Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 239209, 18 pages
doi:10.1155/2009/239209
Research Article

Multiple Positive Periodic Solutions for Delay Differential System

Zhao-Cai Hao,1,2 Ti-Jun Xiao,2,3 and Jin Liang2,4

1Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China
2Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
3School of Mathematical Sciences, Fudan University, Shanghai 200433, China
4Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 11 September 2009; Accepted 6 December 2009

Academic Editor: Binggen Zhang

Copyright © 2009 Zhao-Cai Hao et al. 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.

Abstract

We obtain some existence results for multiple positive periodic solutions of some delay differential systems. Examples are presented as applications. For a general positive integer 𝑚 ≥ 2 , main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our results are new also when 𝑚 = 1 .

1. Introduction

It is known that multiple delay Logistic equations

𝑦 ′ ( 𝑡 ) = − 𝑎 ( 𝑡 ) 𝑦 ( 𝑡 ) + 𝑓 𝑡 , 𝑦 𝑡 − 𝜏 0 ( 𝑡 ) , 𝑦 𝑡 − 𝜏 1 ( 𝑡 ) , … , 𝑦 𝑡 − 𝜏 𝑛 ( 𝑡 ) , 𝑦 ′ ( 𝑡 ) = 𝑎 ( 𝑡 ) 𝑦 ( 𝑡 ) − 𝑓 𝑡 , 𝑦 𝑡 − 𝜏 0 ( 𝑡 ) , 𝑦 𝑡 − 𝜏 1 ( 𝑡 ) , … , 𝑦 𝑡 − 𝜏 𝑛 ( 𝑡 ) , ( 1 . 1 ) are generalizations of many biological models, such as Logistic models of Single-species growth (see [1–3]), 𝑦 ′ ( 𝑡 ) = 𝑎 ( 𝑡 ) 𝑦 ( 𝑡 ) 1 − 𝑦 ( 𝑡 − 𝜏 ( 𝑡 ) ) 𝐾 ( 𝑡 ) , 𝑦 ′ ( 𝑡 ) = 𝑦 ( 𝑡 ) ⎡ ⎢ ⎣ 𝑎 ( 𝑡 ) − 𝑛 𝑖 = 1 𝑏 𝑖 ( 𝑡 ) 𝑦 𝑡 − 𝜏 𝑖 ( 𝑡 ) ⎤ ⎥ ⎦ , 𝑦 ′ ( 𝑡 ) = 𝑎 ( 𝑡 ) 𝑦 ( 𝑡 ) ⎡ ⎢ ⎣ 1 − 𝑛 𝑖 = 1 𝑦 𝑡 − 𝜏 𝑖 ( 𝑡 ) 𝐾 ( 𝑡 ) ⎤ ⎥ ⎦ , ( 1 . 2 ) and red blood cell models (see [4–7]),

𝑦 ′ ( 𝑡 ) = − 𝑎 ( 𝑡 ) 𝑦 ( 𝑡 ) + 𝑏 ( 𝑡 ) 𝑒 − 𝑦 ( 𝑡 − 𝜏 ( 𝑡 ) ) , 𝑦 ′ ( 𝑡 ) = − 𝑎 ( 𝑡 ) 𝑦 ( 𝑡 ) + 𝑏 ( 𝑡 ) 1 + 𝑦 𝑛 ( 𝑡 − 𝜏 ( 𝑡 ) ) . ( 1 . 3 ) For biological models, positive periodic solutions are often important and many results have been achieved in this direction, for instance, [8–10].

To the best of our knowledge, few papers concerning the existence of multiple positive solutions of (1.1) can be found in literature. Furthermore, no papers have yet deal with the more general nonautonomous delay differential systems

𝑦 1 ( 𝑡 ) = − 𝑎 1 ( 𝑡 ) 𝑦 1 ( 𝑡 ) + 𝑓 1 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑚 ( 𝑡 ) , 𝑦 2 ( 𝑡 ) = − 𝑎 2 ( 𝑡 ) 𝑦 2 ( 𝑡 ) + 𝑓 2 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑚 ( 𝑡 ) , ⋮ 𝑦 𝑚 ( 𝑡 ) = − 𝑎 𝑚 ( 𝑡 ) 𝑦 𝑚 ( 𝑡 ) + 𝑓 𝑚 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑚 ( 𝑡 ) , ( 1 . 4 ) 𝑦 1 ( 𝑡 ) = 𝑎 1 ( 𝑡 ) 𝑦 1 ( 𝑡 ) − 𝑓 1 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑛 ( 𝑡 ) , 𝑦 2 ( 𝑡 ) = 𝑎 2 ( 𝑡 ) 𝑦 2 ( 𝑡 ) − 𝑓 2 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑛 ( 𝑡 ) , ⋮ 𝑦 𝑛 ( 𝑡 ) = 𝑎 𝑛 ( 𝑡 ) 𝑦 𝑛 ( 𝑡 ) − 𝑓 𝑚 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑛 ( 𝑡 ) , ( 1 . 5 ) where 𝑚 , 𝑛 are all positive integer and

𝑎 𝑖 ( ⋅ ) , 𝑖 ∈ Λ 1 ∶ = { 1 , 2 , … , 𝑚 } , 𝑓 𝑖 ( ⋅ , … , ⋅ ) , 𝑖 ∈ Λ 1 , 𝜏 𝑗 ( ⋅ ) , 𝑗 ∈ Λ 2 ∶ = { 1 , 2 , … , 𝑛 } ( 1 . 6 ) are given functions and signs 𝑌 𝑖 , 𝑖 ∈ Λ 1 are given as follows:

𝑌 𝑖 ( 𝑡 ) ∶ = 𝑦 𝑖 𝑡 − 𝜏 1 ( 𝑡 ) , 𝑦 𝑖 𝑡 − 𝜏 2 ( 𝑡 ) , … , 𝑦 𝑖 𝑡 − 𝜏 𝑛 ( 𝑡 ) , 𝑡 ∈ 𝑅 , 𝑖 ∈ Λ 1 . ( 1 . 7 ) The extension to systems is a natural one; for example, many occurrences in nature involve two or more populations coexisting in an environment, with the model being best described by a system of differential equations (see [11]).

The aim of this paper is to investigate systems (1.4) and (1.5). In what follows we only discuss the existence of positive periodic solutions of system (1.4); similar results can be obtained for system (1.5). By using multiple fixed-point theorems (see Lemmas 2.1 and 2.2), which are different from those used in [8–10], we obtain the existence of multiple positive periodic solutions of system (1.4) (see Theorems 3.1, 4.1, and 4.3). Some examples are given also to illustrate our main theorems. Main results of this paper are new also even if 𝑚 = 1 (see Remark 4.5).

This paper is organized as follows. In Section 2, we make some preliminaries. In Section 3, we derive existence result (see Theorem 3.1) for two positive periodic solutions of system (1.4). Example 3.2 is given below Theorem 3.1. The existence of three positive periodic solutions of system (1.4) is presented in Section 4 (see Theorems 4.1 and 4.3). Applications of Theorems 4.1 and 4.3 may be seen from Examples 4.2 and 4.4.

