In this paper, we investigate the existence and uniqueness of almost periodic solutions for a class of nonlinear Duffing system with time-varying delays. By using theory of exponential dichotomies and contraction mapping principle, we establish some new results and give an example to illustrate the theoretical analysis in this work

Xu, Yanli

Electronic Journal of Qualitative Theory of Differential Equations

Yanli Xu

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Existence and uniqueness of almost periodic solutions for a class of nonlinear Duffing system with time-varying delays

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Electronic Journal of Qualitative Theory of Differential Equations2012, No. 80, 1-9; http://www.math.u-szeged.hu/ejqtde/Existence and uniqueness of almost periodic solutions for aclass of nonlinear Duffing system with time-varying delays*Yanli Xu†Department of Mathematics, Xiangnan College, Chenzhou, Hunan 423000, P.R. ChinaAbstract: In this paper, we investigate the existence and uniqueness of almost periodicsolutions for a class of nonlinear Duffing system with time-varying delays. By using theoryof exponential dichotomies and contraction mapping principle, we establish some new resultsand give an example to illustrate the theoretical analysis in this work.Keywords: Almost periodic solution; Duffing system; exponential dichotomy; contrac-tion mapping principle.MR(2000) Subject Classification: 34C25; 34K13; 34K25.1. IntroductionIn recent years, the dynamic behaviors of nonlinear Duffing equations have been widelyinvestigated in [1-4] due to the application in many fields such as physics, mechanics, engi-neering, other scientific fields. In such applications, it is important to know the existence ofthe almost periodic solutions for nonlinear Duffing equations. Some results on existence ofthe almost periodic solutions were obtained in the literature. We refer the reader to [5−8]and the references cited therein.Recently, L. Q. Peng and W. T. Wang [9] considered the following model for nonlinearDuffing equation with a deviating argumentx′′(t) + cx′(t)− ax(t) + bxm(t− τ(t)) = p(t), (1.1)*Supported by the Construct Program of the Key Discipline in Hunan Province, the Science and Tech-nology Planning project of Chenzhou City in Hunan Province (grant no.[2011]29), and the Natural ScientificResearch Fund of Hunan Provincial Education Department of PR China (grant no. 11C1186).†Corresponding author. Tel.:+86 07352653203; fax: +86 07352653203. E-mail:xuyanliling@yahoo.cn.EJQTDE, 2012 No. 80, p. 1where τ(t) and p(t) are almost periodic functions on R, m > 1, a, b and c are constants. LetQ1(t) be a continuous and differentiable function on R. Definey =dxdt+ ξx−Q1(t), Q2(t) = p(t) + (ξ − c)Q1(t)−Q′1(t), (1.2)where ξ > 1 is a constant, then L. Q. Peng and W. T. Wang [9] transformed (1.1) into thefollowing systemdx(t)dt= −ξx(t) + y(t) +Q1(t),dy(t)dt= −(c− ξ)y(t) + (a− ξ(ξ − c))x(t) − bxm(t− τ(t)) +Q2(t).