AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 702456 10.1155/2012/702456 702456 Research Article Periodic Solutions of a Type of Liénard Higher Order Delay Functional Differential Equation with Complex Deviating Argument Wang Haiqing Chung Jaeyoung 1 School of Science Tianjin Polytechnic University Tianjin, Hebei 300387 China tjpu.edu.cn 2012 31 12 2012 2012 26 09 2012 28 11 2012 2012 Copyright © 2012 Haiqing Wang. 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.

The author has studied the existence of periodic solutions of a type of higher order delay functional differential equations with neutral type by using the theory of coincidence degree, and some new sufficient conditions for the existence of periodic solutions have been obtained.

1. Introduction and Lemma

With the rapid development of modern science and technology, functional differential equation with time delay has been widely applied in many areas such as bioengineering, systems analysis, and dynamics. Functional differential equation with complex deviating argument is an important type of the above function. Because the property of the solution to this kind of equation is impossibly estimated, so the literature on the functional differential equation with complex argument is relatively rare . In recent years, with the maturity of the theory of nonlinear functional analysis and algebraic topology, we have the powerful tools of the study on the functional differential equation with complex deviating argument, so it is possible to study the above equation. Furthermore, the study on the periodic solutions of functional differential equation is always one of the most important subject that people concerned for its widespread use. Many results of the study of Duffing-typed functional differential equation and Liénard-typed functional differential equation have been obtained, for example, the literatures . Hitherto, the literature of the discussion of higher order functional differential equations has not been found a lot . In this paper I have studied and derived some sufficient conditions that guarantee the existence of periodic solutions for a type of higher order functional differential equations with complex deviating argument as the following: (*)i=1maix(i)(t)+f(x(t))x˙(t)+β(t)g(x(x(t)))=p(t)(ai0), and some new results have been obtained.

In order to establish the existence of T-periodic solutions of (*), we make some preparations.

Definition 1.1.

Let X, Y are Banach spaces, and let Ω be an open and bounded subset in X, and let L:Dom(L)XY be linear mapping; the mapping L will be called a Fredholm mapping of index zero if dimkerL=codimImL<+ and ImL is closed in Y.

Definition 1.2.

Let P:Xker(L), let Q:YY/Im(L) be projectors, and let N:Ω-Y be nonlinear mapping; the mapping N will be called L-compact on Ω- if QN:Ω-Y/Im(L) and (L|ker(P))-1(I-Q)N:Ω-X are compact.

Lemma 1.3 (see [<xref ref-type="bibr" rid="B19">20</xref>]).

Let X, Y be Banach spaces; L:DLXY is a Fredholm mapping of index zero P:XX; Q:XY are continuous mapping projectors; Ω is an open bounded set in X; N:Ω-×[0,1]Y is L-Compact on Ω-, furthermore suppose that:

LxλN(x,λ),  for  all  xDLΩ,  λ(0,1);

QN(x,0)0,  for  all  xker(L)Ω;

deg(QN(x,0),  ker(L)Ω,0)0,

then the equation Lx=N(x,1) has at least one solution on Ω-, where deg is Brouwer degree.

2. Main Results and Proof of Theorems Theorem 2.1.

Suppose that f, β, g, p are continuous for their variables, respectively, p(t+T)=p(t),  β(t+T)=β(t)>0, 0Tp(t)dt=0, and furthermore suppose that

A>0,for  all  x, when |x|>A, such that xg(x)>0;

M>0,  for  all  x, such that |g(x)|M;

f1=supx|f(x)|<(am-k(Tm-1-Tm-2--T))/Tm-1,

where k=max{|ai|}, i=1,2,,m-1 and am>k(Tm-1+Tm-2++T), then (*) has at least one T-periodic solution.

Proof of Theorem <xref ref-type="statement" rid="thm1">2.1</xref>.

In order to use continuation theorem to obtain T-periodic solution of (*), we firstly make some required preparations. Let (2.1)X={xcm-1(,)x(t+T)=x(t)},Y={yC(,)y(t+T)=y(t)}, and the norm of X and Y is x=max0im-1{|x(i)|}, |x(i)|=maxt{|x(i)(t)|}, i=1,2,,m-1, and y=maxt{|y(t)|}, respectively; then the X and Y with this norm are Banach spaces.

Firstly, we study the priori bound of T-periodic solution of following equation: (2.2)i=1maix(i)(t)+λf(x(t))x˙(t)+λβ(t)g(x(x(t)))=λ2p(t).

Suppose that x=x(t)X is an arbitrary T-periodic solution of (2.2), put x(t) into, (2.2) and then integrate both sides of (2.2) on [0,T], so (2.3)0Tβ(t)g(x(x(t)))dt=0.

