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International Journal of Differential Equations
Volume 2012 (2012), Article ID 406835, 13 pages
http://dx.doi.org/10.1155/2012/406835
Research Article

Existence of Periodic Solutions to Nonlinear Differential Equations of Third Order with Multiple Deviating Arguments

Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080 Van, Turkey

Received 12 May 2012; Accepted 19 July 2012

Academic Editor: D. D. Ganji

Copyright © 2012 Cemil Tunç. 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 establish certain new sufficient conditions which guarantee the existence of periodic solutions for a nonlinear differential equation of the third order with multiple deviating arguments. Using the Lyapunov functional approach, we prove a specific theorem and provide an example to illustrate the theoretical analysis in this work and the effectiveness of the method utilized here.

1. Introduction

It is known that functional differential equations, in particular, that delay differential equations can be used as models to describe many physical, biological systems, and so forth. In reality, many actual systems have the property aftereffect, that is, the future states depend not only on the present, but also on the past history, and after effect is also known to occur in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, and so forth (Burton [1], Kolmanovskii and Myshkis [2]). Therefore, it is important to investigate the qualitative behaviors of functional differential equations.

In 1978, using the known theorem of Yoshizawa [3, Theorem 37.2], Chukwu [4] found certain sufficient conditions that guarantee the existence of a periodic solution to nonlinear-linear differential of the third order with the constant deviating argument (>0): 𝑥+𝑓𝑥,𝑥,𝑥𝑥𝑥+𝑔(𝑡),𝑥(𝑡)+𝑖(𝑥(𝑡))=𝑝𝑡,𝑥,𝑥,𝑥(𝑡),𝑥(𝑡),𝑥.(1.1)

Later, in 1992, Zhu [5] considered the nonlinear differential equation of the third order with the constant deviating argument 𝑟 (>0): 𝑥+𝑎𝑥𝑥+𝜙(𝑡𝑟)+𝑓(𝑥)=𝑝(𝑡),(1.2) and he discussed the existence of periodic solutions for this equation when 𝑝(𝑡) is a periodic function of period 𝑇, 𝑇>0.

In 2000, Tejumola and Tchegnani [6] considered the nonlinear differential equation of the third order with the constant deviating argument 𝜏 (>0): 𝑥+𝑓𝑡,𝑥,𝑥,𝑥+𝑔𝑡,𝑥(𝑡𝜏),𝑥(𝑡𝜏)+(𝑥(𝑡𝜏))=𝑃1𝑡,𝑥,𝑥,𝑥,𝑥(𝑡𝜏),𝑥(.𝑡𝜏)(1.3) The authors established certain sufficient conditions on the existence of periodic of solutions of this equation.

In 2010, Tunç [7] established certain sufficient conditions for the existence of a periodic solution for the nonlinear differential equation of the third order with the constant deviating argument 𝑟 (>0): 𝑥𝑥+𝜓𝑥𝑥+𝑔(𝑡𝑟)+𝑓(𝑥)=𝑝𝑡,𝑥,𝑥(𝑡𝑟),𝑥,𝑥(𝑡𝑟),𝑥.(1.4)

However, a review to date of the literature indicates that the existence of periodic solutions to the nonlinear differential equation of the third order with multiple deviating arguments has not been investigated. The paper considers the nonlinear differential equation of the third order with multiple constant deviating arguments 𝜏𝑖,(𝑖=1,2,,𝑛): 𝑥𝑥+𝜓𝑥+𝑛𝑖=1𝑔𝑖𝑥𝑡𝜏𝑖+𝑓(𝑥)=𝑝𝑡,𝑥,𝑥𝑡𝜏1,,𝑥,,𝑥𝑡𝜏𝑛,𝑥.(1.5)

