Existence of Positive Periodic Solutions for a Class of Higher-Dimension Functional Differential Equations with Impulses

Abstract

By employing the Krasnoselskii fixed point theorem, we establish some criteria for the existence of positive periodic solutions of a class of 𝑛-dimension periodic functional differential equations with impulses, which improve the results of the literature.

1. Introduction

Some evolution processes are distinguished by the circumstance that the evolutions change very rapidly at certain instants. In mathematical simulations, impulsive delay differential equations may express several simulation processes in real world which depend on their prehistory and are subject to short time disturbances. Such processes occur in the theory of optional control, population dynamics, biotechnologies, economics, and so forth. 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; we refer the reader to [1–4]. For other related works on studying for impulsive delay differential equations, we refer the reader to [5–7].

In [8], Zeng et al. studied the following functional differential equations without impulses: 𝑥̇(𝑡)=𝐴(𝑡,𝑥(𝑡))𝑥(𝑡)+𝜆𝑓𝑡,𝑥_𝑡, (1) and obtained sufficient conditions for the existence of positive periodic solutions of (1).

Zhang et al. [9] investigated the following form: 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓𝑡,𝑥_𝑡,𝑡≠𝜏_𝑘,𝑘∈𝑍₊,Δ𝑥|_{𝑡=𝜏𝑘}=𝐼_𝑘𝑥𝜏_𝑘. (2)

In this paper, we will consider the 𝑛-dimension differential equation with impulses as follows: 𝑥̇(𝑡)=𝐴(𝑡,𝑥(𝑡))𝑥(𝑡)+𝜆𝑓𝑡,𝑥_𝑡,𝑡≠𝜏_𝑘,𝑘∈𝑍₊,Δ𝑥|_{𝑡=𝜏𝑘}=𝐼_𝑘𝑥𝜏_𝑘, (3) where 𝜆>0 is a parameter, 𝐴(𝑡,𝑥(𝑡))=diag[𝑎₁(𝑡,𝑥(𝑡)),𝑎₂(𝑡,𝑥(𝑡)),…,𝑎_𝑛(𝑡,𝑥(𝑡))], 𝑎_𝑖∈𝐶(𝑅×𝑅,𝑅) is 𝜔-periodic, and 𝑓(𝑡,𝑥_𝑡) is an operator defined on 𝑅×𝐵𝐶(𝑅,𝑅^𝑛) (here 𝐵𝐶(𝑅,𝑅^𝑛) denotes the Banach space of bounded continuous operator 𝜙∶𝑅→𝑅^𝑛 with the norm ‖𝜙‖=∑_𝑖=1sup_𝜃∈𝑅|𝜙_𝑖(𝜃)|, where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇). For 𝑥∈𝐵𝐶 and 𝑡∈𝑅, 𝑥_𝑡∈𝐵𝐶 is defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) for 𝜃∈𝑅 (see [10], Zheng). Consider that 𝑓(𝑡+𝜔,𝑥_𝑡)=𝑓(𝑡,𝑥_𝑡) and Δ𝑥|_{𝑡=𝜏𝑘}=𝑥(𝜏⁺_𝑘)−𝑥(𝜏_𝑘) (here 𝑥(𝜏⁺_𝑘) represents the right limit of 𝑥 at the point 𝜏_𝑘), 𝐼_𝑘=(𝐼¹_𝑘,𝐼²_𝑘,…,𝐼^𝑛_𝑘)∈𝐶(𝑅^𝑛₊,𝑅^𝑛₋), that is, 𝑥 changes decreasingly suddenly at 𝜏_𝑘, 𝜔>0 is a constant, 𝑅₊ and 𝑅₋ are the sets of all nonnegative and nonpositive real numbers, respectively. We assume that there exists an integer 𝑝>0 such that 𝜏_𝑘+𝑝=𝜏_𝑘+𝜔,𝐼_𝑘+𝑝=𝐼_𝑘, where 0<𝜏₁<𝜏₂<⋯<𝜏_𝑝<𝜔.