(1.3)Consequently, some sufficient conditions for the existence of positive almost periodic solutionsof (1.1) and (1.3) were established in [9]. On the other hand, in the real scientific fields, thecoefficients a, b and c in (1.1) and (1.3) are usually time-varying. Hence, the system (1.3)can be naturally extended to the following Duffing system with time-varying coefficients anddelays dx(t)dt= −δ1(t)x(t) + y(t) +Q1(t)dy(t)dt= δ2(t)y(t) + [α(t) − δ22(t)]x(t) − β(t)xm(t− τ(t)) +Q2(t),(1.4)where α(t), β(t), τ(t), δ1(t), δ2(t), Q1(t) and Q2(t) are almost periodic functions on R, m > 1is an integer, α(t) > 0, β(t) 6= 0. However, to the best of our knowledge, few authors haveconsidered the problem for almost periodic solutions of system (1.4). Motivated by the abovearguments, the main purpose of this present paper is to give the conditions to guarantee theexistence of almost periodic solutions of system (1.4).For convenience, we introduce some notations. Throughout this paper adopt the followingnotations:X = (x1, x2)T ∈ R2 to denote a column vector, in which the symbol (T ) denote thetranspose of a vector. We let |X| denote the absolute-value vector given by |X| = (|x1|, |x2|)T ,and define ||X|| = max1≤i≤2|xi|. A vector X ≥ 0 means that all xi are greater than or equal tozero. X > 0 is defined similarly. For vectors X and Y , X ≥ Y (resp. X > Y ) means thatX − Y ≥ 0(resp. X − Y > 0 ). Let δ1, δ2, δ∗, l, θ and q be defined asδ1 = inft∈R|δ1(t)|, δ2 = inft∈R|δ2(t)|, δ∗ = min{δ1, δ2}, (1.5)l = max{supt∈R|Q1(t)|δ∗,supt∈R|Q2(t)|δ∗}, θ = max{ 1δ∗,supt∈R[|α(t)− δ22(t)|+ |β(t)|]δ∗}, (1.6)q = max{ 1δ∗,supt∈R[|α(t) − δ22(t)|+ |β(t)|m( 2l1−θ )m−1]δ∗}. (1.7)EJQTDE, 2012 No. 80, p. 2SetB = {ϕ|ϕ = (ϕ1(t), ϕ2(t))T },where ϕ is an almost periodic function on R. For ∀ϕ ∈ B, we define the induced modulus||ϕ||B = supt∈R||ϕ(t)|| , then B is a Banach space.Definition 1.[10,11] Let u(t) : R → Rn be continuous in t. u(t) is said to be almostperiodic on R if, for any ε > 0, the set T (u, ε) = {λ : ||u(t + λ) − u(t)|| < ε,∀t ∈ R} isrelatively dense, i.e., for ε > 0, it is possible to find a real number L = L(ε) > 0 where,for any interval with length L(ε) , there exists a number λ = λ(ε) in this interval such that||u(t+ λ)− u(t)|| < ε, for all t ∈ R.Definition 2.[10,11] Let z ∈ Rn and Q(t) be an n × n continuous matrix defined on R.The linear systemdz(t)dt= Q(t)z(t), (1.8)is said to admit an exponential dichotomy on R if there exist positive constants k, α, projec-tion P and the fundamental solution matrix X(t) of (1.8) satisfying||X(t)PX−1(s)|| ≤ ke−α(t−s), for all t ≥ s,||X(t)(I − P )X−1(s)|| ≤ ke−α(s−t), for all t ≤ s.Lemma 1.1. [10,11] Let Q(t) = (qij) be an n × n almost periodic matrix defined on Rand let there exist a positive constant ν such that|qii(t)| −n∑j=1,j 6=i|qij(t)| ≥ ν, i = 1, 2, . . . , n.Then the linear system (1.8) admits an exponential dichotomy on R.Lemma 1.2. [10,11] If the linear system (1.8) admits an exponential dichotomy, then thealmost periodic systemdz(t)dt= Q(t)z(t) + g(t), (1.9)has a unique almost periodic solution z(t) andz(t) =∫ t−∞X(t)PX−1(s)g(s)ds −∫ +∞tX(t)(I − P )X−1(s)g(s)ds, (1.10)2. Existence and uniqueness of almost periodic solutionsEJQTDE, 2012 No. 80, p. 3Theorem 2.1. Assume δi(t) > 0, i = 1, 2, and let positive constants l, θ and q satisfyθ < 1,l1− θ < 1, q < 1. (2.1)Then there exists a unique almost periodic solution of system (1.4) in the regionB∗ = {ϕ|ϕ ∈ B, ||ϕ− ϕ0|| ≤ θl1− θ},whereϕ0 = (∫ t−∞e−∫tsδ1(w)dwQ1(s)ds,−∫ +∞te−∫stδ2(w)dwQ2(s)ds)T .Proof. For any ϕ ∈ B, we consider the almost periodic solution of the nonlinear almostperiodic two-dimensional system with time-varying delaysdx(t)dtdy(t)dt = −δ1(t) 00 δ2(t) x(t)y(t)+ ϕ2(t) +Q1(t)ϕ˜1(t) , (2.2)whereϕ˜1(t) = (α(t)− δ22(t))ϕ1(t)− β(t)ϕm1 (t− τ(t)) +Q2(t).Then, notice that the linear systemdx(t)dtdy(t)dt = −δ1(t) 00 δ2(t) x(t)y(t) , (2.3)admits an exponential dichotomy on R. Define a projection P by settingP = 1 00 0 .Thus, by Lemma 1.2, we obtain that the system (2.2) has exactly