For the continuity of β, g, x, there must exist a number t0[0,T] such that (2.4)β(t0)g(x(x(t0)))=0, that is, (2.5)g(x(x(t0)))=0.

For the condition (a) of Theorem 2.1, we have (2.6)|x(x(t0))|A.

Let (2.7)x(t0)=nT-t1,nN,t1[0,T], so (2.8)|x(t1)|=|x(x(t0))|A.

In view of (2.9)t[0,T],x(t)=x(t1)+t1tx˙(s)ds, we have (2.10)|x(t)|=|x(t1)+t1tx˙(s)ds|A+t1t|x˙(s)|dsA+0T|x˙(t)|dt, that is, (2.11)|x(0)|=|x|A+0T|x˙(t)|dt.

Noting x(t)=x(t+T), so there must exist the number ξi[0,T] such that x(i)(ξi)=0, where i=1,2,3,,m-1.

For all  t[0,T], (2.12)x(i)(t)=x(i)(ξi)+ξitx(i+1)(s)ds=ξitx(i+1)(s)ds,

we have (2.13)|x(i)(t)|=|ξitx(i+1)(s)ds|0T|x(i+1)(t)|dtT0T|x(i+2)(t)|dtT2·0T|x(i+3)(t)|dtTm-(i+1)0T|x(i+m-i)(t)|dt=Tm-(i+1)0T|x(m)(t)|dt, that is, (2.14)|x(i)|Tm-(i+1)0T|x(m)(t)|dt,i=1,2,,m-1.

Combining (2.11), (2.14), we get (2.15)|x(0)|=|x|A+Tm-10T|x(m)(t)|dt.

By (2.2), we get (2.16)0T|amxm(t)|dt0T|λf(x(t))x˙(t)|dt+0T|λβ(t)g(x(x(t)))|dt+0T|λ2p(t)|dt+0T|a1x˙(t)|dt+0T|a2x¨(t)|dt++0T|am-3x(m-3)(t)|dt+0T|am-2x(m-2)(t)|dt++0T|am-1x(m-1)(t)|dt, where β1=maxtβ(t),  p1=maxt{|p(t)|}, and k=max{|ai|},  i=1,2,3,,m-1.

Noting (2.14) and the conditions (b), (c) of Theorem 2.1, we have (2.17)0T|amx(m)(t)|dtf1T·Tm-20T|x(m)(t)|dt+β1TM+p1T+kT·Tm-(1+1)0T|x(m)(t)|dt+kT·Tm-(2+1)0T|x(m)(t)|dt++kT·Tm-(m-1+1)0T|x(m)(t)|dt,

so (2.18)am0T|x(m)(t)|dt(kTm-1+kTm-2++kT+f1Tm-1)0T|x(m)(t)|dt+β1TM+p1T, where am>Tm-1i=1mfi+kTm-1+kTm-2++kT.

Let (2.19)β1TM+p1Ta0-(kTm-1+kTm-2++kT+f1Tm-1)A1, that is, (2.20)0T|x(m)(t)|dtA1.

Noting (2.14), (2.15), and (2.20), we have (2.21)|x(0)|=|x|A+Tm-1A1ω0,|x(i)|Tm-(i+1)A1ωi,i=1,2,,m-1.

Let ω=max0im{ωi+1}, and let Ω={xxX:x<ω}; then Ω is an open and bounded set in X.

Let (2.22)L:DLXY:xLx=i=1maix(i)(t),N:X×IY:xN(x,λ)=-f(x(t))x˙(t)-β(t)g(x(x(t)))+λp(t); then the corresponding equation of Lx=λN(x,λ) is (2.2).

Now, we define projection operators as follows; (2.23)P:Xker(L):xPx=1T0Tx(t)dt,Q:YYIm(L):yQy=1T0Ty(t)dt.

Obviously, P, Q are continuous operators, Im(P)==ker(L), ker(Q)=Im(L), and it is easy to prove that L is a Fredholm mapping of index zero and is L-Compact on Ω-.

From the above discussion and the construction of Ω, we know that for all xDLΩ, λ(0,1), LxλN(x,λ), therefore the condition (a) of Lemma 1.3 holds.

For arbitrary xker(L)Ω, x=ω, by the definition of Q, N, we have (2.24)QN(x,0)=1T0T[-f(x(t))x˙(t)-β(t)g(x(x(t)))]dt=-1T0Tβ(t)g(x(x(t)))dt, so (2.25)xQN(x,0)=-1Tx0Tβ(t)g(x(x(t)))dt=-1Txg(x)0Tβ(t)dt0, therefore the condition (b) of Lemma 1.3 holds.