The equation (1.5) is stated in system form as follows: 𝑥=𝑦,𝑦𝑧=𝑧,=𝜓(𝑦)𝑧𝑛𝑖=1𝑔𝑖(𝑦)𝑓(𝑥)+𝑛𝑖=1𝑡𝑡𝜏𝑖𝑔𝑖(𝑦(𝑠))𝑧(𝑠)𝑑𝑠+𝑝𝑡,𝑥,𝑥𝑡𝜏1,,𝑦𝑡𝜏𝑛,,𝑧(1.6) where 𝜏𝑖 are positive constants, that is, 𝜏𝑖 are constant deviating arguments, which are determined in Section 2. It is assumed that the functions 𝜓,   𝑔𝑖,  𝑓, and 𝑝() are continuous in their respective arguments on , , , and +×2𝑛+3,  (+=[0,), =(,)), respectively; 𝑔𝑖(0)=𝑓(0)=0 and 𝑝() is periodic in 𝑡 of period 𝑇,𝑇𝜏𝑖, the derivatives 𝑔𝑖(𝑦)(𝑑/𝑑𝑦)𝑔𝑖(𝑦) exist and are also continuous; throughout what follows 𝑥(𝑡), 𝑦(𝑡), and 𝑧(𝑡) are abbreviated as 𝑥,𝑦, and 𝑧, respectively.

The motivation for this paper is a result of the research mentioned regarding ordinary differential equations with a deviating argument. Our aim is to achieve the results established in [5, 7] to (1.5) with multiple deviating arguments. Our results generalize the results established on the existence of periodic solution in [5, 7]. This paper is the first known publication regarding the existence of periodic solution for differential equations of the third order with multiple deviating arguments.

In order to reach our main result, this paper offers fundamental information regarding the general nonautonomous delay periodic differential system. Consider the delay periodic system: ̇𝑥(𝑡)=𝐹𝑡,𝑥𝑡,𝑥𝑡=𝑥(𝑡+𝜃),𝑟𝜃0,𝑡0,(1.7) where 𝐹[0,)×𝐶𝐻𝑛 is a continuous mapping, 𝐹(𝑡+𝑇,𝜑)=𝐹(𝑡,𝜑) for all 𝜑𝐶 and for some constant 𝑇>0. We assume that 𝐹 takes closed bounded sets into bounded sets of 𝑛. Here (𝐶,) is the Banach space of continuous function 𝜙[𝑟,0]𝑛 with supremum norm, 𝑟>0; for 𝐻>0, we define 𝐶𝐻𝐶 by 𝐶𝐻={𝜙𝐶𝜙<𝐻},𝐶𝐻 is the open 𝐻-ball in 𝐶,𝐶=𝐶([𝑟,0],𝑛).

Theorem 1.1. Suppose that 𝐹(𝑡,𝜑)𝐶0(𝜑) and 𝐹(𝑡,𝜑) is periodic in 𝑡 of period 𝑇,𝑇𝑟, and consequently for any 𝛼>0 there exists an 𝐿(𝛼)>0 such that 𝜑𝐶𝛼 implies |𝐹(𝑡,𝜑)|𝐿(𝛼). Suppose that a continuous Lyapunov functional 𝑉(𝑡,𝜑) exists, defined on 𝑡+,𝜑𝑆, 𝑆 is the set of 𝜑𝐶 such that with |𝜑(0)|𝐻 (𝐻 may be large), and that 𝑉(𝑡,𝜑) satisfies the following conditions.(i)Continuous increasing functions 𝑎(𝑠) and 𝑏(𝑠) exist, satisfying 𝑎(𝑠)>0,𝑏(𝑠)>0 for 𝑠𝐻 and 𝑎(𝑠) as 𝑠, such that 𝑎||𝜙||||𝜑||(0)𝑉(𝑡,𝜙)𝑏(𝜙),when(0)𝐻.(1.8)(ii)A continuous and positive function 𝑤(𝑠) exists such that ̇𝑉||𝜑||(𝑡,𝜙)𝑤(0)for𝑠𝐻.(1.9)(iii)A constant 𝐻1>0,𝐻1>𝐻, exists such that 𝛾𝑟𝐿<𝐻1𝐻,(1.10)where 𝛾>0 is a constant which is determined in the following way.
Using the condition on 𝑉(𝑡,𝜑), constants 𝛼>0,𝛽>0, and 𝛾>0 exist such that 𝑏(𝐻1)𝑎(𝛼),𝑏(𝛼)𝑎(𝛽), and 𝑏(𝛽)𝑎(𝛾),   𝛾 is defined by 𝑏(𝛾)𝑎(𝛾).
Under these conditions, a periodic solution of (1.7) of period 𝑇 exists. In particular, the relation 𝑟𝐿(𝛾)<𝐻1𝐻 is always satisfied if 𝑟 is sufficiently small (see Yoshizawa [3]).