2. Preliminaries

We make the basic assumption throughout this paper that

𝑇 > 0 i s a fi x e d c o n s t a n t ; 𝑎 𝑖 ∈ 𝐶 ( 𝑅 , [ 0 , ∞ ) ) , 𝑎 ( 𝑡 ) ≢ 0 , 𝑎 𝑖 ( 𝑡 ) = 𝑎 𝑖 ( 𝑡 + 𝑇 ) , 𝑡 ∈ 𝑅 , 𝑖 ∈ Λ 1 ; 𝑓 𝑖 ∈ 𝐶 𝑅 × [ 0 , ∞ ) 𝑚 × 𝑛 , [ 0 , ∞ ) , 𝑖 ∈ Λ 1 ; 𝑓 𝑖 i s 𝑇 - p e r i o d i c f u n c t i o n i n r e l a t i v e t o 𝑡 , 𝑖 ∈ Λ 1 ; 𝜏 𝑗 ∈ 𝐶 ( 𝑅 , [ 0 , ∞ ) ) , 𝜏 𝑗 ( 𝑡 + 𝑇 ) = 𝜏 𝑗 ( 𝑡 ) , 𝑡 ∈ 𝑅 , 𝑗 ∈ Λ 2 . ( 2 . 1 )

Let us now provide some preparations. Let 𝑆 be a real Banach space and let 𝑃 be a cone in 𝑆 . A map 𝛼 is said to be a nonnegative continuous concave functional on cone 𝑃 if 𝛼 ∶ 𝑃 → [ 0 , ∞ ) is continuous and

𝛼 ( 𝑡 𝑥 + ( 1 − 𝑡 ) 𝑦 ) ≥ 𝑡 𝛼 ( 𝑥 ) + ( 1 − 𝑡 ) 𝛼 ( 𝑦 ) ∀ 𝑥 , 𝑦 ∈ 𝑃 , 𝑡 ∈ [ 0 , 1 ] . ( 2 . 2 ) For numbers 𝑀 , 𝑁 such that 0 < 𝑀 < 𝑁 , and a nonnegative continuous concave functional 𝛼 on cone 𝑃 , we define

𝑃 𝑀 ∶ = { 𝑥 ∈ 𝑃 ∶ ‖ 𝑥 ‖ < 𝑀 } , 𝑃 ( 𝛼 , 𝑀 ) ∶ = { 𝑥 ∈ 𝑃 ∶ 𝛼 ( 𝑥 ) < 𝑀 } , 𝑃 ( 𝛼 , 𝑀 , 𝑁 ) ∶ = { 𝑥 ∈ 𝑃 ∶ 𝑀 ≤ 𝛼 ( 𝑥 ) , ‖ 𝑥 ‖ ≤ 𝑁 } . ( 2 . 3 ) Setting 𝑢 = ( 𝑢 1 , 𝑢 2 , … , 𝑢 𝑛 ) ∈ [ 0 , ∞ ) 𝑛 , we define

| 𝑢 | 0 ∶ = m a x 𝑗 ∈ Λ 2 𝑢 𝑗 . ( 2 . 4 ) Write

𝐷 ∶ = { 𝑦 ( 𝑡 ) ∶ 𝑦 ∈ 𝐶 ( 𝑅 , 𝑅 ) , 𝑦 ( 𝑡 + 𝑇 ) = 𝑦 ( 𝑡 ) } , 𝐸 ∶ = 𝐷 × 𝐷 × ⋯ × 𝐷 𝑚 , ‖ 𝑦 ‖ 0 ∶ = s u p 𝑡 ∈ [ 0 , 𝑇 ] | | 𝑦 ( 𝑡 ) | | f o r 𝑦 ∈ 𝐷 , ‖ 𝑦 ‖ ∶ = ∑ 𝑖 ∈ Λ 1 ‖ ‖ 𝑦 𝑖 ‖ ‖ 0 f o r 𝑦 = 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ∈ 𝐸 , 𝑃 ∶ = 𝑦 = 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ∈ 𝐸 ∶ 𝑦 𝑖 ( 𝑡 ) ≥ 𝛿 𝑖 ‖ ‖ 𝑦 𝑖 ‖ ‖ 0 , 𝑖 ∈ Λ 1 , ( 2 . 5 ) where

𝛿 𝑖 = 𝑒 − ∫ 𝑇 0 𝑎 𝑖 ( 𝑠 ) 𝑑 𝑠 , 𝑖 ∈ Λ 1 . ( 2 . 6 ) Then ( 𝐷 , ‖ ⋅ ‖ 0 ) and ( 𝐸 , ‖ ⋅ ‖ ) are all Banach spaces and 𝑃 is a cone in 𝐸 . Set

𝑀 𝑖 1 ∶ = 1 𝑒 ∫ 𝑇 0 𝑎 𝑖 ( 𝑠 ) 𝑑 𝑠 − 1 , 𝑀 𝑖 2 ∶ = 𝑒 ∫ 𝑇 0 𝑎 𝑖 ( 𝑠 ) 𝑑 𝑠 𝑒 ∫ 𝑇 0 𝑎 𝑖 ( 𝑠 ) 𝑑 𝑠 − 1 , 𝑖 ∈ Λ 1 , 𝐺 𝑖 ( 𝑡 , 𝑠 ) ∶ = 𝑒 ∫ 𝑠 𝑡 𝑎 𝑖 ( 𝜉 ) 𝑑 𝜉 𝑒 ∫ 𝑇 0 𝑎 𝑖 ( 𝜉 ) 𝑑 𝜉 − 1 , ( 𝑡 , 𝑠 ) ∈ 𝑅 × [ 𝑡 , 𝑡 + 𝑇 ] , 𝑖 ∈ Λ 1 . ( 2 . 7 ) It is easy to see that for any ( 𝑡 , 𝑠 ) ∈ 𝑅 × [ 𝑡 , 𝑡 + 𝑇 ] , functions 𝐺 𝑖 ( 𝑡 , 𝑠 ) , 𝑖 ∈ Λ 1 have properties

𝑀 𝑖 1 ∶ = 𝐺 𝑖 ( 𝑡 , 𝑡 ) ≤ 𝐺 𝑖 ( 𝑡 , 𝑠 ) ≤ 𝐺 𝑖 ( 𝑡 , 𝑡 + 𝑇 ) ∶ = 𝑀 𝑖 2 , 𝑖 ∈ Λ 1 , 𝛿 𝑖 = 𝑀 𝑖 1 𝑀 𝑖 2 ≤ 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝐺 𝑖 ( 𝑡 , 𝑡 + 𝑇 ) ≤ 1 , 𝑖 ∈ Λ 1 . ( 2 . 8 )

Now we define an operator 𝐴 ∶ 𝐸 → 𝐸 as follows:

𝐴 𝑦 ( 𝑡 ) ∶ = 𝐴 1 ( 𝑡 ) , 𝐴 2 ( 𝑡 ) , … , 𝐴 𝑚 ( 𝑡 ) , 𝑡 ∈ 𝑅 , 𝑦 = 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ∈ 𝐸 , ( 2 . 9 ) where

𝐴 𝑖 ( 𝑡 ) ∶ = 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑛 ( 𝑠 ) 𝑑 𝑠 , 𝑡 ∈ 𝑅 , 𝑖 ∈ Λ 1 , ( 2 . 1 0 ) signs 𝑌 𝑖 , 𝑖 ∈ Λ 1 are given in (1.7) and we often use them in the remainder of this paper. It is easy to say that a 𝑇 -periodic solution of operator equation

𝑦 = 𝐴 𝑦 , ( 2 . 1 1 ) on 𝑃 , that is, a fixed point of operator 𝐴 , is a 𝑇 -positive periodic solution of system (1.4). So, our main results concerning multiple positive solutions of system (1.4) will arise as application of the following fixed-point theorem.