one almost periodic solution: xϕ(t)yϕ(t) =∫ t−∞ e−∫tsδ1(w)dw(ϕ2(s) +Q1(s))ds− ∫+∞t e−∫stδ2(w)dw[(α(s) − δ22(s))ϕ1(s)− β(s)ϕm1 (s − τ(s)) +Q2(s)]ds .Define a mapping T : B → B by settingT (ϕ)(t) = xϕ(t)yϕ(t) , ∀ϕ ∈ B.EJQTDE, 2012 No. 80, p. 4Since B∗ = {ϕ|ϕ ∈ B, ||ϕ−ϕ0|| ≤ θl1−θ}, it is easy to see that B∗ is a closed convex subset ofB . According to the definition of the norm in Banach space B , we derive||ϕ0||B = supt∈Rmax{∫ t−∞ e−∫tsδ1(w)dw|Q1(s)|ds,∫ +∞t e−∫stδ2(w)dw|Q2(s)|ds}≤ supt∈Rmax{supt∈R|Q1(t)|∫ t−∞ e−∫tsδ1(w)dwds, supt∈R|Q2(t)|∫ +∞t e−∫stδ2(w)dwds}≤ max{supt∈R|Q1(t)|δ1,supt∈R|Q2(t)|δ2}≤ max{supt∈R|Q1(t)|δ∗,supt∈R|Q2(t)|δ∗} = l.Therefore, for any ϕ ∈ B∗, we have||ϕ||B ≤ ||ϕ − ϕ0||B + ||ϕ0||B ≤ θl1− θ + l =l1− θ < 1. (2.4)Now, we prove that the mapping T is a self-mapping from B∗ to B∗ . In fact, for any ϕ ∈ B∗,from (2.4), we obtain||Tϕ− ϕ0||B= supt∈Rmax{| ∫ t−∞ e−∫tsδ1(w)dwϕ2(s)ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s)− δ22(s))ϕ1(s)− β(s)ϕm1 (s− τ(s))]ds|}≤ supt∈Rmax{||ϕ||B∫ t−∞ e−∫tsδ1(w)dwds,∫ +∞t e−∫stδ2(w)dw[|α(s)− δ22(s)|||ϕ||B+ |β(s)|||ϕ||mB ]ds}≤ supt∈Rmax{∫ t−∞ e−∫tsδ1(w)dwds,∫+∞t e−∫stδ2(w)dw[|α(s) − δ22(s)|+ |β(s)|]ds}||ϕ||B≤ supt∈Rmax{∫ t−∞ e−∫tsδ1(w)dwds, supt∈R[|α(t) − δ22(t)|+ |β(t)|]∫ +∞t e−∫stδ2(w)dwds}||ϕ||B≤ max{ 1δ∗,supt∈R[|α(t)−δ22(t)|+|β(t)|]δ∗}||ϕ||B= θ||ϕ||B≤ θl1−θ ,which implies that Tϕ ∈ B∗. So, the mapping T is a self-mapping from B∗ to B∗ .Next, we prove that the mapping T is a contraction mapping of the B∗. In deed, in viewof (1.5), (1.6), (1.7), (2.1), (2.4) and differential mean-value theorem, for all ϕ,ψ ∈ B∗, weEJQTDE, 2012 No. 80, p. 5have|T (ϕ(t)) − T (ψ(t))|= (|(T (ϕ(t)) − T (ψ(t)))1|, |(T (ϕ(t)) − T (ψ(t)))2|)T= (| ∫ t−∞ e−∫tsδ1(w)dw[ϕ2(s)− ψ2(s)]ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))(ϕ1(s)− ψ1(s))−β(s)(ϕm1 (s− τ(s))− ψm1 (s− τ(s)))]ds|)T= (| ∫ t−∞ e−∫tsδ1(w)dw[ϕ2(s)− ψ2(s)]ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))(ϕ1(s)− ψ1(s))−β(s)m(ψ1(s− τ(s)) + h(s)(ϕ1(s− τ(s))− ψ1(s − τ(s)))m−1(ϕ1(s− τ(s))− ψ1(s− τ(s)))]ds|)T= (| ∫ t−∞ e−∫tsδ1(w)dw[ϕ2(s)− ψ2(s)]ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))(ϕ1(s)− ψ1(s))−β(s)m((1− h(s))ψ1(s− τ(s)) + h(s)ϕ1(s− τ(s)))m−1(ϕ1(s− τ(s))− ψ1(s− τ(s)))]ds|)T ,where h(s) ∈ (0, 1) is the mean point in Lagrange’s mean value theorem. Then,|T (ϕ(t))− T (ψ(t))|≤ (∫ t−∞ e−∫tsδ1(w)dwds sups∈R|ϕ2(s)− ψ2(s)|,∫ +∞t e−∫stδ2(w)dw{|α(s)− δ22(t)|·sups∈R|ϕ1(s)− ψ1(s)|+ |β(s)|m[sups∈R|ψ1(s− τ(s))|+ sups∈R|ϕ1(s− τ(s))|]m−1·sups∈R|ϕ1(s− τ(s))− ψ1(s− τ(s))|}ds|)T≤ (∫ t−∞ e−∫tsδ1(w)dwds||ϕ− ψ||B ,∫+∞t e−∫stδ2(w)dwds sups∈R[|α(s) − δ22(s)|+|β(s)|m( 2l1−θ )m−1]||ϕ− ψ||B)T ,which yields that||T (ϕ)− T (ψ)||B ≤ max{ 1δ∗,supt∈R[|α(t) − δ22(t)|+ |β(t)|m( 2l1−θ )m−1]δ∗}||ϕ − ψ||B = q||ϕ− ψ||B .It follow from q < 1 that the mapping T is a contraction. Therefore, the mapping T possessesa unique fixed point z∗ = (x∗(t), y∗(t))T ∈ B∗ ,Tz∗ = z∗. By (2.2), z∗ satisfies (1.4). Soz∗ is an almost periodic solution of system (1.4) in B∗ . The proof of Theorem 2.1 is nowcompleted.If δi(t) < 0, i = 1, 2, we define a projection P by settingP = 0 00 1 ,then using similar arguments to those above, we can show the followingEJQTDE, 2012 No. 80, p. 6Theorem 2.2. If δi(t) < 0, i = 1, 2 , and assume that (2.1) holds. Then, there exists aunique almost periodic solution of system (1.4) in the regionB∗1 = {ϕ|ϕ ∈ B, ||ϕ− ϕ1|| ≤θl1− θ},whereϕ1 = (−∫ +∞te−∫stδ1(w)dwQ1(s)ds,∫ t−∞e−∫tsδ2(w)dwQ2(s)ds)T .3. An exampleExample 3.1. The nonlinear Duffing equation