2. Main Result

The main result is the following theorem:

let 𝜏=max1𝑖𝑛𝜏𝑖.

Theorem 2.1. Suppose that positive constants 𝑎,𝑏𝑖,  𝑐,  𝑚,  𝛿,  𝐿𝑖, and  𝜏exist such that the following conditions hold: 𝑓𝑎𝑏𝑐>0,𝑓(0)=0,𝑓(𝑥)sgn𝑥>0𝑥0,sup𝑔(𝑥)=𝑐,𝑓(𝑥)sgn𝑥as|𝑥|,𝑖𝑔(0)=0,𝑖(𝑦)𝑦𝑏𝑖||𝑔,(𝑦0),𝑖||(𝑦)𝐿𝑖||||,0𝜓(𝑦)𝑎𝛿y,𝑝()𝑚.(2.1) If 𝜏<min𝑎𝑏𝑐(2+𝜇)𝑏𝑁1,𝑎𝑏𝑐4𝑀1,(2.2) then (1.5) has a periodic solution of period 𝑇, where 𝜇=(𝑎𝑏+𝑐)/2𝑏, 𝑏=𝑛𝑖=1𝑏𝑖, 𝑀1=𝑛𝑖=1(𝜇𝐿𝑖/2) and 𝑁1=𝑛𝑖=1𝐿𝑖.

