient criteria for the existence of positive periodic solutions of system (6). For simplicity, we prefer to list below the corresponding criteria for system (6) without proof, since the proofs are very similar to those in Section 2. For (𝑡, 𝑠) ∈ 𝑅2 , 1 ≤ 𝑖 ≤ 𝑛, we define 𝑠

𝐺𝑖∗

(𝑡, 𝑠) =

𝑒∫𝑡 𝑎𝑖 (𝜉)𝑑𝜉

𝜔

𝑒∫0

𝑎𝑖 (𝜉)𝑑𝜉

−1

,

Lemma 27. Assume that (𝑃1 )–(𝑃4 ) hold. The existence of positive 𝜔-periodic solution of system (6) is equivalent to that of nonzero fixed point of 𝐴 in 𝑃. Lemma 28. Assume that (𝑃1 )–(𝑃4 ) hold. Then the solutions of system (6) are defined on [−𝜏, ∞) and are positive. Lemma 29. Assume that (𝑃1 )–(𝑃4 ) hold. Then 𝐴 : 𝑃 → 𝑃 is well defined. Lemma 30. Assume that (𝑃1 )–(𝑃4 ) hold, and there exists 𝜂 > 0 such that 𝜔

(84)

󵄩 󵄩 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 ≥ 𝜂 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 , 0

≤ ≤ 𝐺∗ (𝑡, 𝑡 + 𝜔), and it is clear ∗ ∗ 𝐺𝑖 (𝑡, 𝑡 + 𝜔) − 𝐺𝑖∗ (𝑡, 𝑡) = 1. In 𝜕𝐺𝑖 (𝑡, 𝑠)/𝜕𝑡 = view of (𝑃1 ), we also define for 1 ≤ 𝑖 ≤ 𝑛

𝑓𝑜𝑟 𝑎𝑛𝑦 𝑦 ∈ 𝑃,

∗

𝐺 (𝑡, 𝑠) =

diag [𝐺1∗

(𝑡, 𝑠) , 𝐺2∗

(𝑡, 𝑠) , . . . , 𝐺𝑛∗

that 𝐺𝑖∗ (𝑡, 𝑡) 𝑎𝑖 (𝑡)𝐺𝑖∗ (𝑡, 𝑠),

𝛼𝑖∗

󵄨 󵄨 := min 󵄨󵄨󵄨𝐺𝑖∗ (𝑡, 𝑠)󵄨󵄨󵄨 = 0≤𝑡≤𝑠≤𝜔

󵄨 󵄨 𝛽𝑖∗ := max 󵄨󵄨󵄨𝐺𝑖∗ (𝑡, 𝑠)󵄨󵄨󵄨 = 0≤𝑡≤𝑠≤𝜔 ∗

𝛼 =

min 𝛼∗ 1≤𝑖≤𝑛 𝑖

= 𝛼,

(𝑡, 𝑠)] ,

𝐺𝑖∗ (𝑡, 𝑠) 1

𝜔

𝑒∫0

𝑎𝑖 (𝜉)𝑑𝜉

−1

𝜔 ∫0 𝑎𝑖 (𝜉)𝑑𝜉

𝑒

𝜔 ∫0 𝑎𝑖 (𝜉)𝑑𝜉

𝑒

−1

󵄩󵄩 󵄩󵄩 󵄩 󵄩 𝐿 󵄩󵄩𝜓𝑦󵄩󵄩 ≥ 𝜆𝑏 𝛼𝜂 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,

1≤𝑖≤𝑛

1≤𝑖≤𝑛

(85)

󵄩󵄩 󵄩󵄩 󵄩 󵄩 𝑀 󵄩󵄩𝐴𝑦󵄩󵄩 ≤ 𝜆𝑏 𝛽𝜖 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,

(𝑡)} . 𝑛

Let 𝑋 = {𝑦 = (𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑛 (𝑡)) ∈ 𝑃𝐶(𝑅, 𝑅 ) | 𝑦(𝑡 + 𝜔) = 𝑦(𝑡)} with the norm ‖𝑦‖ = ∑𝑛𝑖=1 |𝑦𝑖 |0 , |𝑦𝑖 |0 = sup𝑡∈[0,𝜔] |𝑦𝑖 (𝑡)|, and it is easy to verify that (𝑋, ‖⋅‖) is a Banach space. Define 𝑃 to be a cone in 𝑋 by 𝑇

(86)

󵄩 󵄩 ∈ 𝑋 : 𝑦𝑖 (𝑡) ≥ 𝛿󵄩󵄩󵄩𝑦𝑖 󵄩󵄩󵄩0 , 𝑡 ∈ [0, 𝜔]} .