Lemma 2.1 (see [12]). Let 𝑃 be a cone in a real Banach space 𝐵 . Let 𝛼 and 𝛾 be increasing, nonnegative, continuous functionals on 𝑃 , and let 𝜃 be a nonnegative continuous functional on 𝑃 with 𝜃 ( 0 ) = 0 such that, for some 𝑐 > 0 and 𝑀 > 0 , 𝛾 ( 𝑥 ) ≤ 𝜃 ( 𝑥 ) ≤ 𝛼 ( 𝑥 ) , ‖ 𝑥 ‖ ≤ 𝑀 𝛾 ( 𝑥 ) , ∀ 𝑥 ∈ 𝑃 ( 𝛾 , 𝑐 ) . ( 2 . 1 2 ) Suppose there exists a completely continuous operator 𝐴 ∶ 𝑃 ( 𝛾 , 𝑐 ) → 𝑃 and 0 < 𝑎 < 𝑏 < 𝑐 such that 𝜃 ( 𝜆 𝑥 ) ≤ 𝜆 𝜃 ( 𝑥 ) , f o r 0 ≤ 𝜆 ≤ 1 , 𝑥 ∈ 𝜕 𝑃 ( 𝜃 , 𝑏 ) , ( 2 . 1 3 ) and (i) 𝛾 ( 𝐴 𝑥 ) > 𝑐 , for all 𝑥 ∈ 𝜕 𝑃 ( 𝛾 , 𝑐 ) ; (ii) 𝜃 ( 𝐴 𝑥 ) < 𝑏 , for all 𝑥 ∈ 𝜕 𝑃 ( 𝜃 , 𝑏 ) ; (iii) 𝑃 ( 𝛼 , 𝑎 ) ≠ 𝜙 , and 𝛼 ( 𝐴 𝑥 ) > 𝑎 , for all 𝑥 ∈ 𝜕 𝑃 ( 𝛼 , 𝑎 ) .
Then 𝐴 has at least two fixed points 𝑥 1 and 𝑥 2 belonging to 𝑃 ( 𝛾 , 𝑐 ) such that 𝑎 < 𝛼 𝑥 1 , w i t h 𝜃 𝑥 1 < 𝑏 , 𝑏 < 𝜃 𝑥 2 , w i t h 𝛾 𝑥 2 < 𝑐 . ( 2 . 1 4 )

Lemma 2.2 (see [13]). Let 𝑃 be a cone in a real Banach space 𝐸 , let 𝐴 ∶ 𝑃 𝑐 → 𝑃 𝑐 be completely continuous, and let 𝛼 be a nonnegative continuous concave functional on 𝑃 with 𝛼 ( 𝑥 ) ≤ ‖ 𝑥 ‖ for all 𝑥 ∈ 𝑃 𝑐 . Suppose that there exists 0 < 𝑑 < 𝑀 < 𝑁 ≤ 𝑐 such that (i) { 𝑥 ∈ 𝑃 ( 𝛼 , 𝑀 , 𝑁 ) ∶ 𝛼 ( 𝑥 ) > 𝑀 } ≠ 𝜙 and 𝛼 ( 𝐴 𝑥 ) > 𝑀 for 𝑥 ∈ 𝑃 ( 𝛼 , 𝑀 , 𝑁 ) ;(ii) ‖ 𝐴 𝑥 ‖ < 𝑑 for all ‖ 𝑥 ‖ ≤ 𝑑 ;(iii) 𝛼 ( 𝐴 𝑥 ) > 𝑀 for 𝑥 ∈ 𝑃 ( 𝛼 , 𝑀 , 𝑐 ) with ‖ 𝐴 𝑥 ‖ > 𝑁 .
Then 𝐴 has at least three fixed points 𝑥 1 , 𝑥 2 , 𝑥 3 satisfying ‖ ‖ 𝑥 1 ‖ ‖ < 𝑑 , 𝑀 < 𝛼 𝑥 2 , ‖ ‖ 𝑥 3 ‖ ‖ > 𝑑 , 𝛼 𝑥 3 < 𝑀 . ( 2 . 1 5 )

3. Existence of Two Positive Solutions of System (1.4)

In this section, we apply Lemma 2.1 to establish Theorem 3.1, the existence result of two positive solutions of system (1.4). Example 3.2 will be given as an application of Theorem 3.1.

Theorem 3.1. Assume that there exist numbers 0 < 𝑎 < 𝑏 < 𝑐 such that the following three assumptions are satisfied. ( 𝐻 1 ) One has 𝑓 𝑘 0 𝑡 , 𝑈 1 , 𝑈 2 , … , 𝑈 𝑚 > 𝑐 + 1 𝑇 𝑀 1 , 𝑡 ∈ 𝑅 , 𝜑 ∈ 𝑐 , 𝑐 𝛿 − 1 , ( 3 . 1 ) where 𝑘 0 ∈ Λ 1 is fixed and 𝑈 𝑖 ∶ = 𝑈 1 𝑖 , 𝑈 2 𝑖 , … , 𝑈 𝑛 𝑖 , 𝑈 𝑗 𝑖 ∈ [ 0 , ∞ ) , 𝑖 ∈ Λ 1 , 𝑗 ∈ Λ 2 , 𝑀 1 ∶ = m i n 𝑀 1 1 , 𝑀 2 1 , … , 𝑀 𝑚 1 , 𝛿 ∶ = m i n 𝛿 1 , 𝛿 2 , … , 𝛿 𝑚 , 𝜑 ∶ = | | 𝑈 1 | | 0 + | | 𝑈 2 | | 0 + ⋯ + | | 𝑈 𝑚 | | 0 , ( 3 . 2 ) ( 𝐻 2 ) 𝑡 ∈ 𝑅 , 𝑈 𝑖 ∈ [ 0 , ∞ ) 𝑛 , 𝑖 ∈ Λ 1 , and 𝜑 ∈ [ 𝑏 , 𝑏 𝛿 − 1 ] imply 𝑚 𝑖 = 1 𝑓 𝑖 𝑡 , 𝑈 1 , 𝑈 2 , … , 𝑈 𝑚 < 𝑏 ( 1 + 𝛿 ) 𝑇 𝑀 2 , ( 3 . 3 ) where 𝑀 2 ∶ = m a x 𝑀 1 2 , 𝑀 2 2 , … , 𝑀 𝑚 2 , ( 3 . 4 ) ( 𝐻 3 ) 𝑡 ∈ 𝑅 , 𝑈 𝑖 ∈ [ 0 , ∞ ) 𝑛 , 𝑖 ∈ Λ 1 , and 𝜑 ∈ [ 𝛿 𝑎 , 𝑎 ] imply 𝑚 𝑖 = 1 𝑓 𝑖 𝑡 , 𝑈 1 , 𝑈 2 , … , 𝑈 𝑚 > 𝑎 + 1 𝑇 𝑀 1 . ( 3 . 5 )
Then system (1.4) has at least two 𝑇 -positive periodic solutions.

