AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 968541 10.1155/2013/968541 968541 Research Article Existence of Periodic Solutions to Multidelay Functional Differential Equations of Second Order Tunç Cemil Yazgan Ramazan Mohiuddine S. A. Department of Mathematics Faculty of Sciences Yüzüncü Yıl University 65080 Van Turkey yyu.edu.tr 2013 10 11 2013 2013 23 08 2013 03 10 2013 2013 Copyright © 2013 Cemil Tunç and Ramazan Yazgan. 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.

Using Lyapunov-Krasovskii functional approach, we establish a new result to guarantee the existence of periodic solutions of a certain multidelay nonlinear functional differential equation of second order. By this work, we extend and improve some earlier result in the literature.

1. Introduction

It is well known that the problem of the existence of periodic solutions of retarded functional differential equations of second order is not only very important in the background applications, but also of considerable significance in theory of differential equations. Besides, the scope of retarded functional differential equations is very general. For example, it contains ordinary differential equations, differential-difference equations, integrodifferential equations, and so on. The motivation of this paper is that in recent years the study of the existence of periodic solutions to various kinds of retarded functional differential equations of second order has become one of the most attractive topics in the literature. Especially, by using the famous continuation theorem of degree theory (see Gaines and Mawhin ), many authors have made a lot of interesting contributions to the topic for retarded functional differential equations of second order. Here, we would not like to give the details of these works.

On the other hand, amongst the achieved excellent results, one of them is the famous Yoshizawa’s theorem  for existence of periodic solutions of retarded functional differential equations, which has vital influence and has been widely used in the literature. This theorem has also been generally one of the best results in the literature from the past till now. It should be noted that in 1994, Zhao et al.  proved four sufficiency theorems on the existence of periodic solutions for a class of retarded functional differentials to have the existence of an ω-periodic solution. By this work, the authors proved that their theorems are better than Yoshizawa’s  periodic solutions theorem primarily by removing restrictions of the size of the constant delay h. An example of application was given at the end of the paper. Namely, in the same paper, the authors applied the following Theorem A to discuss the existence of an ω-periodic solution of the nonlinear delay differential equation of the second order: (1)  x′′(t)+ax(t)+g(x(t-τ))=p(t), where p(t) is an external force, g(x(t-τ)) is a delayed restoring force, the delay τ is a positive constant and the friction is proportional to the velocity, and a is a positive constant. It should be noted that a feedback system with friction proportional to velocity, an external force p(t), and a delayed restoring force g(x(t-τ)), (τ>0) may be written as (1) (see Burton ).

Consider the following general nonautonomous delay differential equation: (2)x˙=F(t,xt),xt=x(t+θ),-hθ0,t0, where F:×Cn, C=C([-h,0],n), h is a positive constant, and we suppose that F is continuous, ω-periodic, and takes closed bounded sets into bounded sets of n; and such that solutions of initial value problems are unique, h can be either larger than ω, or equal to or smaller than ω. Here (C,·) is the Banach space of continuous function ϕ:[-h,0]n with supremum norm; h>0,   CH is the open H-ball in C; CH={ϕC([-h,0],n):ϕ<H}. Standard existence theory, see Burton , shows that if ϕCH and t0, then there is at least one continuous solution x(t,t0,ϕ) such that on [t0,t0+α) satisfying (2) for t>t0, xt(t,ϕ)=ϕ and α is a positive constant. If there is a closed subset BCH such that the solution remains in B, then α=. Further, the symbol |·| will denote a convenient norm in n with |x|= max1in|xi|. Let us assume that C(t)={ϕ:[t-α,t]nϕ is continuous} and ϕt denotes the ϕ in the particular C(t), and that ϕt=maxt-αst|ϕ(t)|. Finally, by the periodicity, we mean that that there is an ω>0 such that F(t,ϕ) is ω-periodic in the sense that if x(t) is a solution of (2) so is x(t+ω).

Definition 1.

Solutions of (2) are uniform bounded at t=0 if for each B1 there exists B2 such that [ϕC,ϕ<B1,t0] imply that |x(t,0,ϕ)|<B2 (see Burton ).

Definition 2.

Solutions of (2) are uniform ultimate bounded for bound B at t=0 if for each B3>0 there exists a K>0 such that [ϕC,ϕ<B3,tK] imply that |x(t,0,ϕ)|<B (see Burton ).

The first theorem given in Zhao et al.  is the following.

Theorem A.