with a deviating argumentx′′(t) + (sin t− cos t)x′(t) + (626 + 27 cos t+ 25 sin t+ cos2 t+ cos t sin t)x(t)+ (1 + sin2 t)x3(t− cos t) = cos√2t+ cos√3t− 50 cos t− 2 cos2 t, (3.1)has exactly one almost periodic solution x∗(t) such thatmax{supt∈R|x∗(t)|, supt∈R|dx∗(t)dt+ (25 + sin t)x∗(t)− 2 cos t|} ≤ 111.Proof. Setδ1(t) = 25 + sin t, δ2(t) = 25 + cos t, y(t) =dx(t)dt+ (25 + sin t)x(t)− 2 cos t, (3.2)we can transform (3.1) into the following system:dx(t)dt= −(25 + sin t)x(t) + y(t) + 2 cos t,dy(t)dt= (25 + cos t)y(t) + (2− sin2 t)x(t) sin2 t− (2− cos2 t)x3(t− cos t)+ cos√2t+ cos√3t.(3.3)Since α(t) = 626 + 50 cos t+ 2cos2 t , β(t) = 2− cos2 t , Q1(t) = −2 cos t, Q2(t) = cos√2t+cos√3t, m = 3 , δ∗ = 25, thenl = max{supt∈R|Q1(t)|δ∗,supt∈R|Q2(t)|δ∗} = 225θ = max{ 1δ∗,supt∈R[|α(t) − δ22(t)|+ |β(t)|]δ∗} = 325q = max{ 1δ∗,supt∈R[|α(t) − δ22(t)|+ |β(t)|m( 2l1−θ )m−1]δ∗} = 2543025EJQTDE, 2012 No. 80, p. 7It is straight forward to check that all assumptions needed in Theorem 2.1 are satisfied.Hence, equation (3.1) has exactly one almost periodic solution x∗(t) such thatmax{supt∈R|x∗(t)|, supt∈R|dx∗(t)dt+ (25 + sin t)x∗(t)− 2 cos t|} ≤ l1− θ =111.The proof of example 3.1 is now completed.Remark 3.1. Since equation (3.1) is a very simple form of nonlinear Duffing equationwith time-varying coefficientc(t) = sin t− cos t, a(t) = −(626+27 cos t+25 sin t+cos2 t+cos t sin t) and b(t) = 1+sin2 t,it is clear that the results obtained in [9] are invalid for the above example. Moreover, onecan find that main theorem in [8] is a special one of Theorem 2.1 when δ1(t) and δ2(t) areconstants. Thus, the results of this paper substantially extend and improve the main resultsof refs. [8-9].AcknowledgmentThe author would like to express the sincere appreciation to the reviewer for his/herhelpful comments in improving the presentation and quality of this paper. Moreover, theauthor expresses her sincere gratitude to Professor Jianli Li and Doctor Jing Xiao for thehelpful discussion when this work is carried out.References[1] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Aca-demic Press, Orlando, FL, 1985.[2] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.[3] T. Yashizaw, Asymptotic behavior of solutions of differential equations, in: Differential Equation: Qualita-tive Theory (Szeged, 1984), in: Colloq. Math. Soc. Ja´nos Bolyai, vol. 47, North-Holland,Amsterdam,1987, pp. 1141-1172.[4] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, NewYork, 1993.[5] M. S. Berger, Y. Y. Chen, Forced quasi periodic and almost periodic oscillations of nonlinear Duffingequations, Nonlinear Anal. 19 (3) (1992) 249-257.[6] W. Y. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica, New Ser. 13(3) (1997) 373-380.EJQTDE, 2012 No. 80, p. 8[7] Q. Y. Wang, The existence and uniqueness of almost periodic solutions for nonlinear differential equationswith time lag, Acta Math. Sinica 42 (3) (1999) 511-518 (in Chinese).[8] Q. Y. Zhou, B. W. Liu, New results on almost periodic solution for a class of nonlinear Duffing equationswith a deviating argument, Applied Mathematics Letters. 22 (2009) 6-11.[9] L. Q. Peng, W. T. Wang, Positive Almost Periodic Solutions for a Class of Nonlinear Duffing Equationswith a Deviating Argument, Electronic Journal of Qualitative Theory of Differential Equations 2010,No. 6, 1-12; http://www.math.u-szeged.hu/ejqtde/.[10] A. M. Fink, Almost periodic differential equations, in: Lecture Notes in Mathematics, vol. 377, Springer,Berlin, 1974, pp. 80-112.[11] C. Y. He, Almost Periodic Differential Equation, Higher Education Publishing House, Beijing, 1992, pp.90-100 (in Chinese).(Received July 21, 2012)EJQTDE, 2012 No. 80, p. 9