Proof. Define a Lyapunov functional 𝑉=𝑉(𝑥𝑡,𝑦𝑡,𝑧𝑡)by: 𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡=𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡+𝑉2(𝑥,𝑦,𝑧)+1+𝑛𝑖=1𝐿𝑖0𝜏𝑖𝑡𝑡+𝑠||||𝑧(𝜃)𝑑𝜃𝑑𝑠,(2.3) where 𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡=𝜇𝑥0𝑓(𝜉)𝑑𝜉+𝑓(𝑥)𝑦+𝜇𝑦0𝜓(𝜂)𝜂𝑑𝜂+𝑛𝑖=1𝑦0𝑔𝑖1(𝜂)𝑑𝜂+𝜇𝑦𝑧+2𝑧2+𝑛𝑖=1𝛾𝑖0𝜏𝑖𝑡𝑡+𝑠𝑧2𝑉(𝜃)𝑑𝜃𝑑𝑠,2𝑧(𝑥,𝑦,𝑧)=𝑀sgn𝑥,|𝑥|1,|𝑧|𝑀sgn𝑧sgn𝑥,|𝑥|1,|𝑧|𝑀𝑥𝑧𝑀,|𝑥|1,|𝑧|𝑀𝑥sgn𝑧,|𝑥|1,|𝑧|𝑀,(2.4)𝑀,(𝑀>1), and 𝛾𝑖 are certain positive constants; the constants 𝛾𝑖 will be determined later in the proof.
It follows that 𝑉1(0,0,0)=0. In view of the assumptions 𝜓(𝑦)𝑎,𝑔𝑖(𝑦)/𝑦𝑏𝑖,(𝑦0),𝑓(0)=0, 𝑓(𝑥)sgn𝑥>0,(𝑥0), and sup{𝑓(𝑥)}=𝑐, obtain 𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡𝜇𝑥01𝑓(𝜉)𝑑𝜉+𝑓(𝑥)𝑦+2𝜇𝑎𝑦2+𝑛𝑖=1𝑦0𝑔𝑖(𝜂)𝜂1𝜂𝑑𝜂+𝜇𝑦𝑧+2𝑧2+𝑛𝑖=1𝛾𝑖0𝜏𝑖𝑡𝑡+𝑠𝑧21(𝜃)𝑑𝜃𝑑𝑠[]2𝑏𝑏𝑦+𝑓(𝑥)2+12𝑏𝑦24𝑥0𝑓(𝜉)𝑦0𝜇𝑏𝑓+1(𝜉)𝜂𝑑𝜂𝑑𝜉2(𝜇𝑦+𝑧)2+12𝜇(𝑎𝜇)𝑦2+𝑛𝑖=1𝛾𝑖0𝜏𝑖𝑡𝑡+𝑠𝑧2(𝜃)𝑑𝜃𝑑𝑠.(2.5) Using the assumptions of Theorem 2.1, have 𝑎𝜇=𝑎𝑎𝑏+𝑐2𝑏>0,𝜇𝑏𝑓(𝑥)𝑎𝑏𝑐2>0(2.6) so that 𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡𝐷1𝑥2+𝐷2𝑦2+𝐷3𝑧2𝐷4𝑥2+𝑦2+𝑧2,(2.7) where 𝐷4=min{𝐷1,𝐷2,𝐷3}.
It is also clear that the function 𝑉2 is continuous and satisfies ||𝑉2||1.(2.8)
In view of (2.3), (2.7), (2.8), and the assumptions of Theorem 2.1, it can be shown that 𝑉 satisfies the condition (i) of Theorem 1.1.
Using a basic calculation, the time derivative of 𝑉1 along solutions of (1.6) results in ̇𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡=𝑓(𝑥)𝑦2+𝜇𝑧2𝜇𝑦𝑛𝑖=1𝑔𝑖(𝑦)𝜓(𝑦)𝑧2+(𝜇𝑦+𝑧)𝑛𝑖=1𝑡𝑡𝜏𝑖𝑔𝑖(𝑦(𝑠))𝑧(𝑠)𝑑𝑠+𝑛𝑖=1𝛾𝑖𝜏𝑖𝑧2𝑛𝑖=1𝛾𝑖𝑡𝑡𝜏𝑖𝑧2(𝑠)𝑑𝑠+(𝜇𝑦+𝑧)𝑝().(2.9)