We easily verify that 𝑃 is a cone in 𝑋. We define an operator 𝐴 : 𝑋 → 𝑋 as follows: 𝑇

(𝐴𝑦) (𝑡) = ((𝐴 1 𝑦) (𝑡) , (𝐴 2 𝑦) (𝑡) , . . . , (𝐴 𝑛 𝑦) (𝑡)) , (87) where 𝑡+𝜔

𝑡

𝑓𝑜𝑟 𝑎𝑛𝑦 𝑦 ∈ 𝑃 ∩ 𝜕Ω𝑟 , and then

𝑇

(𝐴 𝑖 𝑦) (𝑡) = 𝜆 ∫

󵄩 󵄩 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑓 (𝑡, 𝑦 (𝑡 − 𝜏1 (𝑡)) , . . . , 𝑦 (𝑡 − 𝜏𝑛 (𝑡)))󵄨󵄨󵄨 𝑑𝑡 ≤ 𝜖 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 , 0 (91)

min {𝐵𝑖󸀠 1≤𝑖≤𝑛

𝑃 = {𝑦 = (𝑦1 (𝑡) , 𝑦2 (𝑡) , . . . , 𝑦𝑛 (𝑡))

(90)

𝜔

𝛽 = max 𝛽𝑖∗ = 𝛽,

𝐵 (𝑡) =

𝑓𝑜𝑟 𝑎𝑛𝑦 𝑦 ∈ 𝑃.

Lemma 31. Assume that (𝑃1 )–(𝑃4 ) hold, and let 𝑟 > 0. If there exists a sufficiently small 𝜖 > 0 such that

= 𝛽𝑖 ,

𝛼∗ 𝛿 = ∗ ∈ (0, 1) = 𝜎, 𝛽 󵄨 󵄨 󵄨 󵄨 𝐵𝑖 (𝑡) = max {󵄨󵄨󵄨𝑏1𝑖 (𝑡)󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑏2𝑖 (𝑡)󵄨󵄨󵄨} , 󵄨 󵄨 󵄨 󵄨 𝐵𝑖󸀠 (𝑡) = min {󵄨󵄨󵄨𝑏1𝑖 (𝑡)󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑏2𝑖 (𝑡)󵄨󵄨󵄨} , 𝐵 (𝑡) = max {𝐵𝑖 (𝑡)} ,

and then

= 𝛼𝑖 ,

∗

󸀠

(89)

𝐺𝑖∗ (𝑡, 𝑠) 𝑏𝑖 (𝑠) 𝑓𝑖

× (𝑠, 𝑦 (𝑠 − 𝜏1 (𝑠)) , . . . , 𝑦 (𝑠 − 𝜏𝑛 (𝑠))) 𝑑𝑠. (88) The proof of the following lemmas and theorems is similar to those in Section 2, and we all omit the details here.

𝑓𝑜𝑟 𝑎𝑛𝑦 𝑦 ∈ 𝑃 ∩ 𝜕Ω𝑟 .

(92)

Theorem 32. Assume that (𝑃1 )–(𝑃4 ) and (𝐻9 ) hold. Moreover, if one of the following conditions holds: (𝐻3 ) 𝑎𝑛𝑑 (𝐻4 ); (𝐻5 ) 𝑎𝑛𝑑 (𝐻6 ),

(𝐻3 ) 𝑎𝑛𝑑 (𝐻6 );

(𝐻4 ) 𝑎𝑛𝑑 (𝐻5 );

then system (6) has two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑅 < ‖𝑥2 ‖, where 𝑅 is defined in (𝐻9 ). Theorem 33. Assume that (𝑃1 )–(𝑃4 ), and (𝐻10 ) hold. Moreover, if one of the following conditions holds: (𝐻1 ) 𝑎𝑛𝑑 (𝐻2 ); (𝐻7 ) 𝑎𝑛𝑑 (𝐻8 ),

(𝐻1 ) 𝑎𝑛𝑑 (𝐻8 );

