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Mathematical Problems in Engineering
Volume 2011, Article ID 610812, 12 pages
http://dx.doi.org/10.1155/2011/610812
Research Article

Existence Results for a Nonlinear Semipositone Telegraph System with Repulsive Weak Singular Forces

1College of Science, Hohai University, Nanjing 210098, China
2College of Aeronautics and Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
3Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 10 September 2011; Accepted 9 November 2011

Academic Editor: Sebastian Anita

Copyright © 2011 Fanglei Wang 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

Using the fixed point theorem of cone expansion/compression, we consider the existence results of positive solutions for a nonlinear semipositone telegraph system with repulsive weak singular forces.

1. Introduction

In this paper, we are concerned with the existence of positive solutions for the nonlinear telegraph system:𝑢𝑡𝑡𝑢𝑥𝑥+𝑐1𝑢𝑡+𝑎1𝑣(𝑡,𝑥)𝑢=𝑓(𝑡,𝑥,𝑣),𝑡𝑡𝑣𝑥𝑥+𝑐2𝑣𝑡+𝑎2(𝑡,𝑥)𝑣=𝑔(𝑡,𝑥,𝑢),(1.1) with doubly periodic boundary conditions𝑢(𝑡+2𝜋,𝑥)=𝑢(𝑡,𝑥+2𝜋)=𝑢(𝑡,𝑥),(𝑡,𝑥)𝑅2,𝑣(𝑡+2𝜋,𝑥)=𝑣(𝑡,𝑥+2𝜋)=𝑣(𝑡,𝑥),(𝑡,𝑥)𝑅2.(1.2) In particular, the function 𝑓(𝑡,𝑥,𝑣) may be singular at 𝑣=0 or superlinear at 𝑣=+, and 𝑔(𝑡,𝑥,𝑢) may be singular at 𝑢=0 or superlinear at 𝑢=+.

In the latter years, the periodic problem for the semilinear singular equation𝑥+𝑎(𝑡)𝑥=𝑏(𝑡)𝑥𝜆+𝑐(𝑡),(1.3) with 𝑎, 𝑏, 𝑐𝐿1[0,𝑇] and 𝜆>0, has received the attention of many specialists in differential equations. The main methods to study (1.3) are the following three common techniques:(i)the obtainment of a priori bounds for the possible solutions and then the applications of topological degree arguments;(ii)the theory of upper and lower solutions;(iii)some fixed point theorems in a cone.

We refer the readers to see [17] and the references therein.

Equation (1.3) is related to the stationary version of the telegraph equation𝑢𝑡𝑡𝑢𝑥𝑥+𝑐𝑢𝑡+𝜆𝑢=𝑓(𝑡,𝑥,𝑢),(1.4) where 𝑐>0 is a constant and 𝜆𝑅. Because of its important physical background, the existence of periodic solutions for a single telegraph equation or telegraph system has been studied by many authors; see [816]. Recently, Wang utilize a weak force condition to enable the achievement of new existence criteria for positive doubly periodic solutions of nonlinear telegraph system through a basic application of Schauder’s fixed point theorem in [17]. Inspired by these papers, here our interest is in studying the existence of positive doubly periodic solutions for a semipositone nonlinear telegraph system with repulsive weak singular forces by using the fixed point theorem of cone expansion/compression.

Lemma 1.1 (see [18]). Let 𝐸 be a Banach space, and let 𝐾𝐸 be a cone in 𝐸. Assume that Ω1, Ω2 are open subsets of 𝐸 with 0Ω1, Ω1Ω2, and let 𝑇𝐾(Ω2Ω1)𝐾 be a completely continuous operator such that either(i)𝑇𝑢𝑢,𝑢𝐾𝜕Ω1 and 𝑇𝑢𝑢,𝑢𝐾𝜕Ω2; or(ii)𝑇𝑢𝑢,𝑢𝐾𝜕Ω1 and 𝑇𝑢𝑢,𝑢𝐾𝜕Ω2.Then, 𝑇 has a fixed point in 𝐾(Ω2Ω1).

This paper is organized as follows: in Section 2, some preliminaries are given; in Section 3, we give the main results.