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Xu Yanli

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2012, No. 80, 1-9; http://www.math.u-szeged.hu/ejqtde/Existence and uniqueness of almost periodic solutions for aclass of nonlinear Duffing system with time-varying delays*Yanli Xu†Department of Mathematics, Xiangnan College, Chenzhou, Hunan 423000, P.R. ChinaAbstract: In this paper, we investigate the existence and uniqueness of almost periodicsolutions for a class of nonlinear Duffing system with time-varying delays. By using theoryof exponential dichotomies and contraction mapping principle, we establish some new resultsand give an example to illustrate the theoretical analysis in this work.Keywords: Almost periodic solution; Duffing system; exponential dichotomy; contrac-tion mapping principle.MR(2000) Subject Classification: 34C25; 34K13; 34K25.1. IntroductionIn recent years, the dynamic behaviors of nonlinear Duffing equations have been widelyinvestigated in [1-4] due to the application in many fields such as physics, mechanics, engi-neering, other scientific fields. In such applications, it is important to know the existence ofthe almost periodic solutions for nonlinear Duffing equations. Some results on existence ofthe almost periodic solutions were obtained in the literature. We refer the reader to [5−8]and the references cited therein.Recently, L. Q. Peng and W. T. Wang [9] considered the following model for nonlinearDuffing equation with a deviating argumentx′′(t) + cx′(t) − ax(t) + bxm(t− τ(t)) = p(t), (1.1)*Supported by the Construct Program of the Key Discipline in Hunan Province, the Science and Tech-nology Planning project of Chenzhou City in Hunan Province (grant no.[2011]29), and the Natural ScientificResearch Fund of Hunan Provincial Education Department of PR China (grant no. 11C1186).†Corresponding author. Tel.:+86 07352653203; fax: +86 07352653203. E-mail:xuyanliling@yahoo.cn.EJQTDE, 2012 No. 80, p. 1where τ(t) and p(t) are almost periodic functions on R, m > 1, a, b and c are constants. LetQ1(t) be a continuous and differentiable function on R. Definey =dxdt+ ξx−Q1(t), Q2(t) = p(t) + (ξ − c)Q1(t) −Q′1(t), (1.2)where ξ > 1 is a constant, then L. Q. Peng and W. T. Wang [9] transformed (1.1) into thefollowing systemdx(t)dt= −ξx(t) + y(t) +Q1(t),dy(t)dt= −(c− ξ)y(t) + (a− ξ(ξ − c))x(t) − bxm(t− τ(t)) +Q2(t).(1.3)Consequently, some sufficient conditions for the existence of positive almost periodic solutionsof (1.1) and (1.3) were established in [9]. On the other hand, in the real scientific fields, thecoefficients a, b and c in (1.1) and (1.3) are usually time-varying. Hence, the system (1.3)can be naturally extended to the following Duffing system with time-varying coefficients anddelaysdx(t)dt= −δ1(t)x(t) + y(t) +Q1(t)dy(t)dt= δ2(t)y(t) + [α(t) − δ22(t)]x(t) − β(t)xm(t− τ(t)) +Q2(t),(1.4)where α(t), β(t), τ(t), δ1(t), δ2(t), Q1(t) and Q2(t) are almost periodic functions on R, m > 1is an integer, α(t) > 0, β(t) 6= 0. However, to the best of our knowledge, few authors haveconsidered the problem for almost periodic solutions of system (1.4). Motivated by the abovearguments, the main purpose of this present paper is to give the conditions to guarantee theexistence of almost periodic solutions of system (1.4).For convenience, we introduce