Making a transformation. (2.26)H(x,μ)=-μx+(1-μ)QN(x,0),xΩker(L),μ[0,1], we have (2.27)xH(x,μ)=-μx2+x(1-μ)QN(x,0)=-μx2-(1-μ)1Tg(x)x0Tβ(t)dt<0.

So xH(x,μ)0, that is, H(x,μ)0 is a homotopy, deg(QN(x,0),ker(L)Ω,0) = deg(-I,ker(L)Ω,0)=deg(-I,Ω,0)0, where I is an identity mapping, and the condition (c) of Lemma 1.3 holds.

From above all, the requirements of Lemma 1.3 are all satisfied, so (*) has at least one T-periodic solution under the condition of Theorem 2.1, so the proof of Theorem 2.1 is completed.

Remark 2.2.

In Theorem 2.1, if β(t)<0 and the condition (a) of Theorem 2.1 is when |x|>A, xg(x)<0, and the rest are unchangeable, then (*) has at least one T-periodic solution.

If the g(x) is not a bounded function, we have the following theorem.

Theorem 2.3.

Suppose that f, β, g, p are continuous for their variables, respectively, p(t+T)=p(t), β(t+T)=β(t)>0, 0Tp(t)dt=0, and furthermore suppose following:

A>0,for  all  x, when |x|>A, such that xg(x)>0;

M>0,  for  all  x, such that |g(x)|M|x|+c;

f1=supx|f(x)|<(am-kTm-1-kTm-2--kT-β1Tm)/Tm-1,

where k=max{|ai|},i=1,2,,m-1, and am>kTm-1+kTm-2++kT+β1Tm, then (*) has at least one T-periodic solution.

Proof of Theorem <xref ref-type="statement" rid="thm2">2.3</xref>.

Banach spaces X,  Y and the mappings L, P, Q, and N are the same to Theorem 2.1, and their property are equal to Theorem 2.1, then the corresponding equation of Lx=λN(x,λ) is (2.28)i=1maix(i)(t)+λf(x(t))x˙(t)+λβ(t)g(x(x(t)))=λ2p(t).

It is similar to Theorem 2.1, there must exist a number t1[0,T], such that (2.29)|x(t1)|A, and it is easy to obtain (2.30)|x(i)|Tm-(i+1)0T|x(m)(t)|dt,i=1,2,,m-1,|x(0)|=|x|A+Tm-10T|x(m)(t)|dt.

Noting (2.28), (2.30) and the conditions (b), (c) of Theorem 2.3, we have (2.31)0T|amx(m)|dt0T|λf(x(t))x˙(t)|dt+0T|λβ(t)g(x(x(t)))|dt+0T|λ2p(t)|dt+kT·Tm-(1+1)0T|xm(t)|dt+kT·Tm-(2+1)0T|xm(t)|dt++kT·Tm-(m-1+1)0T|xm(t)|dtf1T·Tm-20T|xm(t)|dt+β1T[M|x(x(t))|+c]+p1T+kT·Tm-(1+1)0T|xm(t)|dt+kT·Tm-(2+1)0T|xm(t)|dt++kT·Tm-(m-1+1)0T|xm(t)|dt.

So (2.32)am0T|xm(t)|dt(kTm-1+kTm-2++kT+f1Tm-1)0T|xm(t)|dt+β1Tc+β1TM|x|+p1T(kTm-1+kTm-2++kT+f1Tm-1)0T|xm(t)|dt+β1Tc+β1TM(A+Tm-10T|x(m)(t)|dt)+p1T(kTm-1+kTm-2++kT+f1Tm-1+β1Tm)0T|xm(t)|dt+β1Tc+β1TMA+p1T, where k=max{|ai|},i=1,2,3,,m-1, and am>kTm-1+kTm-2++kT+f1Tm-1+β1Tm.

Let (2.33)β1Tc+β1TMA+p1Tam-(kTm-1+kTm-2++kT+f1Tm-1+β1Tm)A1, that is, (2.34)0T|xm(t)|dtA1.

Noting (2.30) and (2.34), we have (2.35)|x(0)|=|x|A+Tm-1A1ω0,|x(i)|Tm-(i+1)A1ωi,i=1,2,,m-1.

Let ω=max0im{ωi+1}, and we take Ω={xxX:x<ω}; then Ω is an open and bounded set in X.

Similarly to Theorem 2.1, we prove easily that L is a Fredholm mapping of index zero and N is L-compact on Ω- and the conditions (a), (b), and (c) of Lemma 1.3 hold.