Proof. We begin by defining 𝛾 ( 𝑦 ) ∶ = 𝜃 ( 𝑦 ) ∶ = 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑦 𝑖 ( 𝑡 ) , 𝑦 = 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ∈ 𝑃 , 𝛼 ( 𝑦 ) ∶ = ‖ 𝑦 ‖ , 𝑦 ∈ 𝑃 . ( 3 . 6 ) Clearly, 𝛼 and 𝛾 are increasing, nonnegative, continuous functionals on 𝑃 , and 𝜃 is nonnegative a continuous functional on 𝑃 with 𝜃 ( 0 ) = 0 . Moreover, we observe that 𝛾 ( 𝑦 ) = 𝜃 ( 𝑦 ) ≤ 𝛼 ( 𝑦 ) , ∀ 𝑦 ∈ 𝑃 , ( 3 . 7 ) ‖ 𝑦 ‖ ≤ 𝛿 − 1 𝛾 ( 𝑦 ) , ∀ 𝑦 ∈ 𝑃 , ( 3 . 8 ) 𝜃 ( 𝜆 𝑥 ) = 𝜆 𝜃 ( 𝑥 ) , f o r 0 ≤ 𝜆 ≤ 1 , 𝑥 ∈ 𝜕 𝑃 ( 𝜃 , 𝑏 ) . ( 3 . 9 ) Now, we proceed to show that other conditions of Lemma 2.1 are also satisfied.
Firstly, we will show that 𝐴 ∶ 𝑃 ( 𝛾 , 𝑐 ) ⟶ 𝑃 i s c o m p l e t e l y c o n t i n u o u s . ( 3 . 1 0 ) In fact, we have from (2.8), for any 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) ∈ 𝑃 ( 𝛾 , 𝑐 ) , ‖ ‖ 𝐴 𝑖 ‖ ‖ 0 ≤ 𝑀 𝑖 2 𝑇 0 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑛 ( 𝑠 ) 𝑑 𝑠 , 𝑖 ∈ Λ 1 , ( 3 . 1 1 ) which yields 𝐴 𝑖 ( 𝑡 ) ≥ 𝑀 𝑖 1 𝑇 0 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑛 ( 𝑠 ) 𝑑 𝑠 ≥ 𝑀 𝑖 1 𝑀 𝑖 2 ‖ ‖ 𝐴 𝑖 ‖ ‖ 0 = 𝛿 𝑖 ‖ ‖ 𝐴 𝑖 ‖ ‖ 0 , 𝑡 ∈ 𝑅 , 𝑦 ∈ 𝑃 ( 𝛾 , 𝑐 ) , 𝑖 ∈ Λ 1 . ( 3 . 1 2 ) Hence 𝐴 𝑦 ∈ 𝑃 for all 𝑦 ∈ 𝑃 ( 𝛾 , 𝑐 ) . Furthermore, we know from the continuity of functions 𝑓 𝑖 ( ⋅ , … , ⋅ ) , 𝑎 𝑖 ( ⋅ ) , Γ 𝑖 ( ⋅ , ⋅ ) , 𝑖 ∈ Λ 1 that the operator 𝐴 is completely continuous. Hence, we conclude that (3.10) holds.
Secondly, let us prove 𝛾 ( 𝐴 𝑦 ) > 𝑐 , ∀ 𝑦 ∈ 𝜕 𝑃 ( 𝛾 , 𝑐 ) . ( 3 . 1 3 ) For any 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) ∈ 𝜕 𝑃 ( 𝛾 , 𝑐 ) , so that 𝛾 ( 𝑦 ) = 𝑐 , we get, in view of (1.7), (2.4) and (3.8), 𝑐 = 𝑚 ∑ 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑦 𝑖 ( 𝑡 ) ≤ 𝑚 ∑ 𝑖 = 1 | | 𝑌 𝑖 ( 𝑡 ) | | 0 , 𝑡 ∈ 𝑅 , 𝑚 𝑖 = 1 | | 𝑌 𝑖 ( 𝑡 ) | | 0 ≤ 𝑚 𝑖 = 1 ‖ ‖ 𝑦 𝑖 ‖ ‖ 0 ≤ 𝛿 − 1 𝛾 ( 𝑦 ) = 𝑐 𝛿 − 1 , 𝑡 ∈ 𝑅 . ( 3 . 1 4 ) Consequently, for any 𝑦 ∈ 𝜕 𝑃 ( 𝛾 , 𝑐 ) , condition ( 𝐻 1 ) and (3.14) imply that 𝛾 ( 𝐴 𝑦 ) = 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≥ m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑘 0 ( 𝑡 , 𝑠 ) 𝑓 𝑘 0 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≥ m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑀 𝑘 0 1 𝑐 + 1 𝑇 𝑀 1 𝑑 𝑠 > 𝑐 , ( 3 . 1 5 ) which gives (3.13).
Thirdly, we verify 𝜃 ( 𝐴 𝑦 ) < 𝑏 , ∀ 𝑦 ∈ 𝜕 𝑃 ( 𝜃 , 𝑏 ) . ( 3 . 1 6 ) As before, 𝜃 ( 𝑦 ) = 𝑏 and (1.7), (2.4), and (3.8) also tell us that 𝑏 ≤ 𝑚 𝑖 = 1 | | 𝑌 𝑖 ( 𝑡 ) | | 0 ≤ 𝑏 𝛿 − 1 , 𝑡 ∈ 𝑅 . ( 3 . 1 7 ) Then condition ( 𝐻 2 ) , (3.17), and the fact that the function m i n is concave imply 𝜃 ( 𝐴 𝑦 ) = 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≤ m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑚 𝑖 = 1 𝑀 𝑖 2 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≤ m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑀 2 𝑏 ( 1 + 𝛿 ) 𝑇 𝑀 2 𝑑 𝑠 < 𝑏 . ( 3 . 1 8 ) Thus (3.16) holds.
Finally, let us prove 𝑃 ( 𝛼 , 𝑎 ) ≠ 𝜙 , 𝛼 ( 𝐴 𝑦 ) > 𝑎 , ∀ 𝑦 ∈ 𝜕 𝑃 ( 𝛼 , 𝑎 ) . ( 3 . 1 9 ) Obviously, 𝑦 = 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 = 𝑎 𝑚 + 1 , 𝑎 𝑚 + 1 , … , 𝑎 𝑚 + 1 ∈ 𝑃 ( 𝛼 , 𝑎 ) . ( 3 . 2 0 ) In addition, for any 𝑦 ∈ 𝜕 𝑃 ( 𝛼 , 𝑎 ) , we get 𝛿 𝑎 ≤ 𝑚 𝑖 = 1 | | 𝑌 𝑖 ( 𝑡 ) | | 0 ≤ 𝑎 , 𝑡 ∈ 𝑅 ( 3 . 2 1 ) since ∑ 𝑚 𝑖 = 1 ‖ 𝑦 𝑖 ‖ 0 = 𝑎 and 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) ∈ 𝑃 . So we have from condition ( 𝐻 3 ) that 𝛼 ( 𝐴 𝑦 ) = 𝑚 𝑖 = 1 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≥ m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑚 𝑖 = 1 𝑀 𝑖 1 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≥ m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑀 1 𝑎 + 1 𝑇 𝑀 1 𝑑 𝑠 > 𝑎 . ( 3 . 2 2 ) Hence (3.19) holds.
To sum up, (3.6)–(3.10), (3.13), (3.16), and (3.19) tell us that conditions of Lemma 2.1 all hold here. Consequently, system (1.4) has at least two 𝑇 -positive periodic solutions 𝑦 1 = ( 𝑦 1 1 , 𝑦 1 2 , … , 𝑦 1 𝑚 ) and 𝑦 2 = ( 𝑦 2 1 , 𝑦 2 2 , … , 𝑦 2 𝑚 ) belonging to 𝑃 ( 𝛾 , 𝑐 ) such that 𝑎 < ‖ ‖ 𝑦 1 ‖ ‖ , w i t h 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑦 1 𝑖 < 𝑏 , 𝑏 < 𝑚 ∑ 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑦 2 𝑖 < 𝑐 . ( 3 . 2 3 )