some notations. Throughout this paper adopt the followingnotations:X = (x1, x2)T ∈ R2 to denote a column vector, in which the symbol (T ) denote thetranspose of a vector. We let |X| denote the absolute-value vector given by |X| = (|x1|, |x2|)T ,and define ||X|| = max1≤i≤2|xi|. A vector X ≥ 0 means that all xi are greater than or equal tozero. X > 0 is defined similarly. For vectors X and Y , X ≥ Y (resp. X > Y ) means thatX − Y ≥ 0(resp. X − Y > 0 ). Let δ1, δ2, δ∗, l, θ and q be defined asδ1 = inft∈R|δ1(t)|, δ2 = inft∈R|δ2(t)|, δ∗ = min{δ1, δ2}, (1.5)l = max{supt∈R|Q1(t)|δ∗,supt∈R|Q2(t)|δ∗}, θ = max{ 1δ∗,supt∈R[|α(t) − δ22(t)| + |β(t)|]δ∗}, (1.6)q = max{ 1δ∗,supt∈R[|α(t) − δ22(t)| + |β(t)|m( 2l1−θ )m−1]δ∗}. (1.7)EJQTDE, 2012 No. 80, p. 2SetB = {ϕ|ϕ = (ϕ1(t), ϕ2(t))T },where ϕ is an almost periodic function on R. For ∀ϕ ∈ B, we define the induced modulus||ϕ||B = supt∈R||ϕ(t)|| , then B is a Banach space.Definition 1.[10,11] Let u(t) : R → Rn be continuous in t. u(t) is said to be almostperiodic on R if, for any ε > 0, the set T (u, ε) = {λ : ||u(t + λ) − u(t)|| < ε,∀t ∈ R} isrelatively dense, i.e., for ε > 0, it is possible to find a real number L = L(ε) > 0 where,for any interval with length L(ε) , there exists a number λ = λ(ε) in this interval such that||u(t+ λ) − u(t)|| < ε, for all t ∈ R.Definition 2.[10,11] Let z ∈ Rn and Q(t) be an n × n continuous matrix defined on R.The linear systemdz(t)dt= Q(t)z(t), (1.8)is said to admit an exponential dichotomy on R if there exist positive constants k, α, projec-tion P and the fundamental solution matrix X(t) of (1.8) satisfying||X(t)PX−1(s)|| ≤ ke−α(t−s), for all t ≥ s,||X(t)(I − P )X−1(s)|| ≤ ke−α(s−t), for all t ≤ s.Lemma 1.1. [10,11] Let Q(t) = (qij) be an n × n almost periodic matrix defined on Rand let there exist a positive constant ν such that|qii(t)| −n∑j=1,j 6=i|qij(t)| ≥ ν, i = 1, 2, . . . , n.Then the linear system (1.8) admits an exponential dichotomy on R.Lemma 1.2. [10,11] If the linear system (1.8) admits an exponential dichotomy, then thealmost periodic systemdz(t)dt= Q(t)z(t) + g(t), (1.9)has a unique almost periodic solution z(t) andz(t) =∫ t−∞X(t)PX−1(s)g(s)ds −∫ +∞tX(t)(I − P )X−1(s)g(s)ds, (1.10)2. Existence and uniqueness of almost periodic solutionsEJQTDE, 2012 No. 80, p. 3Theorem 2.1. Assume δi(t) > 0, i = 1, 2, and let positive constants l, θ and q satisfyθ < 1,l1 − θ < 1, q < 1. (2.1)Then there exists a unique almost periodic solution of system (1.4) in the regionB∗ = {ϕ|ϕ ∈ B, ||ϕ− ϕ0|| ≤θl1 − θ},whereϕ0 = (∫ t−∞e−∫tsδ1(w)dwQ1(s)ds,−∫ +∞te−∫stδ2(w)dwQ2(s)ds)T .Proof. For any ϕ ∈ B, we consider the almost periodic solution of the nonlinear almostperiodic two-dimensional system with time-varying delaysdx(t)dtdy(t)dt =−δ1(t) 00 δ2(t)x(t)y(t) +ϕ2(t) +Q1(t)ϕ̃1(t) , (2.2)whereϕ̃1(t) = (α(t) − δ22(t))ϕ1(t) − β(t)ϕm1 (t− τ(t)) +Q2(t).Then, notice that the linear systemdx(t)dtdy(t)dt =−δ1(t) 00 δ2(t)x(t)y(t) , (2.3)admits an exponential dichotomy on R. Define a projection P by settingP =1 00 0 .Thus, by Lemma 1.2, we obtain that the system (2.2) has exactly one almost periodic solution:xϕ(t)yϕ(t) =∫ t−∞ e−∫tsδ1(w)dw(ϕ2(s) +Q1(s))ds− ∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))ϕ1(s) − β(s)ϕm1 (s − τ(s)) +Q2(s)]ds .Define a mapping T : B → B by settingT (ϕ)(t) =xϕ(t)yϕ(t) , ∀ϕ ∈ B.EJQTDE, 2012 No. 80, p. 4Since B∗ = {ϕ|ϕ ∈ B, ||ϕ−ϕ0|| ≤ θl1−θ}, it is easy to see that B∗ is a closed convex subset ofB . According to the definition of the norm in Banach space B , we