From above all, the requirements of Lemma 1.3 are all satisfied, so (*) has at least one T-periodic solution under the condition of Theorem 2.3, so far the proof of Theorem 2.3 is completed.

Remark 2.4.

In Theorem 2.3, if β(t)<0 and the condition (a) of Theorem 2.3 is when |x|>A, xg(x)<0, and the rest are unchangeable, then (*) has at least one T-periodic solution.

If the 0Tp(t)dt0, we have the following theorem.

Theorem 2.5.

Suppose that f, β, g, p are continuous for their variables, respectively, β(t+T)=β(t)>0, and meet the condition (a) of Theorem 2.1 and furthermore suppose as follows:

lim|x|+|g(x)|=+;

a,b,c>0, such that |g(x)|ag(x)+b|x|+c;

f1=supx|f(x)|(am-kTm-1-kTm-2--kT-f1Tm-1-bβ1Tm)/Tm-1,

where k=max{|ai|},i=1,2,,m-1, and am>kTm-1+kTm-2++kT+f1Tm-1+bβ1Tm, then (*) has at least one T-periodic solution.

Proof of Theorem <xref ref-type="statement" rid="thm3">2.5</xref>.

Banach spaces X,  Y and the mappings L, P, Q, and N are the same to Theorem 2.1, and their property are equal to Theorem 2.1, then the corresponding equation of Lx=λN(x,λ) is (2.36)i=1maix(i)(t)+λf(x(t))x˙(t)+λβ(t)g(x(x(t)))=λ2p(t).

Suppose that x=x(t)X is an arbitrary T-periodic solution of (2.36), put x(t) into (2.36), and then integrate both sides of (2.36) on [0,T], so (2.37)0Tβ(t)g(x(x(t)))dt=0Tλp(t)dt.

For the continuity of β, g, x, there must exist a number t1[0,T], such that (2.38)g(x(x(t1)))=λ0Tp(t)dt0Tβ(t)dt.

Combing the condition (a) of Theorem 2.5, there must exist A1>0, such that (2.39)|x(x(t1))|A1.

Similarly to Theorem 2.1, we have (2.40)|x(i)|Tm-(i+1)0T|x(m)(t)|dt,i=1,2,,m-1,(2.41)|x(0)|=|x|A1+Tm-10T|x(m)(t)|  dt.

By (2.36), (2.37), (2.39), and (2.41) and the conditions (b), (c) of Theorem 2.5, we have (2.42)0T|amx(m)|dt0T|λf(x(t))x˙(t)|dt+0T|λβ(t)g(x(x(t)))|dt+0T|λ2p(t)|dt+kT·Tm-(1+1)0T|xm(t)|dt+kT·Tm-(2+1)0T|xm(t)|dt++kt·Tm-(m-1+1)0T|xm(t)|dtf1T|x˙|+(kTm-1+kTm-2++am-1T)0T|xm(t)|dt+0Taβ(t)g(x(x(t)))dt+0Tbβ(t)[|x(x(t))|+c]dt+p1Tf1TTm-20T|x(m)(t)|dt+(kTm-1+kTm-2++kT)0T|xm(t)|dt+aβ1Tp1+bβ1T|x|+bβ1Tc+p1Tf1Tm-10T|x(m)(t)|dt+(kTm-1+kTm-2++kT)0T|xm(t)|dt+bβ1T(A1+Tm-10T|x(m)(t)|dt)+aβ1Tp1+bβ1Tc+p1T(kTm-1+kTm-2++kT+f1Tm-1+bβ1Tm)0T|xm(t)|dt+bβ1TA1+aβ1Tp1+bβ1Tc+p1T.

So (2.43)[am-(kTm-1+kTm-2++kT+f1Tm-1+bβ1Tm)]0T|xm(t)|dtbβ1TA1+aβ1Tp1+bβ1Tc+p1T.

Let (2.44)bβ1TA1+aβ1Tp1+bβ1Tc+p1Tam-(kTm-1+kTm-2++kT+f1Tm-1+bβ1Tm)A2, that is, (2.45)0T|xm(t)|dtA2.

Noting (2.40), (2.41), and (2.45), we have (2.46)|x(0)|=|x|A+Tm-1A2ω0,|x(i)|Tm-(i+1)A2ωi,i=1,2,,m-1.

For condition (a), there exist M0>0 and A0> 0, such that |x|>M0,  |g(x)|>A0; let ω=max0im{ωi+1,M0}, and we take Ω={xxX:x<ω}; then Ω is an open and bounded set in X.