2. Some Preliminaries

𝑃𝐶(𝐽,𝑅^𝑛)={𝜙∶𝐽→𝑅^𝑛, 𝜙 is continuous everywhere except at a finite number of points 𝜏_𝑘 at which 𝜙(𝜏⁺_𝑘) and 𝜙(𝜏⁻_𝑘) exist and 𝜙(𝜏⁻_𝑘)=𝜙(𝜏_𝑘)}, 𝐽⊂𝑅. For each 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛, the norm of 𝑥 is defined as |𝑥|=∑_𝑖=1|𝑥_𝑖|.

Throughout the paper, we make the following assumptions: (𝐻₁)𝑓(𝑡,𝜑_𝑡)≤0 for all (𝑡,𝜑)∈𝑅×𝐵𝐶(𝑅,𝑅^𝑛₊); (𝐻₂)𝑓_𝑖(𝑡,𝜑_𝑡) is a continuous function of 𝑡 for each 𝜑∈𝐵𝐶(𝑅,𝑅^𝑛₊), 𝑖=1,2,…,𝑛; (𝐻₃) for any 𝐿>0 and 𝜖>0, there exists 𝛿>0 such that for 𝜙,𝜓∈𝐵𝐶(𝑅,𝑅^𝑛₊), ‖𝜙‖≤𝐿, ‖𝜓‖≤𝐿, and ‖𝜙−𝜓‖≤𝛿 imply that 𝑓_𝑖𝑡,𝜙_𝑡−𝑓_𝑖𝑡,𝜓_𝑡<𝜖,∀𝑡∈[0,𝜔],𝑖=1,2,…,𝑛. (4)

To conclude this section, we summarize in the following a few concepts and results that will be needed in our arguments.

Definition 1. Let 𝑋 be a Banach space, and let 𝑃 be a closed, nonempty subset of 𝑋; 𝑃 is a cone if(i)𝛼𝑥+𝛽𝑥∈𝑃 for all 𝑥,𝑦∈𝑃 and all 𝛼,𝛽≥0;(ii)𝑥,−𝑥∈𝑃 imply 𝑥=0.
Let 𝑋={𝑥=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑃𝐶(𝑅,𝑅^𝑛)∣𝑥(𝑡+𝜔)=𝑥(𝑡)} with the norm ‖𝑥‖=∑_𝑖=1|𝑥_𝑖|₀, where |𝑥_𝑖|₀=sup_{𝑡∈[0,𝜔]}|𝑥_𝑖(𝑡)|; then 𝑋 is a Banach space.
If 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑋 is a solution of (3), then 𝑥_𝑖(𝑡)=𝜆_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑢)𝑓_𝑖𝑢,𝑥_𝑢+𝑑𝑢_{𝑗=𝑝𝑗=1}𝐺_𝑖𝑡,𝜏_𝑚𝑗+𝑛𝜔𝐼^𝑖_𝑗𝑥𝜏_𝑚𝑗, (5) where 𝐺_𝑖(𝑡,𝑢)=exp−∫^𝑢_𝑡𝑎_𝑖(𝑠,𝑥(𝑠))𝑑𝑠exp−∫^𝜔₀𝑎_𝑖(𝑠,𝑥(𝑠))𝑑𝑠−1,𝑖=1,2,…,𝑛. (6) See [9], Zhang et al.
It is clear that 𝐺_𝑖(𝑡+𝜔,𝑢+𝜔)=𝐺_𝑖(𝑡,𝑢), for all (𝑡,𝑢)∈𝑅², and by (𝐻₁), 𝐺_𝑖(𝑡,𝑢)𝑓_𝑖𝑢,𝜑_𝑢≥0 (7) for (𝑡,𝑢)∈𝑅² and (𝑢,𝜑)∈𝑅×𝐵𝐶(𝑅,𝑅^𝑛₊).
Define for 𝑖=1,2,…,𝑛, 𝐴_𝑖∶=min_{0≤𝑡≤𝑢≤𝜔}𝐺_𝑖(𝑡,𝑢)=exp−∫^𝜔₀𝑎_𝑖(𝑠,𝑥(𝑠))𝑑𝑠1−exp−∫^𝜔₀𝑎_𝑖((𝑠,𝐵𝑠,𝑥))𝑑𝑠_𝑖∶=max_{0≤𝑡≤𝑢≤𝜔}𝐺_𝑖(=1𝑡,𝑢)1−exp−∫^𝜔₀𝑎_𝑖((𝑠,𝑠,𝑥))𝑑𝑠𝐴∶=min_{1≤𝑖≤𝑛}𝐴_𝑖,𝐵∶=max_{1≤𝑖≤𝑛}𝐵_𝑖. (8) Let 𝐾=𝑥(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)^𝑇∈𝑋∶𝑥_𝑖(𝑡)≥𝜎𝑥_𝑖0,𝑖=1,2,…,𝑛, (9) where 𝜎=𝐴/𝐵∈(0,1). It is not difficult to verify that 𝐾 is a cone in 𝑋. We define an operator Φ∶𝑋→𝑋 as follows: (Φ𝑥)(𝑡)=Φ₁𝑥(𝑡),Φ₂𝑥(𝑡),…,Φ_𝑛𝑥(𝑡)^𝑇, (10) where Φ_𝑖𝑥(𝑡)=𝜆_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑢)𝑓_𝑖𝑢,𝑥_𝑢+𝑑𝑢_{𝑗=𝑝𝑗=1}𝐺_𝑖𝑡,𝜏_𝑚𝑗+𝑛𝜔𝐼^𝑖_𝑗𝑥𝜏_𝑚𝑗. (11) Then, it can be immediately obtained from the assumptions (𝐻₂) and (𝐻₃) that the operator Φ is completely continuous. On the other hand, it is not difficult to check that 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇 is a positive 𝜔-periodic solution of (3) if and only if 𝑥^∗(𝑡) is a fixed point of the operator Φ.