The assumption |𝑔(𝑦)|𝐿𝑖 and the estimate 2|𝑚||𝑛|𝑚2+𝑛2 imply 𝜇𝑦𝑛𝑖=1𝑡𝑡𝜏𝑖𝑔𝑖||𝑦||(𝑦(𝑠))𝑧(𝑠)𝑑𝑠𝜇𝑛𝑖=1𝑡𝑡𝜏𝑖||𝑔𝑖||||||||𝑦||(𝑦(𝑠))𝑧(𝑠)𝑑𝑠𝜇𝑛𝑖=1𝑡𝑡𝜏𝑖𝐿𝑖||||𝜇𝑧(𝑠)𝑑𝑠2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2+𝜇2𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖𝑧2𝑧(𝑠)𝑑𝑠,𝑛𝑖=1𝑡𝑡𝜏𝑖𝑔𝑖(𝑦(𝑠))𝑧(𝑠)𝑑𝑠|𝑧|𝑛𝑖=1𝑡𝑡𝜏𝑖||𝑔𝑖||||||(𝑦(𝑠))𝑧(𝑠)𝑑𝑠|𝑧|𝑛𝑖=1𝑡𝑡𝜏𝑖𝐿𝑖||||1𝑧(𝑠)𝑑𝑠2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑧2+12𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖𝑧2(𝑠)𝑑𝑠(2.10) so that ̇𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡𝑓(𝑥)𝑦2+𝜇𝑧2𝜇𝑦𝑛𝑖=1𝑔𝑖(𝑦)𝜓(𝑦)𝑧2+𝑛𝑖=1𝛾𝑖𝜏𝑖𝑧2+𝜇2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2+12𝑛𝑖=1𝐿𝑖𝜏𝑖𝑧2+𝑛𝑖=112𝐿𝑖+12𝜇𝐿𝑖𝛾𝑖𝑡𝑡𝜏𝑖𝑧2(𝑠)𝑑𝑠+(𝜇𝑦+𝑧)𝑝().(2.11) Using the assumptions sup{𝑓(𝑥)}=𝑐>0,𝜓(𝑦)𝑎 and 𝑎𝑏𝑐>0, and the estimation 𝜇=(𝑎𝑏+𝑐)/2𝑏, it follows that ̇𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡𝜇𝑛𝑖=1𝑔𝑖(𝑦)𝑦𝜇𝑐2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦21𝑎𝜇2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑛𝑖=1𝛾𝑖𝜏𝑖𝑧2+𝑛𝑖=112𝐿𝑖+12𝜇𝐿𝑖𝛾𝑖𝑡𝑡𝜏𝑖𝑧2||||||||𝜇(𝑠)𝑑𝑠(𝜇𝑦+𝑧)𝑝()𝑛𝑖=1𝑔𝑖(𝑦)𝑦𝜇𝑐2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2𝑎𝑏𝑐12𝑏2𝑛𝑖=1𝐿𝑖+2𝛾𝑖𝜏𝑖𝑧2+12𝑛𝑖=1(1+𝜇)𝐿𝑖2𝛾𝑖𝑡𝑡𝜏𝑖𝑧2||||||||.(𝑠)𝑑𝑠+(𝜇𝑦+𝑧)𝑝()(2.12) If we choose 𝛾𝑖=(1/2)𝑛𝑖=1(1+𝜇)𝐿𝑖 and use the assumption |𝑝()|𝑚, then ̇𝑉1𝑥𝑡,𝑦𝑡,𝑧𝑡𝜇𝑛𝑖=1𝑔𝑖(𝑦)𝑦𝜇𝑐2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2𝑎𝑏𝑐12𝑏2𝑛𝑖=1(2+𝜇)𝐿𝑖𝜏𝑖𝑧2||𝑦||+𝜇𝑚+𝑚|𝑧|.(2.13)
An easy calculation from 𝑉2(𝑥,𝑦,𝑧) and (1.6) leads to ̇𝑉21()=𝑀𝜓(𝑦)𝑧𝑛𝑖=1𝑔𝑖(𝑦)𝑓(𝑥)+𝑛𝑖=1𝑡𝑡𝜏𝑖𝑔𝑖(𝑦(𝑠))𝑧(𝑠)𝑑𝑠+𝑝()sgn𝑥,|𝑥|1,|𝑧|𝑀0,|𝑥|1,|𝑧|𝑀𝑦𝑧𝑀+𝑥𝑀𝜓(𝑦)𝑧𝑛𝑖=1𝑔𝑖(𝑦)𝑓(𝑥)+𝑛𝑖=1𝑡𝑡𝜏𝑖𝑔𝑖|(𝑦(𝑠))𝑧(𝑠)𝑑𝑠+𝑝(),|𝑥|1,|𝑧|𝑀𝑦sgn𝑧,𝑥|1,|𝑧|𝑀(2.14) so that ̇𝑉21()𝑀𝑓(𝑥)sgn𝑥+𝑎+𝛿+𝑚+𝑛𝑖=1||𝑔𝑖||+(𝑦)𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||||𝑦||𝑧(𝑠)𝑑𝑠,|𝑥|1,|𝑧|𝑀0,|𝑥|1,|𝑧|𝑀+𝑎+𝛿+𝑚+𝑛𝑖=1||𝑔𝑖||+(𝑦)𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||||𝑦||,𝑧(𝑠)𝑑𝑠,|𝑥|1,|𝑧|𝑀|𝑥|1,|𝑧|𝑀,(2.15) using the assumptions of Theorem 2.1.
First, we consider 𝑉 in the domainmax{|𝑦|𝐾,|𝑧|𝑀}0, where the constants 𝐾 and 𝑀 are large enough, which will be determined later. We have to discuss the following two cases.