derive||ϕ0||B = supt∈Rmax{∫ t−∞ e−∫tsδ1(w)dw|Q1(s)|ds,∫ +∞t e−∫stδ2(w)dw|Q2(s)|ds}≤ supt∈Rmax{supt∈R|Q1(t)|∫ t−∞ e−∫tsδ1(w)dwds, supt∈R|Q2(t)|∫ +∞t e−∫stδ2(w)dwds}≤ max{supt∈R|Q1(t)|δ1,supt∈R|Q2(t)|δ2}≤ max{supt∈R|Q1(t)|δ∗,supt∈R|Q2(t)|δ∗} = l.Therefore, for any ϕ ∈ B∗, we have||ϕ||B ≤ ||ϕ − ϕ0||B + ||ϕ0||B ≤θl1 − θ + l =l1 − θ < 1. (2.4)Now, we prove that the mapping T is a self-mapping from B∗ to B∗ . In fact, for any ϕ ∈ B∗,from (2.4), we obtain||Tϕ− ϕ0||B= supt∈Rmax{| ∫ t−∞ e−∫tsδ1(w)dwϕ2(s)ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))ϕ1(s)− β(s)ϕm1 (s− τ(s))]ds|}≤ supt∈Rmax{||ϕ||B∫ t−∞ e−∫tsδ1(w)dwds,∫ +∞t e−∫stδ2(w)dw[|α(s) − δ22(s)|||ϕ||B+ |β(s)|||ϕ||mB ]ds}≤ supt∈Rmax{∫ t−∞ e−∫tsδ1(w)dwds,∫ +∞t e−∫stδ2(w)dw[|α(s) − δ22(s)| + |β(s)|]ds}||ϕ||B≤ supt∈Rmax{∫ t−∞ e−∫tsδ1(w)dwds, supt∈R[|α(t) − δ22(t)| + |β(t)|]∫ +∞t e−∫stδ2(w)dwds}||ϕ||B≤ max{ 1δ∗,supt∈R[|α(t)−δ22(t)|+|β(t)|]δ∗}||ϕ||B= θ||ϕ||B≤ θl1−θ ,which implies that Tϕ ∈ B∗. So, the mapping T is a self-mapping from B∗ to B∗ .Next, we prove that the mapping T is a contraction mapping of the B∗. In deed, in viewof (1.5), (1.6), (1.7), (2.1), (2.4) and differential mean-value theorem, for all ϕ,ψ ∈ B∗, weEJQTDE, 2012 No. 80, p. 5have|T (ϕ(t)) − T (ψ(t))|= (|(T (ϕ(t)) − T (ψ(t)))1|, |(T (ϕ(t)) − T (ψ(t)))2|)T= (|∫ t−∞ e−∫tsδ1(w)dw[ϕ2(s) − ψ2(s)]ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))(ϕ1(s) − ψ1(s))−β(s)(ϕm1 (s− τ(s)) − ψm1 (s− τ(s)))]ds|)T= (|∫ t−∞ e−∫tsδ1(w)dw[ϕ2(s) − ψ2(s)]ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))(ϕ1(s) − ψ1(s))−β(s)m(ψ1(s− τ(s)) + h(s)(ϕ1(s− τ(s)) − ψ1(s − τ(s)))m−1(ϕ1(s− τ(s)) − ψ1(s− τ(s)))]ds|)T= (|∫ t−∞ e−∫tsδ1(w)dw[ϕ2(s) − ψ2(s)]ds|, |∫ +∞t e−∫stδ2(w)dw[(α(s) − δ22(s))(ϕ1(s) − ψ1(s))−β(s)m((1 − h(s))ψ1(s− τ(s)) + h(s)ϕ1(s− τ(s)))m−1(ϕ1(s− τ(s)) − ψ1(s− τ(s)))]ds|)T ,where h(s) ∈ (0, 1) is the mean point in Lagrange’s mean value theorem. Then,|T (ϕ(t)) − T (ψ(t))|≤ (∫ t−∞ e−∫tsδ1(w)dwds sups∈R|ϕ2(s) − ψ2(s)|,∫ +∞t e−∫stδ2(w)dw{|α(s) − δ22(t)|·sups∈R|ϕ1(s) − ψ1(s)| + |β(s)|m[sups∈R|ψ1(s− τ(s))| + sups∈R|ϕ1(s− τ(s))|]m−1·sups∈R|ϕ1(s− τ(s)) − ψ1(s− τ(s))|}ds|)T≤ (∫ t−∞ e−∫tsδ1(w)dwds||ϕ− ψ||B ,∫ +∞t e−∫stδ2(w)dwds sups∈R[|α(s) − δ22(s)|+|β(s)|m( 2l1−θ )m−1]||ϕ− ψ||B)T ,which yields that||T (ϕ) − T (ψ)||B ≤ max{1δ∗,supt∈R[|α(t) − δ22(t)| + |β(t)|m( 2l1−θ )m−1]δ∗}||ϕ − ψ||B = q||ϕ− ψ||B .It follow from q < 1 that the mapping T is a contraction. Therefore, the mapping T possessesa unique fixed point z∗ = (x∗(t), y∗(t))T ∈ B∗ ,Tz∗ = z∗. By (2.2), z∗ satisfies (1.4). Soz∗ is an almost periodic solution of system (1.4) in B∗ . The proof of Theorem 2.1 is nowcompleted.If δi(t) < 0, i = 1, 2, we define a projection P by settingP =0 00 1 ,then using similar arguments to those above, we can show the followingEJQTDE, 2012 No. 80, p. 6Theorem 2.2. If δi(t) < 0, i = 1, 2 , and assume that (2.1) holds. Then, there exists aunique almost periodic solution of system (1.4) in the regionB∗1 = {ϕ|ϕ ∈ B, ||ϕ− ϕ1|| ≤θl1 − θ},whereϕ1 = (−∫ +∞te−∫stδ1(w)dwQ1(s)ds,∫ t−∞e−∫tsδ2(w)dwQ2(s)ds)T .3. An exampleExample 3.1. The nonlinear Duffing equation with a deviating argumentx′′(t) + (sin t− cos t)x′(t) + (626 + 27 cos t+ 25 sin t+ cos2 t+ cos t sin t)x(t)+ (1 + sin2 t)x3(t− cos t) = cos√2t+ cos√3t− 50 cos t− 2 cos2 t, (3.1)has exactly one almost periodic solution x∗(t) such thatmax{supt∈R|x∗(t)|, supt∈R|dx∗(t)dt+ (25 + sin t)x∗(t) − 2 cos t|} ≤ 111.Proof. Setδ1(t) = 25 + sin t, δ2(t) = 25 + cos t, y(t) =dx(t)dt+ (25 + sin t)x(t) − 2 cos t, (3.2)we can transform (3.1) into the following system:dx(t)dt= −(25 + sin t)x(t) + y(t) + 2 cos t,dy(t)dt= (25 + cos t)y(t) + (2 − sin2 t)x(t) sin2 t− (2 − cos2 t)x3(t− cos t)+ cos√2t+ cos√3t.