Before stating the main results, we shall give some important lemmas.

Lemma 2. The mapping Φ maps 𝐾 into 𝐾, that is, Φ𝐾⊂𝐾.

Proof. For any 𝑥∈𝐾, it is easy to see that Φ𝑥∈𝑋. From (11), we have Φ_𝑖𝑥₀≤𝜆𝐵_𝑖^𝜔₀𝑓_𝑖𝑢,𝑥_𝑢𝑑𝑢+𝐵_𝑖^𝑗=𝑝_𝑗=1𝐼^𝑖_𝑗𝑥𝜏_𝑚𝑗. (12) Noting that 𝐺_𝑖(𝑡,𝑢)𝑓_𝑖(𝑢,𝑥_𝑢)≥0, we can also obtain Φ_𝑖𝑥(𝑡)≥𝜆𝐴_𝑖^𝜔₀𝑓_𝑖𝑢,𝑥_𝑢𝑑𝑢+𝐴_𝑖^𝑗=𝑝_𝑗=1𝐼^𝑖_𝑗𝑥𝜏_𝑚𝑗≥𝐴_𝑖𝐵_𝑖Φ_𝑖𝑥₀≥𝜎Φ_𝑖𝑥₀. (13) Hence, Φ𝐾⊂𝐾. The proof is complete.

Lemma 3. Let 𝑋 be a Banach space, and let 𝐾 be a cone in 𝑋. Suppose that Ω₁ and Ω₂ are open subsets of 𝑋 such that 0∈Ω₁⊂Ω₁⊂Ω₂. Suppose that Φ∶𝐾∩Ω₂⧵Ω₁⟶𝐾 (14) is a completely continuous operator and satisfies either (1)‖Φ𝑢‖≤‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₁;‖Φ𝑢‖≥‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₂; (15) or (2)‖Φ𝑢‖≤‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₂;‖Φ𝑢‖≥‖𝑢‖,∀𝑢∈𝐾∩𝜕Ω₁. (16)Then, Φ has a fixed point in 𝐾∩(Ω₂⧵Ω₁).
The proof of Lemma 3 can be found in [11], Guo et al.