(𝐻2 ) 𝑎𝑛𝑑 (𝐻7 );

then system (6) has two positive 𝜔-periodic solutions 𝑥1 and 𝑥2 satisfying 0 < ‖𝑥1 ‖ < 𝑟 < ‖𝑥2 ‖, where r is defined in (𝐻10 ). Theorem 34. Assume that (𝑃1 )–(𝑃4 ) hold. Moreover, if one of the following conditions holds: (𝐻1 ) 𝑎𝑛𝑑 (𝐻4 ); (𝐻1 ) 𝑎𝑛𝑑 (𝐻6 ); (𝐻2 ) 𝑎𝑛𝑑 (𝐻3 ); (𝐻2 ) 𝑎𝑛𝑑 (𝐻5 ); (𝐻3 ) 𝑎𝑛𝑑 (𝐻8 ); (𝐻4 ) 𝑎𝑛𝑑 (𝐻9 ); (𝐻5 ) 𝑎𝑛𝑑 (𝐻8 ); (𝐻6 ) 𝑎𝑛𝑑 (𝐻7 ); (𝐻9 ) 𝑎𝑛𝑑 (𝐻10 ), then system (6) has at least one positive 𝜔-periodic solution.

12


4. Examples

On the other hand, we have 𝜔

In order to illustrate our results, we take the following examples.

∫ 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏𝑖 (𝑡))) 𝑑𝑡 0

𝑛

Example 35. We consider the following generalized so-called Michaelis-Menton type single species growth model with impulse:

𝑖=1 0

𝑛

𝛼𝑖 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡)) ], 𝑖=1 1 + 𝛽𝑖 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡)) (93)

𝑘 ∈ 𝑍+ ,

which is a special case of system (5), and where 𝑎(𝑡), 𝛼𝑖 (𝑡), 𝛽𝑖 (𝑡), 𝜏𝑖 (𝑡) ∈ 𝐶(𝑅, 𝑅+ ) (𝑖 = 1, 2, . . . , 𝑛) are 𝜔-periodic, and 𝜆 > 0, 𝜇 > 0 are two parameters. Theorem 36. Assume that (𝑃1 )–(𝑃4 ) hold. Moreover, if the following condition holds: 𝑛

𝜔

∑ ∫ 𝛼𝑖 (𝑡) 𝑑𝑡 > 𝑖=1 0

𝛽𝑖𝑀 , 𝜆𝛼𝜎3 𝑏𝐿

(99)

𝜔

𝑖=1 0

𝑦󸀠 (𝑡) = 𝑦 (𝑡) [𝑎 (𝑡) − 𝜆∑

Δ𝑦 (𝑡𝑘 ) = 𝜇𝑐𝑘 𝑦 (𝑡𝑘 ) ,

𝛼𝑖 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡)) 𝑑𝑡 1 + 𝛽𝑖 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡))

󵄩 󵄩2 ≤ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ∑ ∫ 𝛼𝑖 (𝑡) 𝑑𝑡.

𝑛

𝑡 ∈ 𝑅, 𝑡 ≠ 𝑡𝑘 ,

𝜔

= ∑ ∫ 𝑦 (𝑡)

(94)

then system (93) has at least one positive 𝜔-periodic solution.

This can lead to 𝜔

∫0 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏𝑖 (𝑡))) 𝑑𝑡 󵄩 󵄩 𝑛 𝜔 ≤ 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ∑ ∫ 𝛼𝑖 (𝑡) 𝑑𝑡 󳨀→ 0, 󵄩󵄩 󵄩󵄩 (100) 󵄩󵄩𝑦󵄩󵄩 𝑖=1 0 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 󳨀→ 0. 󵄩 󵄩 That is 𝑓0 = 0.

(101)

By Theorem 21, it follows that system (93) has at least one positive 𝜔-periodic solution. The proof of Theorem 36 is complete. Example 37. We consider the following generalized hematopoiesis model with impulse: 𝑦󸀠 (𝑡) = −𝛼 (𝑡) 𝑦 (𝑡) + 𝜆𝛽 (𝑡) exp {−𝛾 (𝑡) 𝑦 (𝑡 − 𝜏 (𝑡))} ,

Proof. Note that

𝑡 ∈ 𝑅, 𝑡 ≠ 𝑡𝑘 ,

𝑛

𝛼 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡)) 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏𝑖 (𝑡))) = 𝑦 (𝑡) ∑ 𝑖 . 1 + 𝛽𝑖 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡)) 𝑖=1