As an application of Theorem 3.1, we provide the following example. For convenience, all examples in this paper are given when 𝑛 = 1 , 𝑚 = 2 .

Example 3.2. Assume that 𝜏 1 > 0 is a fixed constant. Consider the following system: 𝑦 1 ( 𝑡 ) = − ( l n 2 ) | | s i n 𝑡 | | 𝑦 1 ( 𝑡 ) + 1 4 𝜋 𝑒 | c o s 𝑡 | − 1 𝑔 1 𝑡 , 𝑦 1 𝑡 − 𝜏 1 , 𝑦 2 𝑡 − 𝜏 1 , 𝑦 2 ( 𝑡 ) = − 1 2 ( l n 3 ) | c o s 𝑡 | 𝑦 2 ( 𝑡 ) + 1 4 𝜋 𝑒 | s i n 𝑡 | − 1 𝑔 2 𝑡 , 𝑦 1 𝑡 − 𝜏 1 , 𝑦 2 𝑡 − 𝜏 1 , ( 3 . 2 4 ) where 𝑔 1 𝑡 , 𝑢 1 , 𝑢 2 ∶ = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 2 𝑒 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 0 , 1 6 0 ] , 2 4 0 1 𝑒 𝑢 1 + 𝑢 2 − 1 6 0 + 1 2 𝑒 1 9 9 − 𝑢 1 − 𝑢 2 3 9 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 1 6 0 , 1 9 9 ] , 2 4 0 1 𝑒 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 1 9 9 , ∞ ) . 𝑔 2 𝑡 , 𝑢 1 , 𝑢 2 ∶ = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 4 2 2 3 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 0 , 1 6 0 ] , 𝑢 1 + 𝑢 2 − 1 6 0 + 4 2 ( 2 / 3 ) 1 9 9 − 𝑢 1 − 𝑢 2 3 9 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 1 6 0 , 1 9 9 ] , 1 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 1 9 9 , ∞ ) . ( 3 . 2 5 ) We set 𝑎 = 1 , 𝑏 = 4 0 , 𝑐 = 1 9 9 , 𝑎 1 ( 𝑡 ) ∶ = ( l n 2 ) | | s i n 𝑡 | | , 𝑎 2 ( 𝑡 ) ∶ = 1 2 ( l n 3 ) | c o s 𝑡 | , 𝑡 ∈ 𝑅 , 𝑓 1 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) = 1 4 𝜋 𝑒 | c o s 𝑡 | − 1 𝑔 1 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) , 𝑡 ∈ 𝑅 , 𝑓 2 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) = 1 4 𝜋 𝑒 | s i n 𝑡 | − 1 𝑔 2 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) , 𝑡 ∈ 𝑅 ( 3 . 2 6 ) Then 𝑇 = 𝜋 , 𝛿 = 1 4 , 𝑀 1 = 1 3 , 𝑀 2 = 3 2 . ( 3 . 2 7 ) We may verify that conditions ( 𝐻 1 ) , ( 𝐻 2 ) , and ( 𝐻 3 ) are all satisfied. Hence, Theorem 3.1 tells us that system (3.24) has at least two 𝜋 -positive periodic solutions 𝑦 1 = ( 𝑦 1 1 , 𝑦 1 2 ) and 𝑦 2 = ( 𝑦 2 1 , 𝑦 2 2 ) such that 1 < m a x 𝑡 ∈ [ 0 , 𝜋 ] 𝑦 1 1 + m a x 𝑡 ∈ [ 0 , 𝜋 ] 𝑦 1 2 , m i n 𝑡 ∈ [ 0 , 𝜋 ] 𝑦 1 1 + m i n 𝑡 ∈ [ 0 , 𝜋 ] 𝑦 1 2 < 4 0 , 4 0 < m i n 𝑡 ∈ [ 0 , 𝜋 ] 𝑦 2 1 + m i n 𝑡 ∈ [ 0 , 𝜋 ] 𝑦 2 2 < 1 9 9 . ( 3 . 2 8 )

4. Existence of Three Positive Solutions of System (1.4)

For the sake of convenience we list the assumptions to be used in this section as follows.

( 𝐻 4 ) There exists a number 𝐶 1 > 0 such that

𝑓 𝑖 𝑡 , 𝑈 1 , 𝑈 2 , … , 𝑈 𝑚 < 𝐶 1 𝑚 𝑇 𝑀 2 , 𝑡 ∈ 𝑅 , 𝑈 𝑖 ∈ [ 0 , ∞ ) 𝑛 , 𝜑 ≤ 𝐶 1 , 𝑖 ∈ Λ 1 , ( 4 . 1 ) where 𝜑 and 𝑈 𝑖 , 𝑖 ∈ Λ 1 are given in ( 𝐻 1 ) .

( 𝐻 5 ) There exist numbers 𝐶 2 > 𝐶 1 and 𝑖 0 ∈ Λ 1 such that

𝑓 𝑖 0 𝑡 , 𝑈 1 , 𝑈 2 , … , 𝑈 𝑚 > 𝐶 3 𝑇 𝑀 1 , 𝑡 ∈ 𝑅 , 𝑈 𝑖 ∈ [ 0 , ∞ ) 𝑛 , 𝐶 2 ≤ 𝜑 ≤ 𝐶 3 , ( 4 . 2 ) where

𝐶 3 ∶ = 𝐶 2 ( 1 + 𝛿 ) 𝛿 . ( 4 . 3 )

( 𝐻 6 ) One has

l i m 𝜑 → ∞ m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑓 𝑖 𝑡 , 𝑈 1 , 𝑈 2 , … , 𝑈 𝑚 𝜑 < 1 𝑚 𝑇 𝑀 2 , 𝑈 𝑖 ∈ [ 0 , ∞ ) 𝑛 , 𝑖 ∈ Λ 1 . ( 4 . 4 )

( 𝐻 7 ) There exists a number 𝐶 4 > 𝐶 3 such that,

𝑓 𝑖 𝑡 , 𝑈 1 , 𝑈 2 , … , 𝑈 𝑚 ≤ 𝐶 4 𝑚 𝑇 𝑀 2 , 𝑡 ∈ 𝑅 , 𝑈 𝑖 ∈ [ 0 , ∞ ) 𝑛 , 𝜑 ≤ 𝐶 4 , 𝑖 ∈ Λ 1 . ( 4 . 5 )

Let us now state the first existence result of three positive solutions of system (1.4).