2. Preliminaries

Let 2 be the torus defined as2=𝑅×𝑅2𝜋𝑍.2𝜋𝑍(2.1) Doubly 2𝜋-periodic functions will be identified to be functions defined on 2. We use the notations𝐿𝑝2,𝐶2,𝐶𝛼2,𝐷2=𝐶2,(2.2) to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space 𝐷(2)denotes the space of distributions on 2.

By a doubly periodic solution of (1.1)-(1.2) we mean that a (𝑢,𝑣)𝐿1(2)×𝐿1(2) satisfies (1.1)-(1.2) in the distribution sense; that is,2𝑢𝜑𝑡𝑡𝜑𝑥𝑥𝑐1𝜑𝑡+𝑎1(𝑡,𝑥)𝜑𝑑𝑡𝑑𝑥=2𝑓(𝑡,𝑥,𝑣)𝜑𝑑𝑡𝑑𝑥,2𝑣𝜑𝑡𝑡𝜑𝑥𝑥𝑐2𝜑𝑡+𝑎2(𝑡,𝑥)𝜑𝑑𝑡𝑑𝑥=2𝑔(𝑡,𝑥,𝑢)𝜑𝑑𝑡𝑑𝑥,𝜑𝐷2.(2.3) First, we consider the linear equation𝑢𝑡𝑡𝑢𝑥𝑥+𝑐𝑖𝑢𝑡𝜆𝑖𝑢=𝑖(𝑡,𝑥),in𝐷2,(2.4) where 𝑐𝑖>0, 𝜆𝑖𝑅, and 𝑖(𝑡,𝑥)𝐿1(2), (𝑖=1,2).

Let £𝜆𝑖 be the differential operator£𝜆𝑖=𝑢𝑡𝑡𝑢𝑥𝑥+𝑐𝑖𝑢𝑡𝜆𝑖𝑢,(2.5) acting on functions on 2. Following the discussion in [14], we know that if 𝜆𝑖<0, then £𝜆𝑖 has the resolvent 𝑅𝜆𝑖:𝑅𝜆𝑖𝐿12𝐶2,𝑖𝑢𝑖,(2.6) where 𝑢𝑖 is the unique solution of (2.4), and the restriction of 𝑅𝜆𝑖 on 𝐿𝑝(2)(1<𝑝<) or 𝐶(2) is compact. In particular, 𝑅𝜆𝑖𝐶(2)𝐶(2) is a completely continuous operator.

For 𝜆𝑖=𝑐2𝑖/4, the Green function 𝐺𝑖(𝑡,𝑥) of the differential operator £𝜆𝑖 is explicitly expressed; see lemma  5.2 in [14]. From the definition of 𝐺𝑖(𝑡,𝑥), we have𝐺𝑖=essinf𝐺𝑖𝑒(𝑡,𝑥)=3𝑐𝑖𝜋/2(1𝑒𝑐𝑖𝜋)2,𝐺𝑖=esssup𝐺𝑖(𝑡,𝑥)=(1+𝑒𝑐𝑖𝜋)2(1𝑒𝑐𝑖𝜋)2.(2.7) Let 𝐸 denote the Banach space 𝐶(2) with the norm 𝑢=max(𝑡,𝑥)2|𝑢(𝑡,𝑥)|, then 𝐸 is an ordered Banach space with cone𝐾0=𝑢𝐸𝑢(𝑡,𝑥)0,(𝑡,𝑥)2.(2.8) For convenience, we assume that the following condition holds throughout this paper:(H1)𝑎𝑖(𝑡,𝑥)𝐶(2,𝑅+), 0<𝑎𝑖(𝑡,𝑥)𝑐2𝑖/4 for (𝑡,𝑥)2, and 2𝑎𝑖(𝑡,𝑥)𝑑𝑡𝑑𝑥>0.

Next, we consider (2.4) when 𝜆𝑖 is replaced by 𝑎𝑖(𝑡,𝑥). In [10], Li has proved the following unique existence and positive estimate result.