Lemma 4. Assume that (𝐻₁)–(𝐻₃) hold and there exists 𝜂>0, such that ^𝜔₀𝑓𝑠,𝜙_𝑠𝑑𝑠≥𝜂𝜙,𝑓𝑜𝑟𝜙∈𝐾. (17) Then, ‖Φ𝑥‖≥𝜆𝐴𝜂‖𝑥‖. (18)

Proof. If 𝑥∈𝐾, then Φ_𝑖𝑥(𝑡)≥𝜆𝐴_𝑖𝑡^𝑡+𝜔𝑓_𝑖𝑢,𝑥_𝑢𝑑𝑢+𝐴_𝑖^𝑗=𝑝_𝑗=1𝐼^𝑖_𝑗𝑥𝜏_𝑚𝑗≥𝜆𝐴_𝑖𝑡^𝑡+𝜔𝑓_𝑖𝑢,𝑥_𝑢𝑑𝑢. (19) Thus, we have ‖Φ𝑥‖=sup_{𝑡∈𝑅𝑖=1}Φ_𝑖𝑥(𝑡)≥_𝑖=1𝜆𝐴_𝑖^𝜔₀𝑓_𝑖𝑢,𝑥_𝑢𝑑𝑢≥𝜆𝐴^𝜔₀𝑓𝑢,𝑥_𝑢𝑑𝑢≥𝜆𝐴𝜂‖𝑥‖. (20)

Lemma 5. Assume that (𝐻₁)–(𝐻₃) hold and let 𝑟>0, if there exists a sufficiently small 𝜖>0 such ^𝜔₀𝑓𝑠,𝜙_𝑠𝑑𝑠≤𝜖𝑟,^𝑗=𝑝_𝑗=1𝐼_𝑗𝜙≤𝜖𝑟,𝑓𝑜𝑟𝜙∈𝐾∩𝜕Ω_𝑟. (21) Then, ‖Φ𝑥‖≤(𝜆+1)𝐵𝜖‖𝑥‖,𝑓𝑜𝑟𝑥∈𝐾∩𝜕Ω_𝑟. (22)

Proof. For any 𝑥∈𝐾∩𝜕Ω_𝑟, ‖Φ𝑥‖=sup_{𝑡∈𝑅𝑖=1}Φ_𝑖𝑥(𝑡)≤_𝑖=1𝜆𝐵_𝑖^𝜔₀𝑓_𝑖𝑠,𝑥_𝑠𝑑𝑠+_𝑖=1𝐵_𝑖^𝑗=𝑝_𝑗=1𝐼^𝑖_𝑗𝑥𝜏_𝑚𝑗≤𝜆𝐵^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠+𝐵^𝑗=𝑝_𝑗=1𝐼_𝑗𝑥𝜏_𝑚𝑗≤(𝜆+1)𝐵𝜖‖𝑥‖. (23)

3. Main Results

For the sake of convenience, we introduce the following notations: 𝑓^𝛼=lim_𝑥∈𝐾sup_{‖𝑥‖→𝛼}∫^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠‖𝑥‖,𝑓_𝛼=lim_𝑥∈𝐾inf_{‖𝑥‖→𝛼}∫^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠‖𝑥‖,𝐼^𝛼=lim_𝑥∈𝐾sup_{‖𝑥‖→𝛼}∑^𝑗=𝑝_𝑗=1𝐼_𝑗(𝑥)‖𝑥‖,𝐼_𝛼=lim_𝑥∈𝐾inf_{‖𝑥‖→𝛼}∑^𝑗=𝑝_𝑗=1𝐼_𝑗(𝑥)‖𝑥‖, (24) where 𝛼 denotes either 0 or ∞.

Theorem 6. Assume that (𝐻₁)–(𝐻₃) hold and 𝑃₁𝑓_∞𝑃=∞,₂𝑓⁰=𝐼⁰=0; (25) then (3) has at least one positive 𝜔-periodic solution.

