Fabao Gao

FB Gao

HM Cheung

Minghui Yao

P Amster

R Manásevich

RE Gaines

S Aizicovici

Shiping Lu

Advances in Difference Equations

English

Crossref

Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 DOI 10.1186/s13662-015-0465-zRESEARCH Open AccessPeriodic solutions for Liénard typeequation with time-variable coefﬁcientFabao Gao1,2*, Shiping Lu3 and Minghui Yao4*Correspondence:gaofabao@sina.com;gaofabao@vt.edu1School of Mathematical Science,Yangzhou University, Yangzhou,225002, China2Department of EngineeringScience and Mechanics, VirginiaTech, Blacksburg, VA 24061, USAFull list of author information isavailable at the end of the articleAbstractBy means of topological theory, a class of Liénard type equations with a deviatingargument of the form (ϕp(x′(t)))′ + f (x(t – τ ))x′(t – τ ) + β(t)g(x(t – τ )) = e(t) is studied. Itis notable that the coeﬃcient β(t) in front of g(x(t – τ )) is allowed to change sign inthis paper. Moreover, a numerical simulation is carried out to verify the validity of theobtained results. In addition, the generalized form of the above equation withtime-varying delays is also discussed brieﬂy.Keywords: periodic solution; p-Laplacian; topological theory; time-varyingcoeﬃcient1 IntroductionConsider the Liénard type p-Laplacian diﬀerential equation(ϕp(x′(t)))′ + f(x(t – τ ))x′(t – τ ) + β(t)g(x(t – τ ))= e(t), (.)where p >  and ϕp : R → R is given by ϕp(s) = |s|p–s for s =  and ϕp() = ,∫ T β(s)ds = ,f , g, e ∈ C(R,R), and e(t) is T-periodic, ∫ T e(s)ds = , T , τ >  are given constants. Theequation can be applied to the dynamics of ﬂuids. An example is Euler’s equation, govern-ing the ﬂow of an ideal ﬂuid in a conservative force ﬁeld. While there are plenty of resultson the existence of periodic solutions for the p-Laplacian equation (see [–] and refer-ences therein), studying delay Liénard equations with a variable coeﬃcient in front of thenonlinear term is relatively uncommon. The main diﬃculty lies in ﬁnding a priori boundsfor (.) requiring the coeﬃcient of the nonlinear term to keep a ﬁxed sign.In the aforementioned literature, based on the Leray-Schauder degree theory, Amsteret al. [] considered the existence of at least one periodic solution for the p-Laplacian-likesystemwith a ﬁxed delay. Gao and Zhang [] discussed an n-dimensional p-Laplacian-likeneutral functional diﬀerential equation, and they established some criteria to guaranteethe existence of periodic solutions for the equation by using Mawhin’s continuation the-orem. Manásevich and Se¸dziwy [] were concerned with the existence and uniqueness ofa limit circle for a generalized Liénard type equation, which involves the one-dimensionalp-Laplacian operator and a positive small parameter. Manásevich and Mawhin [] stud-ied the existence of periodic solutions to some system cases involving the fairly generalvector-valued p-Laplacian operator. Aizicovici et al. [] investigated a nonlinear periodic© 2015 Gao et al.; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commonslicense, and indicate if changes were made.Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 2 of 9problem driven by the scalar p-Laplacian with a nonsmooth potential via the degree map;they proved the existence of at least three distinct nontrivial solutions, two of which havea constant sign. In Gaines and Mawhin’s monograph [], coincidence degree theory wasdescribed and used to study alternative problems. Aizicovici et al. [] conducted periodicproblems driven by the scalar p-Laplacianwith amultivalued right-hand side nonlinearity.Gao et al. [, ] considered the existence of periodic solutions for two kinds of Rayleightype p-Laplacian equations by using the continuation theorem. Cheung [] proposed theexistence of periodic solutions of a p-Laplacian Rayleigh equation with two deviating ar-guments.Based on topological theory and some analysis techniques, the existence of periodic so-lutions for (.) is investigated in the present paper. It is signiﬁcant that the coeﬃcient β(t)of the nonlinear term can change sign, which cannot be achieved in most of the previouspapers. Furthermore, a numerical simulation is performed to validate the feasibility of theobtained results. In addition, the ﬁxed delays in (.) are also extended to the time-varyingdelays and we brieﬂy discuss them. Moreover, the approaches used to estimate a prioribounds of periodic solutions are diﬀerent from the corresponding ones in the literature.2 PreliminariesLet us consider the problem: ﬁnd u ∈ CT such that(ϕp(u′(t)))′ = f˜(t,u(t – τ ),u′(t – τ )). (.)Lemma . (see Amster et al. []) Let  ∈ CT be an open set. Assume that:(A) For λ ∈ (, ] the problem(ϕp(u′(t)))′ = λ˜f(t,u(t – τ ),u′(t – τ ))has no solution on ∂.(A) The equationF˜(a) T∫ Tf˜ (t,a, ) = has no solution on ∂ ∩ Rn.(A) The Brouwer degreedeg(F˜ , ∩ Rn, ) = .Then the problem (.) has at least one solution in CT .3 Main resultsFor the sake of convenience, we only study the periodic solutions of (.) in the case∫ T β(t)dt >  (∫ T β(t)dt <  can be discussed in the same way).Theorem . Assume that the following conditions hold:Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 3 of 9(H) There are positive constants m,m, and d such thatm|u|p– ≤∣∣g(u)∣∣ ≤m|u|p–, ∀|u| > d.(H) There is a constant r ≥  such that lim|u|→∞ |F(u)||u|p– ≤ r, where F(x) =∫ x f (s)ds.(H)A :=⎧⎪⎨⎪⎩[ (β∞+ε)mTm∫ T (β(t)+β∞+ε)dt]p– –pp– < ,  < p≤ ,[ (β∞+ε)mTm∫ T (β(t)+β∞+ε)dt]p– < , p > ,where ε > , β∞ = maxt∈[,T] |β(t)|.(H) ug(u) > , ∀|u| > d.Then (.) has at least one T-periodic solution, if Cpr[ T(–A) ]p– +β∞mT(–A)p < , whereCp ={,  < p≤ ,p–, p > .Proof Consider the homotopic equation of (.) as follows:(ϕp(x′(t)))′ + λf(x(t – τ ))x′(t – τ ) + λβ(t)g(x(t – τ ))= λe(t), λ ∈ (, ]. (.)For ∀ε > , β∞ = maxt∈[,T] |β(t)|, (.) can be written in the following form:(ϕp(x′(t)))′ + λf(x(t – τ ))x′(t – τ ) + λ(β(t) + β∞ + ε)g(x(t – τ ))= λ(β∞ + ε)g(x(t – τ ))+ λe(t), λ ∈ (, ]. (.)Integrating both sides of (.) from  to T and using the integral mean value theorem,there exists a constant ξ ∈ (,T) such thatg(x(ξ – τ ))∫ T(β(t) + β∞ + ε)dt =∫ T(β∞ + ε)g(x(t – τ ))dt. (.)Now, we claim that∣∣x(ξ – τ )∣∣ ≤ A|x|∞ + B, (.)whereA =⎧⎪⎨⎪⎩[ (β∞+ε)mTm∫ T (β(t)+β∞+ε)dt]p– –pp– ,  < p≤ ,[ (β∞+ε)mTm∫ T (β(t)+β∞+ε)dt]p– , p > ,B =⎧⎪⎨⎪⎩[ (β∞+ε)MgTm∫ T (β(t)+β∞+ε)dt]p– –pp– + d,  < p≤ ,[ (β∞+ε)MgTm∫ T (β(t)+β∞+ε)dt]p– + d, p > ,Mg = max|u|≤d∣∣g(u)∣∣.Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 4 of 9Case : If |x(ξ – τ )| ≤ d, then (.) holds clearly.Case : If |x(ξ – τ )| > d. DenoteE ={t : t ∈ [,T], ∣∣x(t – τ )∣∣ ≤ d}, E ={t : t ∈ [,T], ∣∣x(t – τ )∣∣ > d}.From (.), we have∣∣g(x(ξ – τ ))∣∣∫ T(β(t) + β∞ + ε)dt ≤ (β∞ + ε)(∫E+∫E)∣∣g(x(t – τ ))∣∣dt,which, together with assumption (H), leads to∣∣x(ξ – τ )∣∣p–m∫ T(β(t) + β∞ + ε)dt ≤ (β∞ + ε)(∫Em∣∣x(t – τ )∣∣p– dt +MgT)≤ (β∞ + ε)mT |x|p–∞ + (β∞ + ε)MgT ,i.e.,∣∣x(ξ – τ )∣∣ ≤[ (β∞ + ε)mTm∫ T (β(t) + β∞ + ε)dt] p––pp– |x|∞+[ (β∞ + ε)MgTm∫ T (β(t) + β∞ + ε)dt] p––pp– ,  < p≤ ,∣∣x(ξ – τ )∣∣ ≤[ (β∞ + ε)mTm∫ T (β(t) + β∞ + ε)dt] p– |x|∞+[ (β∞ + ε)MgTm∫ T (β(t) + β∞ + ε)dt] p–, p > .Thus, from Case  and Case , we see that (.) holds.Let ξ – τ = kT + ξ¯ , where k is an integer and ξ¯ ∈ [,T]; noticing (.), we get∣∣x(t)∣∣ ≤ A|x|∞ + B +∫ tξ¯∣∣x′(s)∣∣ds, t ∈ [ξ¯ , ξ¯ + T],and∣∣x(t – T)∣∣ ≤ A|x|∞ + B +∫ ξ¯t–T∣∣x′(s)∣∣ds, t ∈ [ξ¯ , ξ¯ + T].Combining the above two inequalities, we obtain|x|∞ = maxt∈[ξ¯ ,ξ¯+T]∣∣x(t)∣∣≤ maxt∈[ξ¯ ,ξ¯+T]{A|x|∞ + B + (∫ tξ¯∣∣x′(s)∣∣ds +∫ ξ¯t–T∣∣x′(s)∣∣ds)}≤ A|x|∞ + B + ∫ T∣∣x′(s)∣∣ds.Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 5 of 9In view of (H), we have|x|∞ ≤ B –A +∫ T |x′(s)|ds( –A) . (.)On the other hand, noticing Cpr[ T(–A) ]p– +β∞mT(–A)p < , we easily see that there is a suﬃ-ciently small constant ε >  such thatCp(r + ε)[ T( –A)]p–+ β∞mT( –A)p < . (.)By assumption (H) and for such ε, we know there exists a constant ρ > d (independentof λ) such that∣∣F(u)∣∣ ≤ (r + ε)|u|p–, |u| > ρ. (.)Let ={t : t ∈ [,T], ∣∣x(t – τ )∣∣ ≤ ρ},  ={t : t ∈ [,T], ∣∣x(t – τ )∣∣ > ρ}.Multiplying both sides of (.) by x(t) and integrating them with [,T], noticing (.), weget∫ T∣∣x′(t)∣∣p dt = –∫ T(ϕp(x′(t)))′x(t)dt= λ∫ Tf(x(t – τ ))x′(t – τ )x(t)dt+ λ∫ Tβ(t)g(t,x(t – τ ))x(t)dt – λ∫ Te(t)x(t)dt≤(∫+∫)∣∣F(x(t – τ ))x′(t)∣∣dt+ β∞|x|∞(∫E+∫E)∣∣g(x(t – τ ))∣∣dt + e∞|x|∞T≤ [Mρ + (r + ε)|x|p–∞] ∫ T∣∣x′(t)∣∣dt + β∞mT |x|p∞+ β∞MgT |x|∞ + e∞T |x|∞, (.)whereMρ = max|u|≤ρ |F(u)|, e∞ = maxt∈[,T] |e(t)|.Substituting (.) into (.) yields∫ T∣∣x′(t)∣∣p dt ≤ (r + ε)[ B –A +∫ T |x′(t)|dt( –A)]p– ∫ T∣∣x′(t)∣∣dt+ β∞mT[ B –A +∫ T |x′(t)|dt( –A)]p+[Mρ +β∞MgT( –A) +e∞T( –A)]∫ T∣∣x′(t)∣∣p dtGao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 6 of 9+ (β∞Mg + e∞)BT –A≤ Cp(r + ε)[( B –A)p–+(∫ T∣∣x′(t)∣∣dt( –A))p–]∫ T∣∣x′(t)∣∣dt+ β∞mT( –A)p(∫ T∣∣x′(t)∣∣dt)p+[Mρ +β∞MgT( –A) +e∞T( –A)]Tq(∫ T∣∣x′(t)∣∣p dt) p+ (β∞Mg + e∞)BT –A +p–β∞mBpT( –A)p≤[Cp(r + ε)[ T( –A)]p–+ β∞mT( –A)p]∫ T∣∣x′(t)∣∣p dt+ θ(∫ T∣∣x′(t)∣∣p dt) p+ θ, (.)whereCp ={,  < p≤ ,p–, p > ,θ = Cp(r + ε)( B –A)p–Tq +[Mρ +β∞MgT( –A) +e∞T( –A)]Tq ,θ = (β∞Mg + e∞)BT –A +p–β∞mBpT( –A)p ,e∞ = maxt∈[,T]∣∣e(t)∣∣.In view of (.) and p < , it follows from (.) that there is a constantM >  such that∫ T∣∣x′(t)∣∣p dt ≤M, (.)which implies that there exists a constantM >  such that∣∣x′∣∣∞ ≤M. (.)From (.) and (.), we can see that there exists a constantM such that|x|∞ ≤ B –A +TqMp( –A) M. (.)According to (.) and (.), set  = {x : |x′|∞ <M + , |x|∞ <M + }, then we see that(.) has no solution on ∂ for λ ∈ (, ], and when x(t) =M +  or –M – , from (H),we can get– T∫ Tβ(t)g(M + )dt < ,Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 7 of 9– T∫ Tβ(t)g(–M – )dt < .So, condition (A) of Lemma . is also satisﬁed.DenoteH(x,μ) = μx +  –μT∫ Tβ(t)g(x)dt,and when x ∈ ∂ ∩ R, μ ∈ [, ], in view of ∫ T β(s)ds > , we havexH(x,μ) = μx + ( –μ)xT∫ Tβ(t)g(x)dt > .Thus, H(x,μ) is a homotopic transformation anddeg {˜F , ∩ R, } = deg{– T∫ Tβ(t)g(x)dt, ∩ R, }= deg {x, ∩ R, } = .Therefore, condition (A) of Lemma . is also satisﬁed. By using Lemma ., we concludethat (.) has at least one T-periodic solution x(t) on ¯ with |x|∞ ≤ M. This completesthe proof of Theorem .. 4 GeneralizationAs a matter of fact, (.) can also be extended to the time-varying case, which admits thefollowing form:(ϕp(x′(t)))′ + f(x(t – τ ))x′(t – τ ) + β(t)g(x(t – δ(t)))= e(t), (.)where δ(t + T) ≡ δ(t), δ′(t) < , and ∫ T β(s)ds =  (the case of∫ T β(s)ds >  will be con-sidered in this section, and the other case,∫ T β(s)ds < , can be studied in the same way).Then, according to Mawhin’s continuation theorem, the following result is obtained witha similar process to Theorem ..Theorem . Assume that the following conditions hold:(C) There exist constants r > , r > , and d ≥  such thatr|u|p– ≤∣∣g(u)∣∣ ≤ r|u|p– and ug(u) > , ∀|u| > d.(C) There are constant r >  and d ≥  such that |f (u)| ≤ r|u|p–, ∀|u| > d.(C) There is a constant ε >  such thatM :=⎧⎪⎨⎪⎩[ (|β–|∞+ε)rTr∫ T (β++ε)dt]p– –pp– < ,  < p≤ ,[ (|β–|∞+ε)rTr∫ T (β++ε)dt]p– < , p > ,where β+ = maxt∈[,T] {β(t), }, β– = maxt∈[,T] {–β(t), }.Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 8 of 9Figure 1 Phase diagram of the system for τ = 0.02.Then (.) has at least one T-periodic solution ifCprTp– + p–rθ |β|∞Tp < ( –M)p–,where θ = maxt∈[,T] –δ′(γ (t)) and γ (t) denotes the inverse of function t – δ(t).5 ExampleFor convenience, as an application of Theorem ., we consider the following example:(ϕ(x′(t)))′ + f(x(t – τ ))x′(t – τ ) +(sin t + )g(x(t – τ ))= cost, (.)where p = , f (t) = t sin t, , β(t) = sin t + , g(u) =u, + , e(t) = cost, then F(u) =∫ u f (s)ds =, (–u cosu + u sinu +  cosu – ), and β∞ = . So we can choose m =, ,m =, , d = √, r = , , ε =  so that the conditions of Theorem . hold.Therefore, by Theorem . we ﬁnd that (.) has at least one π -periodic solution, whichcan also be illustrated by numerical simulation.By using MATLAB© .. (Ra) toolkit: dde, which can be used to solve delaydiﬀerential equations with constant delays, (.) is simulated on tspan = [., .] withhistory = [.,–.] for t ≤  (see Figure ). It can be found from Figure  that the equationadmits one periodic solution with periodicity ., which is around π . Therefore, theresults achieved in this paper are signiﬁcant.