Lemma 2.1. Let 𝑖(𝑡,𝑥)𝐿1(2);𝐸 is the Banach space 𝐶(2). Then; (2.4) has a unique solution 𝑢𝑖=𝑃𝑖𝑖;𝑃𝑖𝐿1(2)𝐶(2) is a linear bounded operator with the following properties;(i)𝑃𝑖𝐶(2)𝐶(2) is a completely continuous operator;(ii)if 𝑖(𝑡,𝑥)>0,then𝑎.𝑒.(𝑡,𝑥)2,𝑃𝑖[𝑖(𝑡,𝑥)] has the positive estimate𝐺𝑖𝑖𝐿1𝑃𝑖𝑖(𝑡,𝑥)𝐺𝑖𝐺𝑖𝑎𝑖𝐿1𝑖𝐿1.(2.9)

3. Main Result

In this section, we establish the existence of positive solutions for the telegraph system𝑣𝑡𝑡𝑣𝑥𝑥+𝑐1𝑣𝑡+𝑎1𝑣(𝑡,𝑥)𝑣=𝑓(𝑡,𝑥,𝑢),𝑡𝑡𝑣𝑥𝑥+𝑐2𝑣𝑡+𝑎2(𝑡,𝑥)𝑣=𝑔(𝑡,𝑥,𝑢).(3.1) where 𝑎𝑖𝐶(𝑅2,𝑅+) and 𝑓(𝑡,𝑥,𝑣) may be singular at 𝑣=0. In particular, 𝑓(𝑡,𝑥,𝑣) may be negative or superlinear at 𝑣=+. 𝑔(𝑡,𝑥,𝑢) has the similar assumptions. Our interest is in working out what weak force conditions of 𝑓(𝑡,𝑥,𝑣) at 𝑣=0, 𝑔(𝑡,𝑥,𝑢) at 𝑢=0 and what superlinear growth conditions of 𝑓(𝑡,𝑥,𝑣) at 𝑣=+, 𝑔(𝑡,𝑥,𝑢) at 𝑢=+ are needed to obtain the existence of positive solutions for problem (1.1)-(1.2).

We assume the following conditions throughout.(H2)𝑓,𝑔2×(0,)𝑅 is continuous, and there exists a constant 𝑀>0 such that 𝑓1(𝑡,𝑥,𝑢)+𝑀0,𝑓2(𝑡,𝑥,𝑢)+𝑀0,(𝑡,𝑥)2and𝑢,𝑣(0,).(3.2)(H3)𝐹(𝑡,𝑥,𝑣)=𝑓(𝑡,𝑥,𝑣)+𝑀𝑗1(𝑣)+1(𝑣) for (𝑡,𝑥,𝑣)2×(0,) with 𝑗1>0 continuous and nonincreasing on (0,), 10 continuous on (0,) and 1/𝑗1 nondecreasing on (0,).𝐺(𝑡,𝑥,𝑢)=𝑔(𝑡,𝑥,𝑢)+𝑀𝑗2(𝑢)+2(𝑢) for (𝑡,𝑥,𝑢)2×(0,) with 𝑗2>0 continuous and nonincreasing on (0,), 20 continuous on (0,) and 2/𝑗2 nondecreasing on (0,).(H4)𝐹(𝑡,𝑥,𝑣)=𝑓(𝑡,𝑥,𝑣)+𝑀𝑗3(𝑣)+3(𝑣) for all (𝑡,𝑥,𝑣)2×(0,) with 𝑗3>0 continuous and nonincreasing on (0,), 30 continuous on (0,) with 3/𝑗3 nondecreasing on (0,);𝐺(𝑡,𝑥,𝑢)=𝑔(𝑡,𝑥,𝑢)+𝑀𝑗4(𝑢)+4(𝑢) for all (𝑡,𝑥,𝑢)2×(0,) with 𝑗4>0 continuous and nonincreasing on (0,), 40 continuous on (0,) with 4/𝑗4 nondecreasing on (0,).(H5) There exists 𝑀𝜔𝑟>1𝛿1,(3.3) such that 𝑟4𝜋2𝐺1𝐺1𝑎1𝐿1𝐼1𝐼2,(3.4) here 𝐼1=𝑗1𝐺2𝑗4(𝑟)1+4𝛿1𝜔𝑟𝑀1𝑗4𝛿1𝜔𝑟𝑀14𝜋2𝜔𝑀2,𝐼2=1+14𝜋2𝐺2/𝐺2𝑎2𝐿1𝑗2𝛿1𝜔𝑟𝑀11+2(𝑟)/𝑗2(𝑟)𝑗14𝜋2𝐺2/𝐺2𝑎2𝐿1𝑗2𝛿1𝜔𝑟𝑀11+2(𝑟)/𝑗2,(𝑟)(3.5) where 𝛿𝑖=(𝐺𝑖2𝑎𝑖𝐿1/𝐺𝑖)(0,1), and 𝜔𝑖(𝑡,𝑥) is the unique solution to problem: 𝑢𝑡𝑡𝑢𝑥𝑥+𝑐𝑖𝑢𝑡+𝑎𝑖(𝑡,𝑥)𝑢=1,𝑢(𝑡+2𝜋,𝑥)=𝑢(𝑡,𝑥+2𝜋)=𝑢(𝑡,𝑥),(𝑡,𝑥)𝑅2.(3.6)(H6) There exists 𝑅>𝑟, such that 4𝜋2𝐺1𝐼3𝐼4𝛿𝑅,2𝑗4(𝑅)1+4𝛿1𝜔𝑟𝑀1𝑗4𝛿1𝜔𝑟𝑀1>𝑀,(3.7) where 𝐼3=𝐺1𝑗34𝜋2𝐺2𝐺2𝑎2𝐿1𝑗2𝛿1𝜔𝑅𝑀11+2(𝑅)𝑗2,𝐼(𝑅)4=1+3𝐺2𝑗4(𝑅)1+4𝛿1𝜔𝑅𝑀1/𝑗4𝛿1𝜔𝑅𝑀14𝜋2𝜔𝑀2𝑗3𝐺2𝑗4(𝑅)1+4𝛿1𝜔𝑅𝑀1/𝑗4𝛿1𝜔𝑅𝑀14𝜋2𝜔𝑀2.(3.8)