Case 10 (|𝑦|𝐾1, and 𝑥,𝑧 are arbitrary). In this case, it follows that: ̇𝑉2||𝑦||+()(𝑎+𝛿+𝑚)+𝑛𝑖=1||𝑔𝑖||+(𝑦)𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||𝑧(𝑠)𝑑𝑠.(2.16) By the estimates (2.3), (2.13), (2.16), and 𝜏=max1𝑖𝑛𝜏𝑖, we get ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝜇𝑛𝑖=1𝑔𝑖(𝑦)𝑦𝜇𝑐2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2+𝑛𝑖=1||𝑔𝑖||(𝑦)𝑎𝑏𝑐2𝑏(2+𝜇)2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑧2||𝑦||+(𝜇𝑚+1)+𝑚|𝑧|+(𝑎+𝛿+𝑚)+𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||+𝑧(𝑠)𝑑𝑠𝑛𝑖=1𝐿𝑖𝜏𝑖|𝑧|𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||𝜇𝑧(𝑠)𝑑𝑠𝑛𝑖=1𝑔𝑖(𝑦)𝑦𝜇𝑐2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2+𝑛𝑖=1||𝑔𝑖||(𝑦)𝑎𝑏𝑐2𝑏(2+𝜇)2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑧2+||𝑦||||𝑦||+(𝜇𝑚+1)+𝑚|𝑧|+(𝑎+𝛿+𝑚)𝑛𝑖=1𝐿𝑖𝜏𝑖|𝑧|.(2.17)
We now consider the term 𝜇𝑛𝑖=1𝑔𝑖(𝑦)𝑦𝑛𝑖=1||𝑔𝑖||(𝑦)𝑐𝑦2𝜇2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2(2.18) and define =𝑎𝑏+3𝑐2(𝑎𝑏+𝑐)<1.(2.19) Then, there exists a constant 𝐾1, (𝐾1>1),satisfying (11/𝜇|𝑦|) for |𝑦|𝐾1, so that, when |𝑦|𝐾1, 𝜇𝑛𝑖=1𝑔𝑖(𝑦)𝑦𝑛𝑖=1||𝑔𝑖||(𝑦)𝑐𝑦2𝜇2𝑛𝑖=1𝐿𝑖𝜏𝑖𝑦2=𝜇𝑛𝑖=1𝑔𝑖(𝑦)𝑦11𝜇||𝑦||𝑦2𝑐𝑛𝑖=1𝜇𝐿𝑖𝜏𝑖2𝑦2(𝑎𝑏+𝑐)𝑏2𝑏𝑎𝑏+3𝑐2𝑦(𝑎𝑏+𝑐)2𝑐𝑛𝑖=1𝜇𝐿𝑖𝜏𝑖2𝑦2=𝑎𝑏𝑐4𝑛𝑖=1𝜇𝐿𝑖𝜏𝑖2𝑦2𝑎𝑏𝑐4𝑀1𝜏𝑦2.(2.20) Hence ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝑎𝑏𝑐4𝑀1𝜏𝑦2𝑎𝑏𝑐2𝑏(2+𝜇)𝑁12𝜏𝑧2||𝑦||++(𝜇𝑚+1+𝑎+𝛿+𝑚)𝑚+𝑁1𝜏|𝑧|,(2.21) where 𝑀1=𝑛𝑖=1𝜇𝐿𝑖2,𝑁1=𝑛𝑖=1𝐿𝑖.(2.22) If 𝜏<min𝑎𝑏𝑐(2+𝜇)𝑏𝑁1,𝑎𝑏𝑐4𝑀1,(2.23) then the above estimate implies ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝛿1𝑦2+𝑧2||𝑦||++(𝜇𝑚+𝑎+𝛿+𝑚+1)𝑚+𝑁1𝜏|𝑧|(2.24) for a positive constant 𝛿1.
Let 𝜌1=max𝜇𝑚+𝛿+𝑎+𝑚+1,𝑚+𝑁1𝜏(2.25) so that ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝛿1𝑦2+𝑧2+𝜌1||𝑦||𝛿+|𝑧|=12𝑦2+𝑧2𝛿12||𝑦||𝜌1𝛿12+𝜌|𝑧|1𝛿12𝜌221𝛿21𝛿12𝑦2+𝑧2(2.26) provided that |𝑦|(2+1)𝜌1𝛿11.