If the solutions of (2) are ultimately bounded by the bound B, then

equation (2) has an ω-periodic solution and is bounded by B,

if (2) is autonomous, then (2) has an equilibrium solution and is bounded by B (see Zhao et al. ).

Regarding (1) Zhao et al.  proved the following theorem as example of application.

Theorem B.

Assume that the following conditions hold:

p(t) is an ω-periodic continuous function, g is a continuous differentiable function,

lim|x|g(x)sgnx=, and there is a bounded set Ω containing the origin such that |g(x)|c on Ωc; Ωc is the complement of the set Ω,

τc<a.

Then, (1) has an ω-periodic motion.

Moreover, when p(t)=K, K is a constant, under the above conditions (1) has a constant motion x=c0, and the constant c0 satisfies g(c0)=K. In fact, from the condition (2), there is a bounded set Ω1 containing the origin such that |g(x)|c and 0xg(s)ds>0 on Ω1c; Ω1c is the complement of the set Ω1.

In this paper, we consider the following nonlinear differential equation of second order with multiple constant delays, τi(>0): (3)x′′(t)+{f(x(t),x(t))+g(x(t),x(t))x(t)}x(t)+h(x(t))+i=1ngi(x(t-τi))=p(t), where τi are fixed constants delay with t-τi0; the primes in (3) denote differentiation with respect to t, f, g, h, gi, and p are continuous functions in their respective arguments on 2, 2, , , and , respectively, and also depend only on the arguments displayed explicitly. The continuity of these functions is a sufficient condition for existence of the solution of (3). It is also assumed as basic that the functions f, g, h, and gi satisfy a Lipschitz condition in x,x,x(t-τ1),x(t-τ2),,x(t-τn). By this assumption, the uniqueness of solutions of (3) is guaranteed. The derivatives dgi/dxgi(x) exist and are continuous. It should be noted that throughout the paper, sometimes, x(t) and y(t) are abbreviated as x and y, respectively.

We write (3) in system form as follows: (4)x=y,y=-{f(x,y)+g(x,y)y}y-h(x)-i=1ngi(x)+i=1n-τi0gi(x(t+s))y(t+s)ds+p(t), where gi(x(t+s))=dgi/dx.

It is clear that (3) is a particular case of (2). It should be noted that the reason or the motivation for taking into consideration (3) comes from the following modified Liénard type equation of the form: (5)x′′(t)+{f(x(t))+g(x(t))x(t)}x(t)+h(x(t))=e(t). These types of equations have great applications in theory and applications of the differential equations. Therefore, till now, the qualitative behaviors, the stability, boundedness, global existence, existence of periodic solutions, and so forth, of these type differential equations have been studied by many researchers, and the researches on these topics are still being done in the literature. For example, we refer the readers to the books of Ahmad and Rao , Burton , Gaines and Mawhin , and the papers of Constantin , Graef , Huang and Yu , Jin , Liu and Huang , Nápoles Valdés , Qian , Tunç , C. Tunç and E. Tunç , Zhao et al. , Zhou , and the references cited in these works.

We here give certain sufficient conditions to guarantee the existence of an ω-periodic solution of (3). This paper is inspired by the mentioned papers and that in the literature. Our aim is to generalize and improve the application given in  for (3). This paper has also a contribution to the investigation of the qualitative behaviors of retarded functional differential equations of second order, and it may be useful for researchers who work on the above mentioned topics. Finally, without using the famous continuation theorem of degree theory, which belongs to Gaines and Mawhin , we prove the following main result. This case makes the topic of this paper interesting.

2. Main Result

Our main result is the following.

Theorem 3.

We assume that there are positive constants a, a-, b, τ, and ci such that the following conditions hold:

a-f(x,y)+g(x,y)ya,

lim|x|h(x)sgnx=, and there is a bounded set Ω containing the origin such that |h(x)|b on Ωc; Ωc is the complement of the set Ω,

lim|x|gi(x)sgnx=, and there is a bounded set Ω containing the origin such that |gi(x)|ci on Ωc; Ωc is the complement of the set Ω,

p(t) is an ω-periodic continuous function.

If τi=1nci<a, then (3) has an ω-periodic solution.

Moreover, p(t)=K, K-constant, under the above conditions (3) has a constant motion x=c0, and the constant c0 satisfies h(c0)+gi(c0)=K.

Proof.

Define the Lyapunov-Krasovskii functional V=V(xt,yt): (6)V=12y2+0xh(s)ds+i=1n0xgi(s)ds+i=1nλi-τi0t+sty2(θ)dθds, where λi are some positive constants to be determined later in the proof.