Proof. By (𝑃₂), for any 𝜖₁,𝜖₁>0 there exists 𝑟₂>0, such that ^𝜔₀𝑓𝑠,𝜙_𝑠𝑑𝑠≤𝜖₁𝜙≤𝜖₁𝑟₂,_{𝑗=𝑝𝑗=1}𝐼_𝑗𝜙≤𝜖₂𝜙≤𝜖₂𝑟₂. (26) Choose 𝜖=max{𝜖₁,𝜖₁}, satisfying 0<𝜖<(1/(𝜆+1)𝐵), by Lemma 5, we have ‖Φ𝑥‖≤(𝜆+1)𝐵𝜖‖𝑥‖≤‖𝑥‖,for𝑥∈𝐾∩𝜕Ω₂. (27) Next, by (𝑃₂), there exists 𝑟₃>𝑟₂>0, such that ^𝜔₀𝑓𝑠,𝜙_𝑠𝜙,𝜙𝑑𝑠≥𝜂for𝜙∈𝐾,≥𝑟₃, (28) where 𝜂>0 is chosen, so that 𝜆𝐴𝜂>1. It follows from Lemma 4 that ‖Φ𝑥‖≥𝜆𝐴𝜂‖𝑥‖>‖𝑥‖,for𝑥∈𝐾∩𝜕Ω₃. (29) It follows from Lemma 3 that (3) has a positive 𝜔-periodic solution satisfying 𝑟₂≤‖𝑥‖≤𝑟₃.

Theorem 7. Assume that (𝐻₁)–(𝐻₃) hold and 𝑃₃𝑓₀𝑃=∞,₄𝑓^∞=𝐼^∞=0; (30) then (3) has at least one positive 𝜔-periodic solution.

Proof. Since 𝑓₀=∞, one can find an 𝑟₀>0, such that ^𝜔₀𝑓𝑠,𝜙_𝑠𝜙,𝜙𝑑𝑠≥𝜂for𝜙∈𝐾,0<≤𝑟₀, (31) where 𝜂>0 is chosen so that 𝜆𝐴𝜂>1. It follows from Lemma 4 that ‖Φ𝑥‖≥𝜆𝐴𝜂‖𝑥‖>‖𝑥‖,for𝑥∈𝐾∩𝜕Ω_𝑟0. (32) By (𝑃₄), we know that there exists 𝑁₁>𝑟₀ and 𝜖₁,𝜖₂>0 such that ^𝜔₀𝑓𝑠,𝜙_𝑠𝑑𝑠≤𝜖₁𝜙,^𝑗=𝑝_𝑗=1𝐼_𝑗𝜙≤𝜖₂𝜙,for𝜙∈𝐾,𝜙≥𝑁₁. (33) Choose 𝜖=max{𝜖₁,𝜖₁}, satisfying 0<𝜖<1/(2(𝜆+1)𝐵); then ^𝜔₀𝑓𝑠,𝜙_𝑠𝑑𝑠≤𝜖𝜙,^𝑗=𝑝_𝑗=1𝐼_𝑗𝜙≤𝜖𝜙. (34) Take 𝑟₁>𝑁₁+1+2𝐵sup_{𝜙∈𝐾,‖𝜙‖<𝑁1}𝜆^𝜔₀𝑓𝑠,𝜙_𝑠𝑑𝑠+^𝑗=𝑝_𝑗=1𝐼_𝑗𝜙,‖Φ𝑥‖≤𝐵𝜆^𝜔₀𝑓𝑠,𝜙_𝑠𝑑𝑠+^𝑗=𝑝_𝑗=1𝐼_𝑗𝜙=𝐵𝜌𝐼₁+𝜌𝐼₂≤𝑟₁2+‖𝑥‖2=‖𝑥‖,forany𝑥∈𝐾∩𝜕Ω_𝑟1, (35)(36) where 𝜌(𝐼₁)=[𝜆∫^𝜔₀|𝑓(𝑠,𝜙_𝑠)|𝑑𝑠+∑^𝑗=𝑝_𝑗=1|𝐼_𝑗(𝜙)|]|_{𝑥∈𝐼𝑖}, 𝑖=1,2, and 𝐼₁={𝑥∈𝐾,‖𝑥‖<𝑁₁}, 𝐼₂={𝑥∈𝐾,‖𝑥‖≥𝑁₁}.
This implies that ‖Φ𝑥‖≤‖𝑥‖, for any 𝑥∈𝐾∩𝜕Ω_𝑟1.
Therefore, (3) has at least one positive 𝜔-periodic solution.