Let 𝐾=max{(2+1)𝜌1𝛿11,𝐾1}. If |𝑦|𝐾, then ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝛿12𝑦2+𝑧2.(2.27)Case 20 (|𝑧|𝑀, and 𝑥,𝑦 are arbitrary). Then ̇𝑉2||𝑦||.(𝑥,𝑦,𝑧)(2.28) By following a similar method shown in the first case, select 𝛾𝑖=(1/2)𝑛𝑖=1(1+𝜇)𝐿𝑖 and take 𝜏<min𝑎𝑏𝑐(2+𝜇)𝑏𝑁1,𝑎𝑏𝑐4𝑀1,(2.29) one can easily obtain ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝛿2𝑦2+𝑧2||𝑦||++(𝜇𝑚+1)𝑚+𝑁1𝜏|𝑧|𝛿2𝑦2+𝑧2+𝜌2||𝑦||+𝛿|𝑧|22𝑦2+𝑧2(2.30) for some positive constants 𝛿2 and 𝜌2 provided that |𝑧|𝑀=𝐾.
At the end, we consider 𝑉 in max{|𝑦|𝐾,|𝑧|𝑀}0.
Let |𝑥|𝐻>1, where the constant 𝐻 will be determined later. Hence ̇𝑉21()𝑀𝑓(𝑥)sgn𝑥+max||𝑦||𝑛𝐾𝑖=1||𝑔𝑖||+(𝑦)𝑎+𝛿+𝑚+𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||.𝑧(𝑠)𝑑𝑠(2.31) It follows from (2.3), (2.13), and (2.31) that ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝑎𝑏𝑐4𝑀1𝜏𝑦2𝑎𝑏𝑐2𝑏(2+𝜇)𝑁12𝜏𝑧2||𝑦||++𝜇𝑚𝑚+𝑁1𝜏|1𝑧|𝑀𝑓(𝑥)sgn𝑥+max||𝑦||𝑛𝐾𝑖=1||𝑔𝑖(||𝑦)+𝑎+𝛿+𝑚+𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||𝑧(𝑠)𝑑𝑠𝑛𝑖=1𝐿𝑖𝑡𝑡𝜏𝑖||||𝑧(𝑠)𝑑𝑠𝑎𝑏𝑐4𝑀1𝜏𝑦2𝑎𝑏𝑐(2𝑏2+𝜇)𝑁12𝜏𝑧21𝑀𝑓(𝑥)sgn𝑥+max||𝑦||𝑛𝐾𝑖=1||𝑔𝑖||+(𝑦)𝑎+𝛿+𝑚+𝑁1.𝑀𝜏+𝜇𝑚𝐾+𝑚𝑀(2.32) Since 𝑓(𝑥)sgn𝑥 as |𝑥| and |𝑥|𝐻>1, then we have 𝑓(𝑥)sgn𝑥2𝑀max||𝑦||𝑛𝐾𝑖=1||𝑔𝑖||(𝑦)+𝑎+𝛿+𝑚+𝑁1,𝑀𝜏+𝜇𝑚𝐾+𝑚𝑀(2.33) so that 𝑓(𝑥)sgn𝑥+2𝑀max||𝑦||𝑛𝐾𝑖=1||𝑔𝑖||(𝑦)+𝑎+𝛿+𝑚+𝑁1𝑀𝜏+𝜇𝑚𝐾+𝑚𝑀0.(2.34) In view of the above discussion, it follows that: ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝑎𝑏𝑐4𝑀1𝜏𝑦2𝑎𝑏𝑐2𝑏(2+𝜇)𝑁12𝜏𝑧212𝑀𝑓(𝑥)sgn𝑥.(2.35) Subject to the evidence thus far, we can conclude that there exists a positive constant 𝑅, which is large enough, such that ̇𝑉𝑥𝑡,𝑦𝑡,𝑧𝑡𝑤(𝑢)for𝑢2𝑅2,(2.36) where 𝑢=(𝑥2+𝑦2+𝑧2)1/2. Thus, the Lyapunov functional 𝑉(𝑥𝑡,𝑦𝑡,𝑧𝑡) satisfies all the assumptions of Theorem 1.1. The proof for Theorem 2.1 is complete.