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors contributed equally to this work. All authors read and approved the ﬁnal version of the manuscript.Author details1School of Mathematical Science, Yangzhou University, Yangzhou, 225002, China. 2Department of Engineering Scienceand Mechanics, Virginia Tech, Blacksburg, VA 24061, USA. 3College of Mathematics & Statistics, Nanjing University ofInformation Science & Technology, Nanjing, 210044, China. 4College of Mechanical Engineering, Beijing University ofTechnology, Beijing, 100124, China.Gao et al. Advances in Diﬀerence Equations  (2015) 2015:125 Page 9 of 9AcknowledgementsThis work is supported by the National Natural Science Foundation of China (NSFC) through grant Nos. 11302187,11271197, 11172009, 11372015 and 61473340. The authors are extremely grateful to the referees for their careful reviewof the manuscript and insightful comments, which helped to improve the paper greatly. The authors are also verygrateful to Ms. Xiaojing Li for her original insightful ideas.Received: 8 October 2014 Accepted: 10 April 2015References1. Amster, P, De Nápoli, PL, Mariani, MC: Periodic solutions for p-Laplacian-like systems with delay. Dyn. Contin. DiscreteImpuls. Syst., Ser. A Math. Anal. 13, 311-319 (2006)2. Gao, FB, Zhang, W: Periodic solutions for a p-Laplacian-like NFDE system. J. Franklin Inst. 348, 1020-1034 (2011)3. Manásevich, R, Se¸dziwy, S: p-Laplacian and Liénard type equation. Rocky Mt. J. Math. 27, 611-617 (1997)4. Manásevich, R, Mawhin, J: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Diﬀer. Equ. 45,367-393 (1998)5. Aizicovici, S, Papageorgiou, NS, Staicu, V: Multiple nontrivial solutions for nonlinear periodic problems with thep-Laplacian. J. Diﬀer. Equ. 243, 504-535 (2007)6. Gaines, RE, Mawhin, J: Coincidence Degree and Nonlinear Diﬀerential Equations. Springer, Berlin (1977)7. Aizicovici, S, Papageorgiou, NS, Staicu, V: Periodic solutions for second order diﬀerential inclusions with the scalarp-Laplacian. J. Math. Anal. Appl. 322, 913-929 (2006)8. Gao, FB, Lu, SP, Zhang, W: Periodic solutions for a Rayleigh type p-Laplacian equation with sign-variable coeﬃcient ofnonlinear term. Appl. Math. Comput. 216, 2010-2015 (2010)9. Gao, FB, Lu, SP: Periodic solutions for a Rayleigh type equation with a variable coeﬃcient ahead of the nonlinearterm. Nonlinear Anal., Real World Appl. 10, 254-258 (2009)10. Cheung, HM: On p-Laplacian equations with deviating arguments. Master thesis, The University of Hong Kong (2009)

Periodic solutions for Liénard type equation with time-variable coefficient

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Periodic solutions for Liénard type equation with time-variable coefficient

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