Theorem 8. Suppose that (𝑃₅) there exists 𝑑₂>0, such that ∫^𝜔₀|𝑓(𝑠,𝜙_𝑠)|𝑑𝑠<𝑑₁/𝜆𝐴, for 𝜎𝑑₁≤‖𝜙‖≤𝑑₁, (𝑃₆) there  exists 𝑑₂>0, such that∫^𝜔₀|𝑓(𝑠,𝜙_𝑠)|𝑑𝑠<𝑑₂/2𝜆𝐵, ∑^𝑗=𝑝_𝑗=1|𝐼_𝑗(𝜙)|≤𝑑₂/2𝐵, for ‖𝜙‖<𝑑₂hold; then (3) has at least one positive 𝜔-periodic solution.

Proof. Without loss of generality, we may assume that 𝑑₂<𝑑₁. If 𝑥∈𝐾∩𝜕Ω_𝑑2, then by (𝑃₆), one can get ‖‖𝑑Φ𝑥≤𝜆𝐵²𝑑2𝜆𝐵+𝐵²2𝐵=𝑑₂=‖𝑥‖, (37) in particular, ‖Φ𝑥‖<‖𝑥‖, for all𝑥∈𝐾∩𝜕Ω_𝑑2.
On the other hand, by(𝑃₅), one has ‖Φ𝑥‖≥𝜆𝐴^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑑𝑠>𝜆𝐴¹𝜆𝐴=𝑑₁=‖𝑥‖,for𝑥∈𝐾∩𝜕Ω_𝑑1. (38) Therefore, (3) has at least one positive 𝜔-periodic solution.

Theorem 9. If  (𝑃₇)𝑓⁰=𝛼₁∈[0,1/2𝜆𝐵), 𝐼₀=𝛼₂∈[0,1/2𝐵); (𝑃₈)𝑓_∞=𝛽₁∈(1/𝜆𝐴𝜎,∞)hold, then (3) has at least one positive 𝜔-periodic solution.

Proof. By assumption (𝑃₇), for 𝜖=min{(1/2𝜆𝐵)−𝛼₁,1/2𝐵−𝛼₂}>0, there exists a sufficiently small 𝑑₂>0 such that ∫^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠‖𝑥‖<𝛼₁1+𝜖<,∑2𝜆𝐵_{𝑗=𝑝𝑗=1}𝐼_𝑗(𝑥)‖𝑥‖<𝛼₂1+𝜖<‖𝑥‖2𝐵,for≤𝑑₂; (39) that is ^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑑𝑠<²,2𝜆𝐵_{𝑗=𝑝𝑗=1}𝐼_𝑗(𝑥<𝑑)²‖𝑥‖2𝐵,for≤𝑑₂. (40) So, (𝑃₆) is satisfied.
By assumption (𝑃₈), for 𝜖=𝛽₁−1/𝜆𝐴𝜎, there exists a sufficiently large 𝑑₁>0 such that ∫^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠‖𝑥‖>𝛽₁1−𝜖=𝜆𝐴𝜎,for𝜎𝑑₁≤‖𝑥‖≤𝑑₁, (41) that is ^𝜔₀𝑓𝑠,𝑥_𝑠1𝑑𝑠>‖𝑥‖≥1𝜆𝐴𝜎𝜆𝐴𝜎𝜎𝑑₁=𝑑₁,𝜆𝐴 (42) therefore, (𝑃₅) holds. By Theorem 8, we complete the proof.