Similarly to Theorem 2.1, we prove easily that L is a Fredholm mapping of index zero and N is L-compact on Ω- and the conditions (a), (b), and (c) of Lemma 1.3 hold.

From above all, the requirements of Lemma 1.3 are all satisfied, so (*) has at least one T-periodic solution under the condition of Theorem 2.5, so the proof of Theorem 2.5 is completed.

Remark 2.6.

In Theorem 2.5, if β(t)<0 and the condition (a) of Theorem 2.1 is when |x|>A, xg(x)<0, and the rest are unchangeable, then (*) has at least one T-periodic solution.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (11101305).

Zheng Z. Theory of Functional Diffrential Equation 1994 Hefei, China Anhui Education Press Wang H. Suo X. Periodic solutions of a type of second order functional differential equation with complex deviating argument Journal of Hebei Normal University 28 6 Liu X. Jia M. Ge W. Periodic solutions to a type of Duffing equation with complex deviating argument Applied Mathematics A 2003 181 51 56 Xiang Z. G. Liu C. M. Huang X. K. Periodic solutions of Liénard delay equations Journal of Jishou University 1998 19 4 35 40 1685292 Pascale E. Iannacci R. Periodic Solution of a GenerAlized Linard Equation with Delay 1983 1017 Berlin, Germany Springer Lecture Notes in Mathematics Liu B. Huang L. Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations Journal of Mathematical Analysis and Applications 2006 322 1 121 132 10.1016/j.jmaa.2005.08.069 2238153 ZBL1101.34054 Wang G. Yan J. Existence of periodic solution for first order nonlinear neutral delay equations Journal of Applied Mathematics and Stochastic Analysis 2001 14 2 189 194 10.1155/S1048953301000144 1838346 ZBL1009.34073 Lu S. Ge W. Existence of periodic solutions for a kind of second-order neutral functional differential equation Applied Mathematics and Computation 2004 157 2 433 448 10.1016/j.amc.2003.08.044 2088265 ZBL1059.34043 Lu S. Existence of periodic solutions to a p-Laplacian Liénard differential equation with a deviating argument Nonlinear Analysis. Theory, Methods & Applications A 2008 68 6 1453 1461 10.1016/j.na.2006.12.041 2388826 Fan G. Li Y. Existence of positive periodic solutions for a periodic logistic equation Applied Mathematics and Computation 2003 139 2-3 311 321 10.1016/S0096-3003(02)00182-0 1948643 ZBL1031.45005 Yang X. Multiple periodic solutions for a class of second order differential equations Applied Mathematics Letters 2005 18 1 91 99 10.1016/j.aml.2004.06.020 2121559 ZBL1075.34020 Huang C. He Y. Huang L. Tan W. New results on the periodic solutions for a kind of Rayleigh equation with two deviating arguments Mathematical and Computer Modelling 2007 46 5-6 604 611 10.1016/j.mcm.2006.11.024 2329595 ZBL1161.34345 Zhang Z. Wang Z. Periodic solutions of the third order functional differential equations Journal of Mathematical Analysis and Applications 2004 292 1 115 134 10.1016/j.jmaa.2003.11.059 2050220 ZBL1057.34098 Zhou J. Sun S. Liu Z. Periodic solutions of forced Liénard-type equations Applied Mathematics and Computation 2005 161 2 656 666 10.1016/j.amc.2003.12.055 2112431 ZBL1071.34041 Chen Y. Periodic solutions of a delayed periodic logistic equation Applied Mathematics Letters 2003 16 7 1047 1051 10.1016/S0893-9659(03)90093-0 2013071 ZBL1118.34327 Hale J. K. Mawhin J. Coincidence degree and periodic solutions of neutral equations Journal of Differential Equations 1974 15 295 307 0336004 10.1016/0022-0396(74)90081-3 ZBL0274.34070 Liu B.-W. Huang L.-H. Periodic solutions for a class of forced Liénard-type equations Acta Mathematicae Applicatae Sinica 2005 21 1 81 92 10.1007/s10255-005-0218-y 2123608 ZBL1093.34020 Mawhin J. Periodic solutions of some vector retarded functional differential equations Journal of Mathematical Analysis and Applications 1974 45 588 603 0333400 10.1016/0022-247X(74)90053-5 ZBL0275.34070 Chen S. Z. Existence of periodic solutions for a higher-order functional differential equation Pure and Applied Mathematics 2006 22 1 108 110 2258980 Gaines R. E. Mawhin J. L. Coincidence Degree, and Nonlinear Differential Equations 1977 568 Berlin, Germany Springer Lecture Notes in Mathematics 0637067