Theorem 3.1. Assume that (H1)–(H6) hold. Then, the problem (1.1)-(1.2) has a positive doubly periodic solution (𝑢,𝑣).

Proof. To show that (1.1)-(1.2) has a positive solution, we will proof that 𝑢𝑡𝑡𝑢𝑥𝑥+𝑐1𝑢𝑡+𝑎1(𝑡,𝑥)𝑢=𝐹𝑡,𝑥,𝑣𝑀𝜔2,𝑣𝑡𝑡𝑣𝑥𝑥+𝑐2𝑣𝑡+𝑎2(𝑡,𝑥)𝑣=𝐺𝑡,𝑥,𝑢𝑀𝜔1(3.9) has a solution ̃(̃𝑢,𝑣)=(𝑢+𝑀𝜔1,𝑣+𝑀𝜔2) with ̃𝑢>𝑀𝜔1, ̃𝑣>𝑀𝜔2 for (𝑡,𝑥)2. In addition, by Lemma 2.1, it is clear to see that (𝑢,𝑣)𝐶2(2)×𝐶2(2) is a solution of (3.9) if and only if (𝑢,𝑣)𝐶(2)×𝐶(2) is a solution of the following system: 𝑢=𝑃1𝐹𝑡,𝑥,𝑣𝑀𝜔2,𝑣=𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1.(3.10) Evidently, (3.10) can be rewritten as the following equation: 𝑢=𝑃1𝐹𝑡,𝑥,𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2.(3.11)
Define a cone 𝐾𝐸 as 𝐾=𝑢𝐸𝑢0,𝑢𝛿1𝑢.(3.12) We define an operator 𝑇𝐸𝐾 by (𝑇𝑢)(𝑡,𝑥)=𝑃1𝐹𝑡,𝑥,𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2(3.13) for 𝑢𝐸 and (𝑡,𝑥)2. We have the conclusion that 𝑇𝐸𝐸 is completely continuous and 𝑇(𝐾)𝐾. The complete continuity is obvious by Lemma 2.1. Now, we show that 𝑇(𝐾)𝐾.
For any 𝑢𝐾, we have 𝑇𝑢=𝑃1𝐹𝑡,𝑥,𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2.(3.14) From (H1)–(H3) and Lemma 2.1, we have 𝑇𝑢=𝑃1𝐹𝑡,𝑥,𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝐺1𝐹𝑡,𝑥,𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝐿1,𝑃𝐹𝑇𝑢=𝑡,𝑥,𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝐺1𝐺1𝑎1𝐿1𝐹𝑡,𝑥,𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝐿1.(3.15) So, we get 𝐺𝑇𝑢12𝑎1𝐿1𝐺1𝑇𝑢𝛿1𝑇𝑢,(3.16) namely, 𝑇(𝐾)𝐾.
Let Ω𝑟={𝑢𝐸𝑢<𝑟},Ω𝑅={𝑢𝐸𝑢<𝑅}.(3.17) Since 𝑟𝑢𝑅 for any 𝑢𝐾(Ω𝑅Ω𝑟), we have 0<𝛿1𝑟𝑀𝜔𝑢𝑀𝜔1𝑅.