Evaluating the time derivative of V along system (4), we get (7)V˙=-{f(x,y)+g(x,y)y}y2+yi=1n-τi0gi(x(t+s))y(t+s)ds+yp(t)+i=1nλi-τi0(y2(t)-y2(t+s))ds=-{f(x,y)+g(x,y)y}y2+yi=1n-τi0gi(x(t+s))y(t+s)ds+yp(t)  +i=1n(λiτi)y2-i=1nλi-τi0y2(t+s)ds. By noting the assumption |gi(x)|ci of the theorem and the estimate 2|αγ|α2+γ2, one can obtain the following estimates: (8)yi=1n-τi0gi(x(t+s))y(t+s)dsi=1n-τi0|gi(x(t+s))||y(t)||y(t+s)|ds12i=1n-τi0ci(y2(t)+y2(t+s))ds12i=1n(ciτi)y2+12i=1nci-τi0y2(t+s)ds. Then, it follows that (9)V˙-{f(x,y)+g(x,y)y}y2+yp(t)+12i=1n(ci+2λi)τiy2+12i=1n(ci-2λi)-τi0y2(t+s)ds. Let λi=ci/2 and τ=max{τ1,,τn}. In fact, these choices imply that (10)V˙-{f(x,y)+g(x,y)y-τi=1nci-|p(t)||y|-1}y2-(a-τi=1nci-|p(t)||y|-1)y2.

In view of the continuity and periodicity of the function p and the assumption a-τi=1nci>0, it follows that there is a bounded set Ω2Ω1 with Ω2 containing the origin and a positive constant μ such that (11)μa-cτ-|p(t)||y|for  ×Ω2c. Therefore, we can write (12)V˙-μy2for  (t,x,y)×Ω2c×Ω2c.

From the last estimate, we can arrive that the y-coordinate of the solutions of system (4) is ultimately bounded for a positive constant β. On the other hand, since p is a continuous periodic function, if |y|β and V1=V+y, then, subject to the assumptions of the theorem, it can be easily seen that there is a constant K1>0 on ×Ω2c such that (13)V˙1=V˙+y=-{f(x,y)+g(x,y)y}y2+yi=1n-τi0gi(x(t+s))y(t+s)ds+yp(t)+i=1n(λiτi)y2-i=1nλi-τi0y2(t+s)ds-{f(x,y)+g(x,y)y}y-h(x)-i=1ngi(x)+i=1n-τi0gi(x(t+s))y(t+s)ds+p(t)-ay2+yi=1n-τi0gi(x(t+s))y(t+s)ds+yp(t)+i=1n(λiτi)y2-i=1nλi-τi0y2(t+s)ds+a-|y|-h(x)-i=1ngi(x)+i=1n-τi0gi(x(t+s))y(t+s)ds+p(t)-h(x)-i=1ngi(x)+K1. Since h(x) and gi(x) as x, then it can be chosen a positive constant B1 such that (14)V˙1-0.5for  xB1. Therefore, we can conclude that there is a positive constant α1 such that the x-coordinate of the solutions of system (4) satisfies xα1 for |y|β.

In a similar manner, if |y|β and V2=V-y, then, subject to the assumptions of the theorem, it can be easily followed by the time derivative of the functional V2 that there is a constant K2>0 on ×Ω2c such that (15)V˙2=V˙-y=-{f(x,y)+g(x,y)y}y2+yi=1n-τi0gi(x(t+s))y(t+s)ds+yp(t)+i=1n(λiτi)y2-i=1nλi-τi0y2(t+s)ds+{f(x,y)+g(x,y)y}y+h(x)+i=1ngi(x)-i=1n-τi0gi(x(t+s))y(t+s)ds-p(t)-ay2+yi=1n-τi0gi(x(t+s))y(t+s)ds+yp(t)+i=1n(λiτi)y2-i=1nλi-τi0y2(t+s)ds+a-|y|+h(x)+i=1ngi(x)-i=1n-τi0gi(x(t+s))y(t+s)ds-p(t)h(x)+i=1ngi(x)+K2.

Since h(x)- and gi(x)- as x-,  then it can be chosen a positive constant B2 such that (16)V˙2-0.5for  x-B2.