Theorem 4.1. Assume that conditions ( 𝐻 4 ) , ( 𝐻 5 ) , and ( 𝐻 6 ) hold. Then system (1.4) has at least three 𝑇 -positive periodic solutions.

Proof. Firstly, we set 𝛽 ( 𝑦 ) ∶ = 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑦 𝑖 ( 𝑡 ) , 𝑦 = 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ∈ 𝑃 . ( 4 . 6 ) Obviously, 𝛽 is a nonnegative continuous concave functional on 𝑃 and 𝛽 ( 𝑦 ) ≤ ‖ 𝑦 ‖ , 𝑦 = 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ∈ 𝑃 𝑀 f o r a n y 𝑀 > 0 . ( 4 . 7 )
Secondly, condition ( 𝐻 6 ) implies that there exists a number 𝐶 5 ≥ 𝐶 3 such that 𝐴 ∶ 𝑃 𝐶 5 ⟶ 𝑃 𝐶 5 i s c o m p l e t e l y c o n t i n u o u s . ( 4 . 8 ) In fact, we know from condition ( 𝐻 6 ) that there exist numbers 𝜏 𝑖 > 0 , 0 < 𝜎 𝑖 < 1 𝑚 𝑇 𝑀 2 , 𝑖 ∈ Λ 1 ( 4 . 9 ) satisfying m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑓 𝑖 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑚 ( 𝑡 ) ‖ 𝑦 ‖ ≤ 𝜎 𝑖 , ‖ 𝑦 ‖ ≥ 𝜏 𝑖 , 𝑖 ∈ Λ 1 . ( 4 . 1 0 ) So 𝑓 𝑖 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑚 ( 𝑡 ) ≤ 𝜎 𝑖 ‖ 𝑦 ‖ , 𝑡 ∈ [ 0 , 𝑇 ] , ‖ 𝑦 ‖ ≥ 𝜏 𝑖 , 𝑖 ∈ Λ 1 . ( 4 . 1 1 ) Set 𝐶 𝑖 6 ∶ = m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑓 𝑖 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑚 ( 𝑡 ) , 𝑡 ∈ [ 0 , 𝑇 ] , ‖ 𝑦 ‖ ≤ 𝜏 𝑖 , 𝑖 ∈ Λ 1 . ( 4 . 1 2 ) Then 𝑓 𝑖 𝑡 , 𝑌 1 ( 𝑡 ) , 𝑌 2 ( 𝑡 ) , … , 𝑌 𝑚 ( 𝑡 ) ≤ 𝐶 𝑖 6 + 𝜎 𝑖 ‖ 𝑦 ‖ , 𝑡 ∈ [ 0 , 𝑇 ] , 𝑦 ∈ 𝑃 , 𝑖 ∈ Λ 1 . ( 4 . 1 3 ) Let us choose 𝐶 5 ≥ m a x ⎧ ⎨ ⎩ 𝐶 3 , 𝑚 𝐶 1 6 𝑇 𝑀 1 2 1 − 𝑚 𝑇 𝑀 1 2 𝜎 1 , 𝑚 𝐶 2 6 𝑇 𝑀 2 2 1 − 𝑚 𝑇 𝑀 2 2 𝜎 2 , … , 𝑚 𝐶 𝑚 6 𝑇 𝑀 𝑚 2 1 − 𝑚 𝑇 𝑀 𝑚 2 𝜎 𝑚 ⎫ ⎬ ⎭ . ( 4 . 1 4 ) Then for any 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) ∈ 𝑃 𝐶 5 , we have ‖ 𝐴 𝑦 ‖ = 𝑚 𝑖 = 1 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑡 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≤ 𝑚 𝑖 = 1 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑡 + 𝑇 ) 𝜎 𝑖 ‖ 𝑦 ‖ + 𝐶 𝑖 6 𝑑 𝑠 ≤ 𝑚 𝑖 = 1 𝜎 𝑖 𝐶 5 + 𝐶 𝑖 6 𝑇 𝑀 𝑖 2 ≤ 𝐶 5 , ( 4 . 1 5 ) which implies 𝐴 ( 𝑦 ) ∈ 𝑃 𝐶 5 for all 𝑦 ∈ 𝑃 𝐶 5 . Moreover, we know from the proof of (3.10) that 𝐴 ∶ 𝑃 𝐶 5 → 𝑃 𝐶 5 is completely continuous.
Thirdly, let us show that numbers 0 < 𝐶 1 < 𝐶 2 < 𝐶 3 ≤ 𝐶 5 ( 4 . 1 6 ) satisfy conditions (i), (ii), and (iii) of Lemma 2.2.
Step 1. We prove that 𝑦 ∈ 𝑃 𝛽 , 𝐶 2 , 𝐶 3 ∶ 𝛽 ( 𝑦 ) > 𝐶 2 ≠ 𝜙 , 𝛽 ( 𝐴 𝑦 ) > 𝐶 2 , 𝑦 ∈ 𝑃 𝛽 , 𝐶 2 , 𝐶 3 . ( 4 . 1 7 ) Clearly, 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) = ( 𝐶 3 / 𝑚 , 𝐶 3 / 𝑚 , … , 𝐶 3 / 𝑚 ) ∈ { 𝑃 ( 𝛽 , 𝐶 2 , 𝐶 3 ) ∶ 𝛽 ( 𝑦 ) > 𝐶 2 } . Moreover, for any 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) ∈ 𝑃 ( 𝛽 , 𝐶 2 , 𝐶 3 ) , we have 𝐶 2 ≤ 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑦 𝑖 ( 𝑡 ) ≤ 𝑚 𝑖 = 1 | | 𝑌 𝑖 ( 𝑡 ) | | 0 ≤ ‖ 𝑦 ‖ ≤ 𝐶 3 . ( 4 . 1 8 ) Then condition ( 𝐻 5 ) , (4.6), and (4.18) imply that 𝛽 ( 𝐴 𝑦 ) = 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≥ m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 0 ( 𝑡 , 𝑠 ) 𝑓 𝑖 0 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≥ m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑀 𝑖 0 1 𝐶 3 𝑇 𝑀 1 𝑑 𝑠 ≥ 𝐶 3 > 𝐶 2 , ( 4 . 1 9 ) which gives 𝛽 ( 𝐴 𝑦 ) > 𝐶 2 for 𝑦 ∈ 𝑃 ( 𝛽 , 𝐶 2 , 𝐶 3 ) . And then we arrive at (4.17).Step 2. Condition ( 𝐻 4 ) implies ‖ 𝐴 𝑦 ‖ < 𝐶 1 f o r ‖ 𝑦 ‖ ≤ 𝐶 1 . ( 4 . 2 0 ) In fact, for any 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) ∈ 𝑃 𝐶 1 , that is, ‖ 𝑦 ‖ ≤ 𝐶 1 , from | | 𝑌 𝑖 ( 𝑡 ) | | 0 ≤ ‖ ‖ 𝑦 𝑖 ‖ ‖ 0 , 𝑡 ∈ [ 0 , 𝑇 ] , 𝑖 ∈ Λ 1 , ( 4 . 2 1 ) and condition ( 𝐻 4 ) we have ‖ 𝐴 𝑦 ‖ = 𝑚 𝑖 = 1 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≤ 𝑚 𝑖 = 1 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑀 𝑖 2 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 < 𝑚 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑀 2 𝐶 1 𝑚 𝑇 𝑀 2 𝑑 𝑠 = 𝐶 1 , ( 4 . 2 2 ) which yields (4.20).Step 3. 𝛽 ( 𝐴 𝑦 ) > 𝐶 2 for 𝑦 ∈ 𝑃 ( 𝛽 , 𝐶 2 , 𝐶 5 ) with ‖ 𝐴 𝑦 ‖ > 𝐶 3 . This is the case because 𝐴 𝑦 ∈ 𝑃 implies 𝛽 ( 𝐴 𝑦 ) = 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝐴 𝑖 ( 𝑡 ) ≥ 𝛿 𝑚 𝑖 = 1 ‖ ‖ 𝐴 𝑖 ‖ ‖ 0 = 𝛿 ‖ 𝐴 𝑦 ‖ > 𝛿 𝐶 3 > 𝐶 2 . ( 4 . 2 3 )
At present, we may say that hypotheses of Lemma 2.2 (the Leggett-Willaims theorem) are satisfied. Hence system (1.4) has at least three 𝑇 -positive periodic solutions: 𝑢 = 𝑢 1 , 𝑢 2 , … , 𝑢 𝑚 , 𝑣 = 𝑣 1 , 𝑣 2 , … , 𝑣 𝑚 , 𝑤 = 𝑤 1 , 𝑤 2 , … , 𝑤 𝑚 , ( 4 . 2 4 ) such that ‖ 𝑢 ‖ < 𝐶 1 , 𝐶 2 < 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑣 𝑖 , ‖ 𝑤 ‖ > 𝐶 1 , 𝑚 𝑖 = 1 m i n 𝑡 ∈ [ 0 , 𝑇 ] 𝑤 𝑖 < 𝐶 2 . ( 4 . 2 5 )
We give the following example to illustrate Theorem 4.1.