Example 2.2. Consider the following nonlinear differential equation of the third order with two constant deviating arguments, 𝜏1>0,𝜏2>0: 𝑥+14+1+(𝑥)2𝑥+4𝑥𝑡𝜏1+sin𝑥𝑡𝜏1+4𝑥𝑡𝜏2+sin𝑥𝑡𝜏2=+11𝑥sin𝑡+cos𝑡3+sin𝑡+𝑥2++𝑥2𝑡𝜏2+𝑥2,(2.37) which is a special case of (1.5).
This equation can be written in the system form as follows: 𝑥=𝑦,𝑦𝑧=𝑧,1=4+1+𝑦2+𝑧(8𝑦+2sin𝑦)11𝑥𝑡𝑡𝜏1(4+cos𝑦(𝑠))𝑧(𝑠)𝑑𝑠+𝑡𝑡𝜏2+(4+cos𝑦(𝑠))𝑧(𝑠)𝑑𝑠sin𝑡+cos𝑡3+sin𝑡+𝑥2++𝑦2𝑡𝜏2+𝑧2.(2.38) When we compare the system described to this point with (1.6), it follows the existence of the following estimates: 1𝜓(𝑦)=4+1+𝑦2,0𝜓(𝑦)41,𝑎=4,𝛿=1,𝑔1(𝑦)=4𝑦+sin𝑦,𝑔1𝑔(0)=0,(2.39)1(𝑦)𝑦=4+sin𝑦𝑦3,𝑏1𝑔=3,1||𝑔(𝑦)=4+cos𝑦,1||=||||(𝑦)4+cos𝑦5=𝐿1,𝑔2(𝑦)=4𝑦+sin𝑦,𝑔2𝑔(0)=0,2(𝑦)𝑦=4+sin𝑦𝑦3,𝑏2𝑔=3,2||𝑔(𝑦)=4+cos𝑦,2||=||||(𝑦)4+cos𝑦5=𝐿2,𝑓𝑓(𝑥)=11𝑥,𝑓(0)=0,𝑓(𝑥)sgn𝑥=11𝑥sgn𝑥>0,(𝑥0),𝑓(𝑥)sgn𝑥=11𝑥sgn𝑥as|𝑥|,(𝑥)=11,𝑐=11,𝑎𝑏𝑐=13>0,=𝑎𝑏+3𝑐=2(𝑎𝑏+𝑐)57𝑝70<1,𝑡,𝑥,𝑥𝑡𝜏1,,𝑦𝑡𝜏2=,𝑧sin𝑡+cos𝑡3+sin𝑡+𝑥2++𝑦2𝑡𝜏2+𝑧2𝑝1=𝑚,𝑡+2𝜋,𝑥,𝑥𝑡𝜏1,,𝑦𝑡𝜏2=,𝑧sin(𝑡+2𝜋)+cos(𝑡+2𝜋)3+sin(𝑡+2𝜋)++𝑧2=sin𝑡+cos𝑡3+sin𝑡++𝑧2=𝑝𝑡,𝑥,𝑥𝑡𝜏1,,𝑦𝑡𝜏2,𝑧,𝑇=2𝜋.(2.40) In view of the above estimates, it is seen that all the assumptions of Theorem 2.1 hold. This discussion verifies that (2.37) has a periodic solution of period 𝑇,𝑇=2𝜋.

Acknowledgment

The author would like to expresses his sincere thanks and best regards to the anonymous referees for their many helpful comments, corrections, and suggestions on the paper.

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