(3.3)Since α(t) = 626 + 50 cos t+ 2cos2 t , β(t) = 2 − cos2 t , Q1(t) = −2 cos t, Q2(t) = cos√2t+cos√3t, m = 3 , δ∗ = 25, thenl = max{supt∈R|Q1(t)|δ∗,supt∈R|Q2(t)|δ∗} = 225θ = max{ 1δ∗,supt∈R[|α(t) − δ22(t)| + |β(t)|]δ∗} = 325q = max{ 1δ∗,supt∈R[|α(t) − δ22(t)| + |β(t)|m( 2l1−θ )m−1]δ∗} = 2543025EJQTDE, 2012 No. 80, p. 7It is straight forward to check that all assumptions needed in Theorem 2.1 are satisfied.Hence, equation (3.1) has exactly one almost periodic solution x∗(t) such thatmax{supt∈R|x∗(t)|, supt∈R|dx∗(t)dt+ (25 + sin t)x∗(t) − 2 cos t|} ≤ l1 − θ =111.The proof of example 3.1 is now completed.Remark 3.1. Since equation (3.1) is a very simple form of nonlinear Duffing equationwith time-varying coefficientc(t) = sin t− cos t, a(t) = −(626+27 cos t+25 sin t+cos2 t+cos t sin t) and b(t) = 1+sin2 t,it is clear that the results obtained in [9] are invalid for the above example. Moreover, onecan find that main theorem in [8] is a special one of Theorem 2.1 when δ1(t) and δ2(t) areconstants. Thus, the results of this paper substantially extend and improve the main resultsof refs. [8-9].AcknowledgmentThe author would like to express the sincere appreciation to the reviewer for his/herhelpful comments in improving the presentation and quality of this paper. Moreover, theauthor expresses her sincere gratitude to Professor Jianli Li and Doctor Jing Xiao for thehelpful discussion when this work is carried out.References[1] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Aca-demic Press, Orlando, FL, 1985.[2] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.[3] T. Yashizaw, Asymptotic behavior of solutions of differential equations, in: Differential Equation: Qualita-tive Theory (Szeged, 1984), in: Colloq. Math. Soc. János Bolyai, vol. 47, North-Holland,Amsterdam,1987, pp. 1141-1172.[4] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, NewYork, 1993.[5] M. S. Berger, Y. Y. Chen, Forced quasi periodic and almost periodic oscillations of nonlinear Duffingequations, Nonlinear Anal. 19 (3) (1992) 249-257.[6] W. Y. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica, New Ser. 13(3) (1997) 373-380.EJQTDE, 2012 No. 80, p. 8[7] Q. Y. Wang, The existence and uniqueness of almost periodic solutions for nonlinear differential equationswith time lag, Acta Math. Sinica 42 (3) (1999) 511-518 (in Chinese).[8] Q. Y. Zhou, B. W. Liu, New results on almost periodic solution for a class of nonlinear Duffing equationswith a deviating argument, Applied Mathematics Letters. 22 (2009) 6-11.[9] L. Q. Peng, W. T. Wang, Positive Almost Periodic Solutions for a Class of Nonlinear Duffing Equationswith a Deviating Argument, Electronic Journal of Qualitative Theory of Differential Equations 2010,No. 6, 1-12; http://www.math.u-szeged.hu/ejqtde/.[10] A. M. Fink, Almost periodic differential equations, in: Lecture Notes in Mathematics, vol. 377, Springer,Berlin, 1974, pp. 80-112.[11] C. Y. He, Almost Periodic Differential Equation, Higher Education Publishing House, Beijing, 1992, pp.90-100 (in Chinese).(Received July 21, 2012)EJQTDE, 2012 No. 80, p. 9

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Existence and uniqueness of almost periodic solutions for a class of nonlinear Duffing system with time-varying delays

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