Example 4.2. Consider the following system: 𝑦 1 ( 𝑡 ) = − 1 4 𝜋 ( l n 3 ) ( 2 + c o s 𝑡 ) 𝑦 1 ( 𝑡 ) + 1 4 𝜋 𝑒 ( s i n 𝑡 ) − 1 𝑔 1 𝑡 , 𝑦 1 𝑡 − 𝜏 2 , 𝑦 2 𝑡 − 𝜏 2 , 𝑦 2 ( 𝑡 ) = − 1 4 𝜋 ( l n 2 ) ( 2 + s i n 𝑡 ) 𝑦 2 ( 𝑡 ) + 1 4 𝜋 𝑒 ( c o s 𝑡 ) − 1 𝑔 2 𝑡 , 𝑦 1 𝑡 − 𝜏 2 , 𝑦 2 𝑡 − 𝜏 2 , ( 4 . 2 6 ) where 𝜏 2 > 0 is a fixed constant and 𝑔 1 𝑡 , 𝑢 1 , 𝑢 2 ∶ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 2 5 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 0 , 1 ] , 3 − 𝑢 1 − 𝑢 2 5 + 3 6 𝑒 2 + 1 𝑢 1 + 𝑢 2 − 1 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 1 , 3 ] , 7 2 𝑒 2 + 2 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 3 , 1 2 ] , 5 1 2 𝑢 1 + 𝑢 2 − 3 + 7 2 𝑒 2 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 1 2 , ∞ ) , 𝑔 2 𝑡 , 𝑢 1 , 𝑢 2 ∶ = c o n s t a n t < 7 1 5 , 𝑡 , 𝑢 1 , 𝑢 2 ∈ 𝑅 × [ 0 , ∞ ) × [ 0 , ∞ ) . ( 4 . 2 7 )
We set 𝑎 1 ( 𝑡 ) ∶ = 1 4 𝜋 ( l n 3 ) ( 2 + c o s 𝑡 ) , 𝑎 2 ( 𝑡 ) ∶ = 1 4 𝜋 ( l n 2 ) ( 2 + s i n 𝑡 ) , 𝑡 ∈ 𝑅 , 𝑓 1 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) ∶ = 1 4 𝜋 𝑒 ( s i n 𝑡 ) − 1 𝑔 1 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) , 𝑡 ∈ 𝑅 , 𝑓 2 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) ∶ = 1 4 𝜋 𝑒 ( c o s 𝑡 ) − 1 𝑔 2 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) , 𝑡 ∈ 𝑅 , 𝐶 1 ∶ = 1 , 𝐶 2 ∶ = 3 . ( 4 . 2 8 ) Then 𝑇 = 2 𝜋 , 𝛿 = 1 3 , 𝑀 1 = 1 3 , 𝑀 2 = 2 . ( 4 . 2 9 ) We may verify also that conditions ( 𝐻 4 ) , ( 𝐻 5 ) , and ( 𝐻 6 ) hold. Hence, Theorem 4.1 tells us that system (4.26) has at least three 2 𝜋 -positive periodic solutions: 𝑦 1 = 𝑦 1 1 , 𝑦 1 2 , 𝑦 2 = 𝑦 2 1 , 𝑦 2 2 , 𝑦 3 = 𝑦 3 1 , 𝑦 3 2 ( 4 . 3 0 ) such that ‖ ‖ 𝑦 1 ‖ ‖ < 1 , 3 < m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 2 1 + m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 2 2 , ‖ ‖ 𝑦 3 ‖ ‖ > 1 , m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 3 1 + m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 3 2 < 3 . ( 4 . 3 1 )
The second existence result of three positive solutions of system (1.4) is as follows.

Theorem 4.3. Assume that conditions ( 𝐻 4 ) , ( 𝐻 5 ) , and ( 𝐻 7 ) hold. Then system (1.4) has at least three 𝑇 -positive periodic solutions.