First, we show 𝑇𝑢𝑢,for𝑢𝐾𝜕Ω𝑟.(3.18) In fact, if 𝑢𝐾𝜕Ω𝑟, then 𝑢=𝑟 and 𝑢𝛿1𝑟>𝑀𝜔1 for(𝑡,𝑥)2. By (H3) and (H4), we have 𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝐺2𝐺2𝑎2𝐿1𝐺𝑡,𝑥,𝑢𝑀𝜔1𝐿1𝐺2𝐺2𝑎2𝐿1𝑗2𝑢𝑀𝜔11+2𝑢𝑀𝜔1𝑗2𝑢𝑀𝜔1𝐿1𝐺2𝐺2𝑎2𝐿1𝑗2𝛿1𝜔𝑟𝑀11+2(𝑟)𝑗2(𝑟)4𝜋2,𝑃(3.19)2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝐺2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝐿1𝐺2𝑗4𝑢𝑀𝜔11+4𝑢𝑀𝜔1𝑗4𝑢𝑀𝜔1𝐿1𝐺2𝑗4(𝑟)1+4𝛿1𝜔𝑟𝑀1𝑗4𝛿1𝜔𝑟𝑀14𝜋2.(3.20) In addition, we also have 𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝐺2𝑗4(𝑟)1+4𝛿1𝜔𝑟𝑀1𝑗4𝛿1𝜔𝑟𝑀14𝜋2𝐺2𝑗4(𝑅)1+4𝛿1𝜔𝑟𝑀1𝑗4𝛿1𝜔𝑟𝑀14𝜋2>𝐺2𝐺2𝑎2𝐿1𝑀4𝜋2𝑀𝜔2,(3.21) by (H5), (H6), and (3.20).
So, we have 𝑇𝑢=𝑃1𝐹𝑡,𝑥,𝑣𝑀𝜔2𝐺1𝐺1𝑎1𝐿1𝐹𝑡,𝑥,𝑣𝑀𝜔2𝐿1𝐺1𝐺1𝑎1𝐿1𝑗1𝑣𝑀𝜔21+1𝑣𝑀𝜔2𝑗1𝑣𝑀𝜔2𝐿1𝐺1𝐺1𝑎1𝐿1𝑗1𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2×1+1𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝑗1𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝐿1𝐺1𝐺1𝑎1𝐿1𝑗1𝐺2𝑗4(𝑟)1+4𝛿1𝜔𝑟𝑀1𝑗4𝛿1𝜔𝑟𝑀14𝜋2𝜔𝑀2×1+1𝐺2/𝐺2𝑎2𝐿1𝑗2𝛿1𝜔𝑟𝑀11+2(𝑟)/𝑗2(𝑟)4𝜋2𝑗1𝐺2/𝐺2𝑎2𝐿1𝑗2𝛿1𝜔𝑟𝑀11+2(𝑟)/𝑗2(𝑟)4𝜋24𝜋2𝑟=𝑢(3.22) for (𝑡,𝑥)2, since 𝛿1𝑟𝑀𝜔1𝑢𝑀𝜔1𝑟.
This implies that 𝑇𝑢𝑢; that is, (3.18) holds.
Next, we show 𝑇𝑢𝑢,for𝑢𝐾𝜕Ω𝑅.(3.23) If 𝑢𝐾𝜕Ω𝑅, then 𝑢=𝑅 and 𝑢𝛿𝑅>𝑀𝜔1 for (𝑡,𝑥)2. From (H4) and (H6), we have𝑇𝑢=𝑃1𝐹𝑡,𝑥,𝑣𝑀𝜔1𝐺1𝑗3𝑣𝑀𝜔21+3𝑣𝑀𝜔2𝑗3𝑣𝑀𝜔2𝐿1𝐺1𝑗3𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2×1+3𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝑗3𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝑀𝜔2𝐿1𝐺1𝑗3𝐺2𝐺2𝑎2𝐿1𝑗2𝛿1𝜔𝑅𝑀11+2(𝑅)𝑗2(𝑅)4𝜋2×1+3𝐺2𝑗4(𝑅)1+4𝛿1𝜔𝑅𝑀1/𝑗4𝛿1𝜔𝑅𝑀14𝜋2𝜔𝑀2𝑗3𝐺2𝑗4(𝑅)1+4𝛿1𝜔𝑅𝑀1/𝑗4𝛿1𝜔𝑅𝑀14𝜋2𝜔𝑀2𝐿1𝑅=𝑢(3.24) for (𝑡,𝑥)2, since 𝛿1𝑅𝑀𝜔1𝑢𝑀𝜔1𝑅.
This implies that 𝑇𝑢𝑢; that is, (3.23) holds.
Finally, (3.18), (3.23), and Lemma 1.1 guarantee that 𝑇 has a fixed point 𝑢𝐾Ω𝑅Ω𝑟 with 𝑟𝑢𝑅. Clearly, 𝑢>𝑀𝜔1.
Since 𝑃2𝐺𝑡,𝑥,𝑢𝑀𝜔1𝐺2𝐺𝑡,𝑥,𝑀𝜔1𝐿1𝐺2𝑗4𝑢𝑀𝜔11+4𝑢𝑀𝜔1𝑗4𝑢𝑀𝜔1𝐿1𝐺2𝑗4(𝑅)1+4𝛿1𝜔𝑟𝑀1𝑗4𝛿1𝜔𝑟𝑀14𝜋2>𝐺2𝐺2𝑎2𝐿1𝑀4𝜋2𝑀𝜔2,(3.25) then we have a doubly periodic solution (𝑢,𝑣) of (3.9) with 𝑢>𝑀𝜔1, 𝑣>𝑀𝜔2, namely, (𝑢𝑀𝜔1,𝑣𝑀𝜔2)>(0,0) is a positive solution of (1.1) with (1.2).

Similarly, we also obtain the following result.

Theorem 3.2. Assume that (H1)–(H4) hold. In addition, we assume the following.(H7)There exists𝑀𝜔𝑟>2𝛿2,(3.26) such that 𝑟4𝜋2𝐺2𝐺2𝑎2𝐿1𝐼5𝐼6,(3.27) here 𝐼5=𝑗24𝜋2𝐺1𝑗3(𝑟)1+3𝛿2𝜔𝑟𝑀2𝑗3𝛿2𝜔𝑟𝑀2𝜔𝑀1,𝐼6=1+24𝜋2𝐺1/𝐺1𝑎1𝐿1𝑗1𝛿2𝜔𝑟𝑀21+1(𝑟)/𝑗1(𝑟)𝑗24𝜋2𝐺1/𝐺1𝑎1𝐿1𝑗1𝛿2𝜔𝑟𝑀21+1(𝑟)/𝑗1.(𝑟)(3.28)(H8) There exists 𝑅>𝑟, such that4𝜋2𝐺2𝐼7𝐼8𝛿𝑅,1𝑗3(𝑅)1+3𝛿2𝜔𝑟𝑀2𝑗3𝛿2𝜔𝑟𝑀2>𝑀,(3.29) where 𝐼7=𝑗44𝜋2𝐺1𝐺1𝑎1𝐿1𝑗1𝛿2𝜔𝑅𝑀21+1(𝑅)𝑗1,𝐼(𝑅)8=1+44𝜋2𝐺1𝑗3(𝑅)1+3𝛿2𝜔𝑅𝑀2/𝑗3𝛿2𝜔𝑅𝑀2𝜔𝑀1𝑗44𝜋2𝐺1𝑗3(𝑅)1+3𝛿2𝜔𝑅𝑀2/𝑗3𝛿2𝜔𝑅𝑀2𝜔𝑀1.(3.30) Then, problem (1.1)-(1.2) has a positive periodic solution.

4. An Example

Consider the following system:𝑢𝑡𝑡𝑢𝑥𝑥+2𝑢𝑡+sin2𝑣(𝑡+𝑥)𝑢=𝜇𝛼+𝑣𝛽+𝑘1,𝑣(𝑡,𝑥)𝑡𝑡𝑣𝑥𝑥+2𝑣𝑡+cos2𝑢(𝑡+𝑥)𝑣=𝜆𝜏+𝑢𝜎+𝑘2,(𝑡,𝑥)𝑢(𝑡+2𝜋,𝑥)=𝑢(𝑡,𝑥+2𝜋)=𝑢(𝑡,𝑥),(𝑡,𝑥)𝑅2,𝑣(𝑡+2𝜋,𝑥)=𝑣(𝑡,𝑥+2𝜋)=𝑣(𝑡,𝑥),(𝑡,𝑥)𝑅2,(4.1) where 𝑐1=𝑐2=2, 𝜇,𝜆>0, 𝛼,𝜏>0,𝛽,𝜎>1, 𝑎1(𝑡,𝑥)=sin2(𝑡+𝑥), 𝑎2(𝑡,𝑥)=cos2(𝑡+𝑥)𝐶(2,𝑅+), 𝑘𝑖2𝑅 is continuous. When 𝜇 is chosen such that𝜇<sup𝜔𝑢((𝑀1)/𝛿1,)𝐺𝑎1𝐿1𝐺4𝜋2𝐼1𝐼2,(4.2) here we denote 𝐼1𝐺=𝑢𝜆𝑢𝜏𝛿1+1𝜔𝑢𝑀1𝜎+𝜏4𝜋2𝜔𝑀2𝛼,𝐼2=1+𝐺𝐺𝑎2𝐿1𝜆𝛿1𝜔𝑢𝑀1𝜏1+𝑢𝜎+𝜏+2𝐻𝑢𝜏4𝜋2𝛽+𝛼+2𝐻𝐺𝐺𝑎2𝐿1𝜆𝛿1𝜔𝑢𝑀1𝜏1+𝑢𝜎+𝜏+2𝐻𝑢𝜏4𝜋2,(4.3) where 𝐻=max{𝑘1,𝑘2} and the Green function 𝐺1=𝐺2=𝐺. Then, problem (4.1) has a positive solution.

To verify the result, we will apply Theorem 3.1 with 𝑀=max{𝜇𝐻,𝜆𝐻} and𝑗1(𝑣)=𝑗3(𝑣)=𝜇𝑣𝛼,1𝑣(𝑣)=𝜇𝛽+2𝐻,3(𝑣)=𝜇𝑣𝛽,𝑗2(𝑢)=𝑗4(𝑢)=𝜆𝑢𝜏,2(𝑢)=𝜇(𝑢𝜎+2𝐻),4(𝑢)=𝜇𝑢𝜎.(4.4) Clearly, (H1)–(H4) are satisfied.

Set𝐺𝑇(𝑢)=𝑎1𝐿1𝐺4𝜋2𝐼1𝐼2𝑀𝜔,𝑢1𝛿1,+.(4.5) Obviously, 𝑇((𝑀𝜔1)/𝛿1)=0, 𝑇()=0, then there exists 𝑟((𝑀𝜔1)/𝛿1,+) such that𝑇(𝑟)=sup𝑀𝜔𝑢1/𝛿1,𝐺𝑎1𝐿1𝐺4𝜋2𝐼1𝐼2.(4.6) This implies that there exists𝑀𝜔𝑟1𝛿1,+,(4.7) such that𝜇<sup𝑀𝜔𝑢1/𝛿1,𝐺𝑎1𝐿1𝐺4𝜋2𝐼1𝐼2.(4.8) So, (H5) is satisfied.

Finally, since𝑅𝐺/𝐺𝑎2𝐿1𝜆𝛿1𝜔𝑅𝑀1𝜏1+𝑅𝜎+𝜏+2𝐻𝑅𝜏4𝜋2𝛼𝜇𝐺𝐺1+𝜆𝑅𝜏𝛿1+1𝜔𝑅𝑀1𝜎+𝜏4𝜋2𝜔𝑀2𝛼+𝛽0as𝑅,(4.9) this implies that there exists 𝑅. In addition, for fixed 𝑟,𝑅, choosing 𝜆 sufficiently large, we have𝛿2𝜆𝑅𝜏𝛿1+1𝜔𝑟𝑀1𝜎+𝜏>𝑀.(4.10) Thus, (H6) is satisfied. So, all the conditions of Theorem 3.1 are satisfied.

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