Then, we can conclude that there is a positive constant α2 such that the x-coordinate of the solutions of system (4) satisfies x-α2 for |y|β. On gathering the above whole discussion, one can see that the solutions of system (4) are ultimately bounded. Therefore, (3) has an ω-periodic motion (solution). When p(t)=K, K-constant, (3) has a constant motion x=c0. From (3), it can be seen that the constant x=c0 is given by h(c0)+gi(c0)=K.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Gaines R. E. Mawhin J. L. Coincidence Degree, and Nonlinear Differential Equations 1977 568 Berlin, Germany Springer Lecture Notes in Mathematics MR0637067 Yoshizawa T. Stability theory by Liapunov's second method Publications of the Mathematical Society of Japan 1966 9 Tokyo, Japan The Mathematical Society of Japan Zhao J. M. Huang K. L. Lu Q. S. The existence of periodic solutions for a class of functional-differential equations and their application Applied Mathematics and Mechanics. English Edition 1994 15 1 49 59 translated from Applied Mathematics and Mechanics, vol. 15, no. 1, pp. 49–58, 1994 MR1267133 ZBL0793.34054 Burton T. A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations 2005 Mineola, NY, USA Dover Corrected Version of the 1985 Original MR2761514 Ahmad S. Rao M. R. Theory of Ordinary differential Equations. With Applications in Biology and Engineering 1999 New Delhi, India Affiliated East-West Press MR1720698 Constantin A. A note on a second-order nonlinear differential system Glasgow Mathematical Journal 2000 42 2 195 199 10.1017/S0017089500020048 MR1763738 ZBL0960.34027 Graef J. R. On the generalized Liénard equation with negative damping Journal of Differential Equations 1972 12 34 62 MR0328200 10.1016/0022-0396(72)90004-6 ZBL0254.34038 Huang L. H. Yu J. S. On boundedness of solutions of generalized Liénard's system and its application Annals of Differential Equations 1993 9 3 311 318 MR1244626 ZBL0782.34036 Jin Z. Boundedness and convergence of solutions of a second-order nonlinear differential system Journal of Mathematical Analysis and Applications 2001 256 2 360 374 10.1006/jmaa.2000.7056 MR1821745 ZBL0983.34021 Liu B. Huang L. Boundedness of solutions for a class of retarded Liénard equation Journal of Mathematical Analysis and Applications 2003 286 2 422 434 10.1016/S0022-247X(03)00455-4 MR2008841 ZBL1044.34023 Nápoles Valdés J. E. Boundedness and global asymptotic stability of the forced Liénard equation Revista de la Unión Matemática Argentina 2000 41 4 47 59 MR1853028 Qian C. X. Boundedness and asymptotic behaviour of solutions of a second-order nonlinear system The Bulletin of the London Mathematical Society 1992 24 3 281 288 10.1112/blms/24.3.281 MR1157265 ZBL0763.34021 Tunç C. A note on the bounded solutions Applied Mathematics and Information Sciences 2014 8 1 393 399 Tunç C. A note on boundedness of solutions to a class of non-autonomous differential equations of second order Applicable Analysis and Discrete Mathematics 2010 4 2 361 372 10.2298/AADM100601026T MR2724643 Tunç C. Boundedness results for solutions of certain nonlinear differential equations of second order Journal of the Indonesian Mathematical Society 2010 16 2 115 126 MR2752774 ZBL1232.34057 Tunç C. Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations Applied and Computational Mathematics 2011 10 3 449 462 MR2893512 ZBL05992087 Tunç C. On the boundedness of solutions of a non-autonomous differential equation of second order Sarajevo Journal of Mathematics 2011 7(19) 1 19 29 MR2839316 ZBL1236.34043 Tunç C. Stability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviating argument Acta Mathematica Vietnamica 2012 37 3 311 325 MR3027224 Tunç C. On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order with multiple deviating arguments Afrika Matematika 2012 23 2 249 259 10.1007/s13370-011-0033-y MR2958972 ZBL1266.34117 Tunç C. Stability to vector Lienard equation with constant deviating argument Nonlinear Dynamics 2013 73 3 1245 1251 Tunç C. New results on the existence of periodic solutions for Rayleigh equation with state-dependent delay Journal of Mathematical and Fundamental Sciences 2013 45 2 154 162 Tunç C. Tunç E. On the asymptotic behavior of solutions of certain second-order differential equations Journal of the Franklin Institute 2007 344 5 391 398 10.1016/j.jfranklin.2006.02.011 MR2327928 ZBL1269.34057 Zhou J. Necessary and sufficient conditions for boundedness and convergence of a second-order nonlinear differential system Acta Mathematica Sinica. Chinese Series 2000 43 3 415 420 MR1778806 ZBL1018.34038