Theorem 10. If (𝑃₉)𝑓₀=𝛼₃∈(1/𝜆𝐴𝜎,∞); (𝑃₁₀)𝑓^∞=𝛽₂∈[0,1/2𝜆𝐵), 𝐼_∞=𝛽₃∈[0,1/2𝐵)hold, then (3) has at least one positive 𝜔-periodic solution.

Proof. By (𝑃₉), for 𝜖=𝛼₃−(1/𝜆𝐴𝜎)>0, there exists a sufficiently small 𝑑₁>0, such that ∫^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠‖𝑥‖>𝛼₃1−𝜖=‖𝑥‖𝜆𝐴𝜎,for0<≤𝑑₁, (43) that is ^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠>𝜎𝑑₁=𝑑𝜆𝐴𝜆𝐴¹,for𝜎𝑑₁≤‖𝑥‖≤𝑑₁. (44) Again, By (𝑃₁₀), for 𝜖=min{(1/2𝜆𝐵)−𝛽₂,(1/2𝐵)−𝛽₃}>0, there exists a sufficiently small 𝑑>0 such that ∫^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠‖𝑥‖<𝛽₂∑+𝜖,^{𝑗=𝑝𝑗=1}𝐼_𝑗(𝑥)‖𝑥‖<𝛽₃+𝜖, (45) that is ∫^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠‖𝑥‖<1,∑2𝜆𝐵^{𝑗=𝑝𝑗=1}𝐼_𝑗(𝑥)‖𝑥‖<1,‖𝑥‖2𝐵for>𝑑. (46) In the following, we consider two cases to prove (𝑃₆) to be satisfied:(i)∫^𝜔₀|𝑓(𝑠,𝑥_𝑠)|𝑑𝑠+∑^𝑗=𝑝_𝑗=1|𝐼_𝑗(𝑥)|<∞;(ii)∫^𝜔₀|𝑓(𝑠,𝑥_𝑠)|𝑑𝑠=∞, ∑^𝑗=𝑝_𝑗=1|𝐼_𝑗(𝑥)|=∞.The bounded case is clear. If case (ii) is valid, then there exists 𝑦∈𝐵𝐶(𝑅,𝑅^𝑛₊), ‖𝑦‖=𝑑₂>𝑑 such that ^𝜔₀𝑓𝑠,𝑥_𝑠𝑑𝑠≤^𝜔₀𝑓𝑠,𝑦_𝑠𝑑𝑠,_{𝑗=𝑝𝑗=1}𝐼_𝑗(𝑥)<^𝑗=𝑝_𝑗=1𝐼_𝑗𝑦,for0<‖𝑥‖≤𝑦=𝑑₂. (47) Since ‖𝑦‖=𝑑₂>𝑑, then we have ^𝜔₀𝑓𝑠,𝑥_𝑠𝑦𝑑𝑠<=𝑑2𝜆𝐵²,2𝜆𝐵_{𝑗=𝑝𝑗=1}𝐼_𝑗(𝑥<𝑦)=𝑑2𝐵²,‖𝑥‖2𝐵for0<≤𝑑₂, (48) which implies that condition (𝑃₆) holds. By Theorem 8, we complete the proof.

Corollary 11. If one of the following pairs (𝑃₁) and (𝑃₂); (𝑃₃) and (𝑃₄); (𝑃₅) and (𝑃₆); (𝑃₇) and (𝑃₈); (𝑃₉) and (𝑃₁₀); (𝑃₁) and (𝑃₇); (𝑃₂) and (𝑃₈); (𝑃₃) and (𝑃₁₀); (𝑃₄) and (𝑃₉)is valid, then system (3) has at least one positive 𝜔-periodic solution.

Acknowledgments

This research was supported by the National Nature Science Foundation of China (no. 11371027 and no. 11071001); Special Research Fund for the Doctoral Program of the Ministry of Education of China (no. 20123401120001); National Science Foundation of Educational Committee of Anhui Province of China (no. KJ2011B052); Program of Natural Science of Colleges of Anhui Province (no. KJ2013A032 and no. KJ2012A019).