Proof. If we can get (4.8) with 𝐶 5 replaced by 𝐶 4 in this case, then the proof is complete. In fact, for any 𝑦 = ( 𝑦 1 , 𝑦 2 , … , 𝑦 𝑚 ) ∈ 𝑃 𝐶 4 , condition ( 𝐻 7 ) implies ‖ 𝐴 𝑦 ‖ = 𝑚 𝑖 = 1 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑠 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≤ 𝑚 𝑖 = 1 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝐺 𝑖 ( 𝑡 , 𝑡 + 𝑇 ) 𝑓 𝑖 𝑠 , 𝑌 1 ( 𝑠 ) , 𝑌 2 ( 𝑠 ) , … , 𝑌 𝑚 ( 𝑠 ) 𝑑 𝑠 ≤ 𝑚 m a x 𝑡 ∈ [ 0 , 𝑇 ] 𝑡 + 𝑇 𝑡 𝑀 2 𝐶 4 𝑚 𝑇 𝑀 2 𝑑 𝑠 ≤ 𝐶 4 . ( 4 . 3 2 ) Then 𝐴 ∶ 𝑃 𝐶 4 ⟶ 𝑃 𝐶 4 , ( 4 . 3 3 ) as desired. This ends the proof.

The following example is an application of Theorem 4.3.

Example 4.4. Consider system 𝑦 1 ( 𝑡 ) = − 1 4 𝜋 ( l n 2 ) ( 2 + c o s 𝑡 ) 𝑦 1 ( 𝑡 ) + 1 4 𝜋 𝑒 ( c o s 𝑡 ) − 1 𝑔 1 𝑡 , 𝑦 1 𝑡 − 𝜏 3 , 𝑦 2 𝑡 − 𝜏 3 , 𝑦 2 ( 𝑡 ) = − 1 4 𝜋 ( l n 2 ) ( 2 + s i n 𝑡 ) 𝑦 2 ( 𝑡 ) + 1 4 𝜋 𝑒 ( c o s 𝑡 ) − 1 𝑔 2 𝑡 , 𝑦 1 𝑡 − 𝜏 3 , 𝑦 2 𝑡 − 𝜏 3 , ( 4 . 3 4 ) where 𝜏 3 > 0 is a fixed constant and 𝑔 1 𝑡 , 𝑢 1 , 𝑢 2 ∶ = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 1 + 𝑢 2 3 𝑢 1 + 𝑢 2 + 1 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 0 , 2 ] , 1 + 𝑢 2 4 − 𝑢 1 − 𝑢 2 3 𝑢 1 + 𝑢 2 + 1 + 1 2 2 4 𝑒 2 + 1 𝑢 1 + 𝑢 2 − 2 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 2 , 4 ] , 2 4 𝑒 2 + 1 , 𝑡 ∈ 𝑅 , 𝑢 1 + 𝑢 2 ∈ [ 4 , ∞ ) , 𝑔 2 𝑡 , 𝑢 1 , 𝑢 2 ∶ = c o n s t a n t < 1 , 𝑡 , 𝑢 1 , 𝑢 2 ∈ 𝑅 × [ 0 , ∞ ) × [ 0 , ∞ ) . ( 4 . 3 5 )
If we set 𝑎 1 ( 𝑡 ) ∶ = 1 4 𝜋 ( l n 2 ) ( 2 + c o s 𝑡 ) , 𝑎 2 ( 𝑡 ) ∶ = 1 4 𝜋 ( l n 2 ) ( 2 + s i n 𝑡 ) , 𝑡 ∈ 𝑅 , 𝑓 𝑖 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) = 1 4 𝜋 𝑒 ( c o s 𝑡 ) − 1 𝑔 𝑖 𝑡 , 𝑦 1 ( 𝑡 − 𝜏 ) , 𝑦 2 ( 𝑡 − 𝜏 ) , 𝑡 ∈ 𝑅 , 𝑖 = 1 , 2 , ( 4 . 3 6 ) then 𝑇 = 2 𝜋 , 𝛿 = 1 2 , 𝑀 1 = 1 , 𝑀 2 = 2 . ( 4 . 3 7 ) We choose 𝐶 1 ∶ = 2 , 𝐶 2 ∶ = 4 , 𝐶 4 ∶ = 5 0 𝑒 2 . ( 4 . 3 8 ) Then assumptions ( 𝐻 4 ) , ( 𝐻 5 ) , and ( 𝐻 7 ) hold. Hence we know from Theorem 4.3 that system (4.34) has at least three 2 𝜋 -positive periodic solutions: 𝑦 1 = 𝑦 1 1 , 𝑦 1 2 , 𝑦 2 = 𝑦 2 1 , 𝑦 2 2 , 𝑦 3 = 𝑦 3 1 , 𝑦 3 2 ( 4 . 3 9 ) such that ‖ ‖ 𝑦 1 ‖ ‖ < 2 , 4 < m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 2 1 + m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 2 2 , ‖ ‖ 𝑦 3 ‖ ‖ > 2 , m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 3 1 + m i n 𝑡 ∈ [ 0 , 2 𝜋 ] 𝑦 3 2 < 4 . ( 4 . 4 0 )

We end this paper by the following remark.

Remark 4.5. For a general positive integer 𝑚 ≥ 2 , main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our Theorems 3.1, 4.1, and 4.3 are new also when 𝑚 = 1 .

Acknowledgments

Z.-C. Hao acknowledges support from NSFC (10771117), and PH.D. Programs Foundation of Ministry of Education of China(20093705120002), and NSF of Shandong Province of China (Y2008A24), China Postdoctoral Science Foundation (20090451290), Shandong Province Postdoctoral Foundation (200801001), Foundation of Qufu Normal University (BSQD07026).

References

  1. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at MathSciNet
  2. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  3. S. M. Lenhart and C. C. Travis, “Global stability of a biological model with time delay,” Proceedings of the American Mathematical Society, vol. 96, no. 1, pp. 75–78, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. N. Chow, “Existence of periodic solutions of autonomous functional differential equations,” Journal of Differential Equations, vol. 15, pp. 350–378, 1974. View at Zentralblatt MATH · View at MathSciNet
  5. M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287–289, 1977.
  6. J. Mallet-Paret and R. D. Nussbaum, “Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation,” Annali di Matematica Pura ed Applicata, vol. 145, pp. 33–128, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Wazewska-Czyzwska and A. Lasota, “Mathematical problems of the dynamics of the red blood cells system,” Annals of Polish Mathematical Society, Series III, vol. 17, pp. 23–40, 1988.
  8. K. Gopalsamy, X. Z. He, and L. Z. Wen, “On a periodic neutral logistic equation,” Glasgow Mathematical Journal, vol. 33, no. 3, pp. 281–286, 1991. View at Zentralblatt MATH · View at MathSciNet
  9. K. Gopalsamy and B. G. Zhang, “On a neutral delay logistic equation,” Dynamics and Stability of Systems, vol. 2, no. 3-4, pp. 183–195, 1988. View at Zentralblatt MATH · View at MathSciNet
  10. B. Liu, “Positive periodic solution for a nonautonomous delay differential equation,” Acta Mathematicae Applicatae Sinica, vol. 19, no. 2, pp. 307–316, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. E. Beltrami, Mathematics for Dynamic Modeling, Academic Press, Boston, Mass, USA, 1987. View at MathSciNet
  12. R. I. Avery and J. Henderson, “Two positive fixed points of nonlinear operators on ordered Banach spaces,” Communications on Applied Nonlinear Analysis, vol. 8, no. 1, pp. 27–36, 2001. View at Zentralblatt MATH · View at MathSciNet
  13. R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673–688, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet