Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 610812 | https://doi.org/10.1155/2011/610812

Fanglei Wang, Yunhai Wang, Yukun An, "Existence Results for a Nonlinear Semipositone Telegraph System with Repulsive Weak Singular Forces", Mathematical Problems in Engineering, vol. 2011, Article ID 610812, 12 pages, 2011. https://doi.org/10.1155/2011/610812

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

Academic Editor: Sebastian Anita
Received10 Sep 2011
Accepted09 Nov 2011
Published27 Dec 2011

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:π‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯+𝑐1𝑒𝑑+π‘Ž1𝑣(𝑑,π‘₯)𝑒=𝑓(𝑑,π‘₯,𝑣),π‘‘π‘‘βˆ’π‘£π‘₯π‘₯+𝑐2𝑣𝑑+π‘Ž2(𝑑,π‘₯)𝑣=𝑔(𝑑,π‘₯,𝑒),(1.1) with doubly periodic boundary conditions𝑒(𝑑+2πœ‹,π‘₯)=𝑒(𝑑,π‘₯+2πœ‹)=𝑒(𝑑,π‘₯),(𝑑,π‘₯)βˆˆπ‘…2,𝑣(𝑑+2πœ‹,π‘₯)=𝑣(𝑑,π‘₯+2πœ‹)=𝑣(𝑑,π‘₯),(𝑑,π‘₯)βˆˆπ‘…2.(1.2) In particular, the function 𝑓(𝑑,π‘₯,𝑣) may be singular at 𝑣=0 or superlinear at 𝑣=+∞, and 𝑔(𝑑,π‘₯,𝑒) may be singular at 𝑒=0 or superlinear at 𝑒=+∞.

In the latter years, the periodic problem for the semilinear singular equationπ‘₯ξ…žξ…ž+π‘Ž(𝑑)π‘₯=𝑏(𝑑)π‘₯πœ†+𝑐(𝑑),(1.3) with π‘Ž, 𝑏, π‘βˆˆπΏ1[0,𝑇] and πœ†>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 [1–7] and the references therein.

Equation (1.3) is related to the stationary version of the telegraph equationπ‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯+𝑐𝑒𝑑+πœ†π‘’=𝑓(𝑑,π‘₯,𝑒),(1.4) where 𝑐>0 is a constant and πœ†βˆˆπ‘…. 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 [8–16]. 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 πΎβŠ‚πΈ be a cone in 𝐸. Assume that Ξ©1, Ξ©2 are open subsets of 𝐸 with 0∈Ω1, Ξ©1βŠ‚Ξ©2, and let π‘‡βˆΆπΎβˆ©(Ξ©2⧡Ω1)→𝐾 be a completely continuous operator such that either(i)‖𝑇𝑒‖≀‖𝑒‖,π‘’βˆˆπΎβˆ©πœ•Ξ©1 and ‖𝑇𝑒‖β‰₯‖𝑒‖,π‘’βˆˆπΎβˆ©πœ•Ξ©2; or(ii)‖𝑇𝑒‖β‰₯‖𝑒‖,π‘’βˆˆπΎβˆ©πœ•Ξ©1 and ‖𝑇𝑒‖≀‖𝑒‖,π‘’βˆˆπΎβˆ©πœ•Ξ©2.Then, 𝑇 has a fixed point in 𝐾∩(Ξ©2⧡Ω1).

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 as⊀2=𝑅×𝑅2πœ‹π‘ξ‚.2πœ‹π‘(2.1) Doubly 2πœ‹-periodic functions will be identified to be functions defined on ⊀2. We use the notationsπΏπ‘ξ€·βŠ€2ξ€Έξ€·βŠ€,𝐢2ξ€Έ,πΆπ›Όξ€·βŠ€2ξ€Έξ€·βŠ€,𝐷2ξ€Έ=πΆβˆžξ€·βŠ€2ξ€Έ,…(2.2) to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space π·ξ…ž(⊀2)denotes the space of distributions on ⊀2.

By a doubly periodic solution of (1.1)-(1.2) we mean that a (𝑒,𝑣)∈𝐿1(⊀2)×𝐿1(⊀2) satisfies (1.1)-(1.2) in the distribution sense; that is,ξ€œβŠ€2π‘’ξ€·πœ‘π‘‘π‘‘βˆ’πœ‘π‘₯π‘₯βˆ’π‘1πœ‘π‘‘+π‘Ž1ξ€Έξ€œ(𝑑,π‘₯)πœ‘π‘‘π‘‘π‘‘π‘₯=⊀2ξ€œπ‘“(𝑑,π‘₯,𝑣)πœ‘π‘‘π‘‘π‘‘π‘₯,⊀2π‘£ξ€·πœ‘π‘‘π‘‘βˆ’πœ‘π‘₯π‘₯βˆ’π‘2πœ‘π‘‘+π‘Ž2ξ€Έξ€œ(𝑑,π‘₯)πœ‘π‘‘π‘‘π‘‘π‘₯=⊀2ξ€·βŠ€π‘”(𝑑,π‘₯,𝑒)πœ‘π‘‘π‘‘π‘‘π‘₯,βˆ€πœ‘βˆˆπ·2ξ€Έ.(2.3) First, we consider the linear equationπ‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯+π‘π‘–π‘’π‘‘βˆ’πœ†π‘–π‘’=β„Žπ‘–(𝑑,π‘₯),inπ·ξ…žξ€·βŠ€2ξ€Έ,(2.4) where 𝑐𝑖>0, πœ†π‘–βˆˆπ‘…, and β„Žπ‘–(𝑑,π‘₯)∈𝐿1(⊀2), (𝑖=1,2).

Let Β£πœ†π‘– be the differential operatorΒ£πœ†π‘–=π‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯+π‘π‘–π‘’π‘‘βˆ’πœ†π‘–π‘’,(2.5) acting on functions on ⊀2. Following the discussion in [14], we know that if πœ†π‘–<0, then Β£πœ†π‘– has the resolvent π‘…πœ†π‘–:π‘…πœ†π‘–βˆΆπΏ1ξ€·βŠ€2ξ€Έξ€·βŠ€βŸΆπΆ2ξ€Έ,β„Žπ‘–βŸΌπ‘’π‘–,(2.6) where 𝑒𝑖 is the unique solution of (2.4), and the restriction of π‘…πœ†π‘– on 𝐿𝑝(⊀2)(1<𝑝<∞) or 𝐢(⊀2) is compact. In particular, π‘…πœ†π‘–βˆΆπΆ(⊀2)→𝐢(⊀2) is a completely continuous operator.

For πœ†π‘–=βˆ’π‘2𝑖/4, the Green function 𝐺𝑖(𝑑,π‘₯) of the differential operator Β£πœ†π‘– is explicitly expressed; see lemma  5.2 in [14]. From the definition of 𝐺𝑖(𝑑,π‘₯), we haveπΊπ‘–βˆΆ=essinf𝐺𝑖𝑒(𝑑,π‘₯)=βˆ’3π‘π‘–πœ‹/2(1βˆ’π‘’βˆ’π‘π‘–πœ‹)2,πΊπ‘–βˆΆ=esssup𝐺𝑖(𝑑,π‘₯)=(1+π‘’βˆ’π‘π‘–πœ‹)2(1βˆ’π‘’βˆ’π‘π‘–πœ‹)2.(2.7) Let 𝐸 denote the Banach space 𝐢(⊀2) with the norm ‖𝑒‖=max(𝑑,π‘₯)∈⊀2|𝑒(𝑑,π‘₯)|, then 𝐸 is an ordered Banach space with cone𝐾0=ξ€½π‘’βˆˆπΈβˆ£π‘’(𝑑,π‘₯)β‰₯0,βˆ€(𝑑,π‘₯)∈⊀2ξ€Ύ.(2.8) For convenience, we assume that the following condition holds throughout this paper:(H1)π‘Žπ‘–(𝑑,π‘₯)∈𝐢(⊀2,𝑅+), 0<π‘Žπ‘–(𝑑,π‘₯)≀𝑐2𝑖/4 for (𝑑,π‘₯)∈⊀2, and ∫⊀2π‘Žπ‘–(𝑑,π‘₯)𝑑𝑑𝑑π‘₯>0.

Next, we consider (2.4) when βˆ’πœ†π‘– is replaced by π‘Žπ‘–(𝑑,π‘₯). In [10], Li has proved the following unique existence and positive estimate result.

Lemma 2.1. Let β„Žπ‘–(𝑑,π‘₯)∈𝐿1(⊀2);𝐸 is the Banach space 𝐢(⊀2). Then; (2.4) has a unique solution 𝑒𝑖=π‘ƒπ‘–β„Žπ‘–;π‘ƒπ‘–βˆΆπΏ1(⊀2)→𝐢(⊀2) is a linear bounded operator with the following properties;(i)π‘ƒπ‘–βˆΆπΆ(⊀2)→𝐢(⊀2) is a completely continuous operator;(ii)if β„Žπ‘–(𝑑,π‘₯)>0,thenπ‘Ž.𝑒.(𝑑,π‘₯)∈⊀2,𝑃𝑖[β„Žπ‘–(𝑑,π‘₯)] has the positive estimateπΊπ‘–β€–β€–β„Žπ‘–β€–β€–πΏ1β‰€π‘ƒπ‘–ξ€Ίβ„Žπ‘–ξ€»β‰€(𝑑,π‘₯)πΊπ‘–πΊπ‘–β€–β€–π‘Žπ‘–β€–β€–πΏ1β€–β€–β„Žπ‘–β€–β€–πΏ1.(2.9)

3. Main Result

In this section, we establish the existence of positive solutions for the telegraph systemπ‘£π‘‘π‘‘βˆ’π‘£π‘₯π‘₯+𝑐1𝑣𝑑+π‘Ž1𝑣(𝑑,π‘₯)𝑣=𝑓(𝑑,π‘₯,𝑒),π‘‘π‘‘βˆ’π‘£π‘₯π‘₯+𝑐2𝑣𝑑+π‘Ž2(𝑑,π‘₯)𝑣=𝑔(𝑑,π‘₯,𝑒).(3.1) where π‘Žπ‘–βˆˆπΆ(𝑅2,𝑅+) and 𝑓(𝑑,π‘₯,𝑣) may be singular at 𝑣=0. In particular, 𝑓(𝑑,π‘₯,𝑣) may be negative or superlinear at 𝑣=+∞. 𝑔(𝑑,π‘₯,𝑒) has the similar assumptions. Our interest is in working out what weak force conditions of 𝑓(𝑑,π‘₯,𝑣) at 𝑣=0, 𝑔(𝑑,π‘₯,𝑒) at 𝑒=0 and what superlinear growth conditions of 𝑓(𝑑,π‘₯,𝑣) at 𝑣=+∞, 𝑔(𝑑,π‘₯,𝑒) at 𝑒=+∞ are needed to obtain the existence of positive solutions for problem (1.1)-(1.2).

We assume the following conditions throughout.(H2)𝑓,π‘”βˆΆβŠ€2Γ—(0,∞)→𝑅 is continuous, and there exists a constant 𝑀>0 such that 𝑓1(𝑑,π‘₯,𝑒)+𝑀β‰₯0,𝑓2(𝑑,π‘₯,𝑒)+𝑀β‰₯0,βˆ€(𝑑,π‘₯)∈⊀2and𝑒,π‘£βˆˆ(0,∞).(3.2)(H3)𝐹(𝑑,π‘₯,𝑣)=𝑓(𝑑,π‘₯,𝑣)+𝑀≀𝑗1(𝑣)+β„Ž1(𝑣) for (𝑑,π‘₯,𝑣)∈⊀2Γ—(0,∞) with 𝑗1>0 continuous and nonincreasing on (0,∞), β„Ž1β‰₯0 continuous on (0,∞) and β„Ž1/𝑗1 nondecreasing on (0,∞).𝐺(𝑑,π‘₯,𝑒)=𝑔(𝑑,π‘₯,𝑒)+𝑀≀𝑗2(𝑒)+β„Ž2(𝑒) for (𝑑,π‘₯,𝑒)∈⊀2Γ—(0,∞) with 𝑗2>0 continuous and nonincreasing on (0,∞), β„Ž2β‰₯0 continuous on (0,∞) and β„Ž2/𝑗2 nondecreasing on (0,∞).(H4)𝐹(𝑑,π‘₯,𝑣)=𝑓(𝑑,π‘₯,𝑣)+𝑀β‰₯𝑗3(𝑣)+β„Ž3(𝑣) for all (𝑑,π‘₯,𝑣)∈⊀2Γ—(0,∞) with 𝑗3>0 continuous and nonincreasing on (0,∞), β„Ž3β‰₯0 continuous on (0,∞) with β„Ž3/𝑗3 nondecreasing on (0,∞);𝐺(𝑑,π‘₯,𝑒)=𝑔(𝑑,π‘₯,𝑒)+𝑀β‰₯𝑗4(𝑒)+β„Ž4(𝑒) for all (𝑑,π‘₯,𝑒)∈⊀2Γ—(0,∞) with 𝑗4>0 continuous and nonincreasing on (0,∞), β„Ž4β‰₯0 continuous on (0,∞) with β„Ž4/𝑗4 nondecreasing on (0,∞).(H5) There exists π‘€β€–β€–πœ”π‘Ÿ>1‖‖𝛿1,(3.3) such that π‘Ÿβ‰₯4πœ‹2𝐺1𝐺1β€–β€–π‘Ž1‖‖𝐿1𝐼1⋅𝐼2,(3.4) here 𝐼1=𝑗1𝐺2𝑗4ξƒ―β„Ž(π‘Ÿ)1+4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1‖‖𝑗4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξƒ°4πœ‹2β€–β€–πœ”βˆ’π‘€2β€–β€–ξƒͺ,𝐼2β„Ž=1+1ξ‚€ξ‚€4πœ‹2𝐺2/𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑗2𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξ€½1+β„Ž2(π‘Ÿ)/𝑗2(π‘Ÿ)𝑗1ξ‚€ξ‚€4πœ‹2𝐺2/𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑗2𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξ€½1+β„Ž2(π‘Ÿ)/𝑗2,(π‘Ÿ)(3.5) where 𝛿𝑖=(𝐺𝑖2β€–π‘Žπ‘–β€–πΏ1/𝐺𝑖)∈(0,1), and πœ”π‘–(𝑑,π‘₯) is the unique solution to problem: π‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯+𝑐𝑖𝑒𝑑+π‘Žπ‘–(𝑑,π‘₯)𝑒=1,𝑒(𝑑+2πœ‹,π‘₯)=𝑒(𝑑,π‘₯+2πœ‹)=𝑒(𝑑,π‘₯),(𝑑,π‘₯)βˆˆπ‘…2.(3.6)(H6) There exists 𝑅>π‘Ÿ, such that 4πœ‹2𝐺1𝐼3⋅𝐼4𝛿β‰₯𝑅,2𝑗4ξƒ―β„Ž(𝑅)1+4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1‖‖𝑗4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξƒ°>𝑀,(3.7) where 𝐼3=𝐺1𝑗34πœ‹2𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑗2𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έξ‚»β„Ž1+2(𝑅)𝑗2ξ‚Όξƒͺ,𝐼(𝑅)4β„Ž=1+3𝐺2𝑗4ξ€½(𝑅)1+β„Ž4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έ/𝑗4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έξ€Ύ4πœ‹2β€–β€–πœ”βˆ’π‘€2‖‖𝑗3𝐺2𝑗4ξ€½(𝑅)1+β„Ž4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έ/𝑗4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έξ€Ύ4πœ‹2β€–β€–πœ”βˆ’π‘€2‖‖.(3.8)

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

Proof. To show that (1.1)-(1.2) has a positive solution, we will proof that π‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯+𝑐1𝑒𝑑+π‘Ž1ξ€·(𝑑,π‘₯)𝑒=𝐹𝑑,π‘₯,π‘£βˆ’π‘€πœ”2ξ€Έ,π‘£π‘‘π‘‘βˆ’π‘£π‘₯π‘₯+𝑐2𝑣𝑑+π‘Ž2(𝑑,π‘₯)𝑣=𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έ(3.9) has a solution Μƒ(̃𝑒,𝑣)=(𝑒+π‘€πœ”1,𝑣+π‘€πœ”2) with ̃𝑒>π‘€πœ”1, ̃𝑣>π‘€πœ”2 for (𝑑,π‘₯)∈⊀2. In addition, by Lemma 2.1, it is clear to see that (𝑒,𝑣)∈𝐢2(⊀2)×𝐢2(⊀2) is a solution of (3.9) if and only if (𝑒,𝑣)∈𝐢(⊀2)×𝐢(⊀2) is a solution of the following system: 𝑒=𝑃1𝐹𝑑,π‘₯,π‘£βˆ’π‘€πœ”2,𝑣=𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1.ξ€Έξ€Έ(3.10) Evidently, (3.10) can be rewritten as the following equation: 𝑒=𝑃1𝐹𝑑,π‘₯,𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2ξ€Έξ€Έ.(3.11)
Define a cone πΎβŠ‚πΈ as 𝐾=π‘’βˆˆπΈβˆΆπ‘’β‰₯0,𝑒β‰₯𝛿1‖𝑒‖.(3.12) We define an operator π‘‡βˆΆπΈβ†’πΎ by (𝑇𝑒)(𝑑,π‘₯)=𝑃1𝐹𝑑,π‘₯,𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2ξ€Έξ€Έ(3.13) for π‘’βˆˆπΈ and (𝑑,π‘₯)∈⊀2. We have the conclusion that π‘‡βˆΆπΈβ†’πΈ is completely continuous and 𝑇(𝐾)βŠ†πΎ. The complete continuity is obvious by Lemma 2.1. Now, we show that 𝑇(𝐾)βŠ†πΎ.
For any π‘’βˆˆπΎ, we have 𝑇𝑒=𝑃1𝐹𝑑,π‘₯,𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2ξ€Έξ€Έ.(3.14) From (H1)–(H3) and Lemma 2.1, we have 𝑇𝑒=𝑃1𝐹𝑑,π‘₯,𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2ξ€Έξ€Έβ‰₯𝐺1‖‖𝐹𝑑,π‘₯,𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2‖‖𝐿1,‖‖𝑃𝐹‖𝑇𝑒‖=𝑑,π‘₯,𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2‖‖≀𝐺1𝐺1β€–β€–π‘Ž1‖‖𝐿1‖‖𝐹𝑑,π‘₯,𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2‖‖𝐿1.(3.15) So, we get 𝐺𝑇𝑒β‰₯12β€–β€–π‘Ž1‖‖𝐿1𝐺1‖𝑇𝑒‖β‰₯𝛿1‖𝑇𝑒‖,(3.16) namely, 𝑇(𝐾)βŠ†πΎ.
Let Ξ©π‘Ÿ={π‘’βˆˆπΈβˆΆβ€–π‘’β€–<π‘Ÿ},Ω𝑅={π‘’βˆˆπΈβˆΆβ€–π‘’β€–<𝑅}.(3.17) Since π‘Ÿβ‰€β€–π‘’β€–β‰€π‘… for any π‘’βˆˆπΎβˆ©(Ξ©π‘…β§΅Ξ©π‘Ÿ), we have 0<𝛿1π‘Ÿβˆ’π‘€β€–πœ”β€–β‰€π‘’βˆ’π‘€πœ”1≀𝑅.
First, we show ‖𝑇𝑒‖≀‖𝑒‖,forπ‘’βˆˆπΎβˆ©πœ•Ξ©π‘Ÿ.(3.18) In fact, if π‘’βˆˆπΎβˆ©πœ•Ξ©π‘Ÿ, then ‖𝑒‖=π‘Ÿ and 𝑒β‰₯𝛿1π‘Ÿ>π‘€β€–πœ”1β€– for(𝑑,π‘₯)∈⊀2. By (H3) and (H4), we have 𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1≀𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1‖‖𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1‖‖𝐿1≀𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1‖‖‖‖𝑗2ξ€·π‘’βˆ’π‘€πœ”1ξ€Έξƒ©β„Ž1+2ξ€·π‘’βˆ’π‘€πœ”1𝑗2ξ€·π‘’βˆ’π‘€πœ”1ξ€Έξƒͺ‖‖‖‖𝐿1≀𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑗2𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξ‚»β„Ž1+2(π‘Ÿ)𝑗2ξ‚Ό(π‘Ÿ)4πœ‹2,𝑃(3.19)2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβ‰₯𝐺2‖‖𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1‖‖𝐿1β‰₯𝐺2‖‖‖‖𝑗4ξ€·π‘’βˆ’π‘€πœ”1ξ€Έξƒ©β„Ž1+4ξ€·π‘’βˆ’π‘€πœ”1𝑗4ξ€·π‘’βˆ’π‘€πœ”1ξ€Έξƒͺ‖‖‖‖𝐿1β‰₯𝐺2𝑗4(ξƒ―β„Žπ‘Ÿ)1+4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1‖‖𝑗4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξƒ°4πœ‹2.(3.20) In addition, we also have 𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβ‰₯𝐺2𝑗4ξƒ―β„Ž(π‘Ÿ)1+4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1‖‖𝑗4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξƒ°4πœ‹2β‰₯𝐺2𝑗4ξƒ―β„Ž(𝑅)1+4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1‖‖𝑗4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξƒ°4πœ‹2>𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑀4πœ‹2β‰₯π‘€πœ”2,(3.21) by (H5), (H6), and (3.20).
So, we have 𝑇𝑒=𝑃1𝐹𝑑,π‘₯,π‘£βˆ’π‘€πœ”2≀𝐺1𝐺1β€–β€–π‘Ž1‖‖𝐿1‖‖𝐹𝑑,π‘₯,π‘£βˆ’π‘€πœ”2‖‖𝐿1≀𝐺1𝐺1β€–β€–π‘Ž1‖‖𝐿1‖‖‖‖𝑗1ξ€·π‘£βˆ’π‘€πœ”2ξ€Έξƒ―β„Ž1+1ξ€·π‘£βˆ’π‘€πœ”2𝑗1ξ€·π‘£βˆ’π‘€πœ”2‖‖‖‖𝐿1≀𝐺1𝐺1β€–β€–π‘Ž1‖‖𝐿1‖‖𝑗1𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2ξ€ΈΓ—ξƒ―β„Ž1+1𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2𝑗1𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2‖‖‖‖𝐿1≀𝐺1𝐺1β€–β€–π‘Ž1‖‖𝐿1𝑗1𝐺2𝑗4ξƒ―β„Ž(π‘Ÿ)1+4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1‖‖𝑗4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξƒ°4πœ‹2β€–β€–πœ”βˆ’π‘€2β€–β€–ξƒͺΓ—βŽ§βŽͺ⎨βŽͺβŽ©β„Ž1+1𝐺2/𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑗2𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξ€½1+β„Ž2(π‘Ÿ)/𝑗2ξ€Ύ(π‘Ÿ)4πœ‹2𝑗1𝐺2/𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑗2𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξ€½1+β„Ž2(π‘Ÿ)/𝑗2ξ€Ύ(π‘Ÿ)4πœ‹2ξ‚βŽ«βŽͺ⎬βŽͺ⎭4πœ‹2β‰€π‘Ÿ=‖𝑒‖(3.22) for (𝑑,π‘₯)∈⊀2, since 𝛿1π‘Ÿβˆ’π‘€β€–πœ”1β€–β‰€π‘’βˆ’π‘€πœ”1β‰€π‘Ÿ.
This implies that ‖𝑇𝑒‖≀‖𝑒‖; that is, (3.18) holds.
Next, we show ‖𝑇𝑒‖β‰₯‖𝑒‖,forπ‘’βˆˆπΎβˆ©πœ•Ξ©π‘….(3.23) If π‘’βˆˆπΎβˆ©πœ•Ξ©π‘…, then ‖𝑒‖=𝑅 and 𝑒β‰₯𝛿𝑅>π‘€β€–πœ”1β€– for (𝑑,π‘₯)∈⊀2. From (H4) and (H6), we have𝑇𝑒=𝑃1𝐹𝑑,π‘₯,π‘£βˆ’π‘€πœ”1ξ€Έξ€Έβ‰₯𝐺1‖‖‖‖𝑗3ξ€·π‘£βˆ’π‘€πœ”2ξ€Έξƒ―β„Ž1+3ξ€·π‘£βˆ’π‘€πœ”2𝑗3ξ€·π‘£βˆ’π‘€πœ”2‖‖‖‖𝐿1β‰₯𝐺1‖‖‖‖𝑗3𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2ξ€ΈΓ—ξƒ―β„Ž1+3𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2𝑗3𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβˆ’π‘€πœ”2‖‖‖‖𝐿1β‰₯𝐺1‖‖‖‖𝑗3𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑗2𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έξ‚»β„Ž1+2(𝑅)𝑗2ξ‚Ό(𝑅)4πœ‹2ξƒͺΓ—βŽ§βŽͺ⎨βŽͺβŽ©β„Ž1+3𝐺2𝑗4ξ€½(𝑅)1+β„Ž4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έ/𝑗4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έξ€Ύ4πœ‹2β€–β€–πœ”βˆ’π‘€2‖‖𝑗3𝐺2𝑗4ξ€½(𝑅)1+β„Ž4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έ/𝑗4𝛿1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έξ€Ύ4πœ‹2β€–β€–πœ”βˆ’π‘€2β€–β€–ξ‚βŽ«βŽͺ⎬βŽͺβŽ­β€–β€–β€–β€–β€–πΏ1β‰₯𝑅=‖𝑒‖(3.24) for (𝑑,π‘₯)∈⊀2, since 𝛿1π‘…βˆ’π‘€β€–πœ”1β€–β‰€π‘’βˆ’π‘€πœ”1≀𝑅.
This implies that 𝑇𝑒β‰₯‖𝑒‖; that is, (3.23) holds.
Finally, (3.18), (3.23), and Lemma 1.1 guarantee that 𝑇 has a fixed point π‘’βˆˆπΎβˆ©Ξ©π‘…β§΅Ξ©π‘Ÿ with π‘Ÿβ‰€β€–π‘’β€–β‰€π‘…. Clearly, 𝑒>π‘€πœ”1.
Since 𝑃2𝐺𝑑,π‘₯,π‘’βˆ’π‘€πœ”1ξ€Έξ€Έβ‰₯𝐺2‖‖𝐺𝑑,π‘₯,π‘€πœ”1‖‖𝐿1β‰₯𝐺2‖‖‖‖𝑗4ξ€·π‘’βˆ’π‘€πœ”1ξ€Έξƒ©β„Ž1+4ξ€·π‘’βˆ’π‘€πœ”1𝑗4ξ€·π‘’βˆ’π‘€πœ”1ξ€Έξƒͺ‖‖‖‖𝐿1β‰₯𝐺2𝑗4ξƒ―β„Ž(𝑅)1+4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1‖‖𝑗4𝛿1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€Έξƒ°4πœ‹2>𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1𝑀4πœ‹2β‰₯π‘€πœ”2,(3.25) then we have a doubly periodic solution (𝑒,𝑣) of (3.9) with 𝑒>π‘€πœ”1, 𝑣>π‘€πœ”2, namely, (π‘’βˆ’π‘€πœ”1,π‘£βˆ’π‘€πœ”2)>(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 existsπ‘€β€–β€–πœ”π‘Ÿ>2‖‖𝛿2,(3.26) such that π‘Ÿβ‰₯4πœ‹2𝐺2𝐺2β€–β€–π‘Ž2‖‖𝐿1𝐼5⋅𝐼6,(3.27) here 𝐼5=𝑗24πœ‹2𝐺1𝑗3ξƒ―β„Ž(π‘Ÿ)1+3𝛿2β€–β€–πœ”π‘Ÿβˆ’π‘€2‖‖𝑗3𝛿2β€–β€–πœ”π‘Ÿβˆ’π‘€2β€–β€–ξ€Έξƒ°β€–β€–πœ”βˆ’π‘€1β€–β€–ξƒͺ,𝐼6β„Ž=1+2ξ‚€ξ‚€4πœ‹2𝐺1/𝐺1β€–β€–π‘Ž1‖‖𝐿1𝑗1𝛿2β€–β€–πœ”π‘Ÿβˆ’π‘€2β€–β€–ξ€Έξ€½1+β„Ž1(π‘Ÿ)/𝑗1(π‘Ÿ)𝑗2ξ‚€ξ‚€4πœ‹2𝐺1/𝐺1β€–β€–π‘Ž1‖‖𝐿1𝑗1𝛿2β€–β€–πœ”π‘Ÿβˆ’π‘€2β€–β€–ξ€Έξ€½1+β„Ž1(π‘Ÿ)/𝑗1.(π‘Ÿ)(3.28)(H8) There exists 𝑅>π‘Ÿ, such that4πœ‹2𝐺2𝐼7⋅𝐼8𝛿β‰₯𝑅,1𝑗3ξƒ―β„Ž(𝑅)1+3𝛿2β€–β€–πœ”π‘Ÿβˆ’π‘€2‖‖𝑗3𝛿2β€–β€–πœ”π‘Ÿβˆ’π‘€2β€–β€–ξ€Έξƒ°>𝑀,(3.29) where 𝐼7=𝑗44πœ‹2𝐺1𝐺1β€–β€–π‘Ž1‖‖𝐿1𝑗1𝛿2β€–β€–πœ”π‘…βˆ’π‘€2β€–β€–ξ€Έξ‚»β„Ž1+1(𝑅)𝑗1ξ‚Όξƒͺ,𝐼(𝑅)8β„Ž=1+4ξ‚€4πœ‹2𝐺1𝑗3ξ€½(𝑅)1+β„Ž3𝛿2β€–β€–πœ”π‘…βˆ’π‘€2β€–β€–ξ€Έ/𝑗3𝛿2β€–β€–πœ”π‘…βˆ’π‘€2β€–β€–β€–β€–πœ”ξ€Έξ€Ύβˆ’π‘€1‖‖𝑗4ξ‚€4πœ‹2𝐺1𝑗3ξ€½(𝑅)1+β„Ž3𝛿2β€–β€–πœ”π‘…βˆ’π‘€2β€–β€–ξ€Έ/𝑗3𝛿2β€–β€–πœ”π‘…βˆ’π‘€2β€–β€–β€–β€–πœ”ξ€Έξ€Ύβˆ’π‘€1‖‖.(3.30) Then, problem (1.1)-(1.2) has a positive periodic solution.

4. An Example

Consider the following system:π‘’π‘‘π‘‘βˆ’π‘’π‘₯π‘₯+2𝑒𝑑+sin2𝑣(𝑑+π‘₯)𝑒=πœ‡βˆ’π›Ό+𝑣𝛽+π‘˜1ξ€Έ,𝑣(𝑑,π‘₯)π‘‘π‘‘βˆ’π‘£π‘₯π‘₯+2𝑣𝑑+cos2𝑒(𝑑+π‘₯)𝑣=πœ†βˆ’πœ+π‘’πœŽ+π‘˜2ξ€Έ,(𝑑,π‘₯)𝑒(𝑑+2πœ‹,π‘₯)=𝑒(𝑑,π‘₯+2πœ‹)=𝑒(𝑑,π‘₯),(𝑑,π‘₯)βˆˆπ‘…2,𝑣(𝑑+2πœ‹,π‘₯)=𝑣(𝑑,π‘₯+2πœ‹)=𝑣(𝑑,π‘₯),(𝑑,π‘₯)βˆˆπ‘…2,(4.1) where 𝑐1=𝑐2=2, πœ‡,πœ†>0, 𝛼,𝜏>0,𝛽,𝜎>1, π‘Ž1(𝑑,π‘₯)=sin2(𝑑+π‘₯), π‘Ž2(𝑑,π‘₯)=cos2(𝑑+π‘₯)∈𝐢(⊀2,𝑅+), π‘˜π‘–βˆΆβŠ€2→𝑅 is continuous. When πœ‡ is chosen such thatπœ‡<supβ€–β€–πœ”π‘’βˆˆ((𝑀1β€–β€–)/𝛿1,∞)πΊβ€–β€–π‘Ž1‖‖𝐿1𝐺4πœ‹2𝐼1𝐼2,(4.2) here we denote 𝐼1𝐺=π‘’πœ†π‘’βˆ’πœξ‚†ξ€·π›Ώ1+1β€–β€–πœ”π‘’βˆ’π‘€1β€–β€–ξ€ΈπœŽ+πœξ‚‡4πœ‹2β€–β€–πœ”βˆ’π‘€2‖‖𝛼,𝐼2ξ‚΅=1+πΊπΊβ€–β€–π‘Ž2‖‖𝐿1πœ†ξ€·π›Ώ1β€–β€–πœ”π‘’βˆ’π‘€1β€–β€–ξ€Έβˆ’πœξ€·1+π‘’πœŽ+𝜏+2π»π‘’πœξ€Έ4πœ‹2𝛽+𝛼+2π»πΊπΊβ€–β€–π‘Ž2‖‖𝐿1πœ†ξ€·π›Ώ1β€–β€–πœ”π‘’βˆ’π‘€1β€–β€–ξ€Έβˆ’πœξ€·1+π‘’πœŽ+𝜏+2π»π‘’πœξ€Έ4πœ‹2ξ‚Ά,(4.3) where 𝐻=max{β€–π‘˜1β€–,β€–π‘˜2β€–} 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{πœ‡π»,πœ†π»} and𝑗1(𝑣)=𝑗3(𝑣)=πœ‡π‘£βˆ’π›Ό,β„Ž1𝑣(𝑣)=πœ‡π›½ξ€Έ+2𝐻,β„Ž3(𝑣)=πœ‡π‘£π›½,𝑗2(𝑒)=𝑗4(𝑒)=πœ†π‘’βˆ’πœ,β„Ž2(𝑒)=πœ‡(π‘’πœŽ+2𝐻),β„Ž4(𝑒)=πœ‡π‘’πœŽ.(4.4) Clearly, (H1)–(H4) are satisfied.

Set𝐺𝑇(𝑒)=β€–β€–π‘Ž1‖‖𝐿1𝐺4πœ‹2𝐼1𝐼2ξƒ©ξ€·π‘€β€–β€–πœ”,π‘’βˆˆ1‖‖𝛿1ξƒͺ,+∞.(4.5) Obviously, 𝑇((π‘€β€–πœ”1β€–)/𝛿1)=0, 𝑇(∞)=0, then there exists π‘Ÿβˆˆ((π‘€β€–πœ”1β€–)/𝛿1,+∞) such that𝑇(π‘Ÿ)=supπ‘€β€–β€–πœ”π‘’βˆˆξ€·ξ€·1β€–β€–ξ€Έ/𝛿1ξ€Έ,βˆžπΊβ€–β€–π‘Ž1‖‖𝐿1𝐺4πœ‹2𝐼1𝐼2.(4.6) This implies that there existsξƒ©ξ€·π‘€β€–β€–πœ”π‘Ÿβˆˆ1‖‖𝛿1ξƒͺ,+∞,(4.7) such thatπœ‡<supπ‘€β€–β€–πœ”π‘’βˆˆξ€·ξ€·1β€–β€–ξ€Έ/𝛿1ξ€Έ,βˆžπΊβ€–β€–π‘Ž1‖‖𝐿1𝐺4πœ‹2𝐼1𝐼2.(4.8) So, (H5) is satisfied.

Finally, since𝑅𝐺/πΊβ€–β€–π‘Ž2‖‖𝐿1ξ‚πœ†ξ€·π›Ώ1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€Έβˆ’πœξ€·1+π‘…πœŽ+𝜏+2π»π‘…πœξ€Έ4πœ‹2ξ‚π›Όπœ‡πΊξ‚Έξ‚€πΊ1+πœ†π‘…βˆ’πœξ‚†ξ€·π›Ώ1+1β€–β€–πœ”π‘…βˆ’π‘€1β€–β€–ξ€ΈπœŽ+πœξ‚‡4πœ‹2β€–β€–πœ”βˆ’π‘€2‖‖𝛼+π›½ξ‚ΉβŸΆ0asπ‘…βŸΆβˆž,(4.9) this implies that there exists 𝑅. In addition, for fixed π‘Ÿ,𝑅, choosing πœ† sufficiently large, we have𝛿2πœ†π‘…βˆ’πœξ‚†ξ€·π›Ώ1+1β€–β€–πœ”π‘Ÿβˆ’π‘€1β€–β€–ξ€ΈπœŽ+πœξ‚‡>𝑀.(4.10) Thus, (H6) is satisfied. So, all the conditions of Theorem 3.1 are satisfied.

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