Δ𝑦 (𝑡𝑘 ) = 𝜇𝑐𝑘 𝑦 (𝑡𝑘 ) ,

𝑘 ∈ 𝑍+ ,

(95)

(102)

We can construct the same Banach space 𝑋 and cone 𝐸 as in Section 2. Then for any 𝑦 ∈ 𝐸, we have

which is a special case of system (6), and where 𝑥(𝑡) is the number of red blood cells at time 𝑡, 𝛼(𝑡), 𝛽(𝑡), 𝛾(𝑡) and 𝜏(𝑡) ∈ 𝐶(𝑅, 𝑅+ ) are 𝜔-periodic and 𝜆 > 0, 𝜇 > 0 are two parameters.

𝜔

Theorem 38. Assume that (𝑃1 )–(𝑃4 ) hold. Moreover, if the following condition holds:

∫ 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏𝑖 (𝑡))) 𝑑𝑡 0

𝑛

𝜔

= ∑ ∫ 𝑦 (𝑡) 𝑖=1 0

≥

𝛼𝑖 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡)) 𝑑𝑡 1 + 𝛽𝑖 (𝑡) 𝑦 (𝑡 − 𝜏𝑖 (𝑡))

𝑛

𝑖=1 0

󵄩 󵄩2 𝑛 𝜔 𝜎2 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 󵄩 󵄩 ∑ ∫ 𝛼 (𝑡) 𝑑𝑡. 1 + 𝑐𝑖𝑀 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 𝑖=1 0 𝑖

𝛽𝑖𝑀 , 𝜆𝛼𝜎3 𝑏𝐿

Proof. Note that 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏 (𝑡))) = exp {−𝛾 (𝑡) 𝑦 (𝑡 − 𝜏 (𝑡))} .

󵄩 󵄩 ∫0 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏𝑖 (𝑡))) 𝑑𝑡 𝜎 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 𝑛 𝜔 ≥ 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ∑ ∫ 𝛼 (𝑡) 𝑑𝑡. 1 + 𝛽𝑖𝑀 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 𝑖=1 0 𝑖 󵄩󵄩𝑦󵄩󵄩 2

(104)

We can construct the same Banach space 𝑋 and cone 𝐸 as in Section 2. Then for any 𝑦 ∈ 𝐸, we have (97)

Then we can have 1 𝜎2 𝑛 𝜔 𝑓∞ ≥ 𝑀 ∑ ∫ 𝛼𝑖 (𝑡) 𝑑𝑡 > . 𝐿 𝜆𝛼𝜎𝑏 𝛽𝑖 𝑖=1 0

(103)

then system (102) has at least one positive 𝜔-periodic solution.

This can lead to 𝜔

𝜔

∑ ∫ 𝛼𝑖 (𝑡) 𝑑𝑡 >

(96)

𝜔

∫0 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏 (𝑡))) 𝑑𝑡 𝜔 ≥ 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩; 𝑀 exp {𝛾 󵄩󵄩󵄩𝑦󵄩󵄩󵄩} 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 󵄩󵄩𝑦󵄩󵄩 𝜔

(98)

∫0 𝑓 (𝑡, 𝑦 (𝑡 − 𝜏 (𝑡))) 𝑑𝑡 𝜔 ≤ 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩. 𝐿 exp {𝛾 󵄩󵄩󵄩𝑦󵄩󵄩󵄩} 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 󵄩󵄩𝑦󵄩󵄩

(105)


13

This can lead to 𝑓∞ = 0, 𝑓0 = ∞,

󵄩󵄩 󵄩󵄩 󵄩󵄩𝑦󵄩󵄩 󳨀→ ∞; 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 󳨀→ 0. 󵄩 󵄩

(106)

By Theorem 34, it follows that system (102) has at least one positive 𝜔-periodic solution. The proof of Theorem 38 is complete. Remark 39. We apply the main results obtained in the previous sections to study some examples which have some biological implications. A very basic and important ecological problem associated with the study of population is that of the existence of positive periodic solutions which play the role played by the equilibrium of the autonomous models and means that the species is in an equilibrium state. From Theorems 36 and 38, we see that under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of the systems.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The research is supported by NSF of China (nos. 11161015, 11371367, and 11361012), PSF of China (nos. 2012M512162 and 2013T60934), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, 12C0361, and 13C084), and the Construct Program of the Key Discipline in Hunan Province.

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