Han, Zhenlai

Li, Tongxing

Sun, Shurong

Zhang, Chenghui

Li Tongxing

Han Zhenlai

Sun Shurong

Zhang Chenghui

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2009, No. 60, 1-8; http://www.math.u-szeged.hu/ejqtde/Forced Oscillation of Second-order Nonlinear DynamicEquations on Time Scales ∗Tongxing Li aa School of Science, University of Jinan, Jinan, Shandong 250022, P R Chinae-mail: litongx2007@163.comZhenlai Han a,ba School of Science, University of Jinan, Jinan, Shandong 250022, P R Chinab School of Control Science and Engineering, Shandong University,Jinan, Shandong 250061, P R Chinae-mail: hanzhenlai@163.comShurong Sun a,ca School of Science, University of Jinan, Jinan, Shandong 250022, P R Chinac Department of Mathematics and Statistics, Missouri University ofScience and Technology, Rolla, Missouri 65409-0020, USAe-mail: sshrong@163.comChenghui Zhang bb School of Control Science and Engineering, Shandong University,Jinan, Shandong 250061, P R Chinae-mail: zchui@sdu.edu.cnAbstract: In this paper, by defining a class of functions, we establish some oscillationcriteria for the second order nonlinear dynamic equations with forced termx∆∆(t) + a(t)f(x(q(t))) = e(t)on a time scale T. Our results unify the oscillation of the second order forced differentialequation and the second order forced difference equation. An example is considered toillustrate the main results.Keywords: Oscillation; Forced term; Dynamic equations; Time scalesMathematics Subject Classification 2000: 34N05; 39A10; 39A211 IntroductionThe theory of time scales, which has recently received a lot of attention, was introduced by Hilgerin his PhD thesis in 1988 in order to unify continuous and discrete analysis (see Hilger [19]). Thearea of dynamic equations on time scales is a new component of applied analysis that describesprocesses that feature both continuous and discrete elements. Several authors have expoundedon various aspects of this new theory, see the survey paper by Agarwal et al. [1] and referencescited therein. A book on the subject of time scales, by Bohner and Peterson [7], summarizes andorganizes much of the time scale calculus. We refer also to the last book by Bohner and Peterson[8] for advances in dynamic equations on time scales.∗Corresponding author: Zhenlai Han, e-mail: hanzhenlai@163.com. This research is supported by theNatural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project(20080441126), Shandong Postdoctoral Funded Project (200802018) and supported by Shandong ResearchFunds (Y2008A28), also supported by University of Jinan Research Funds for Doctors (B0621, XBS0843).EJQTDE, 2009 No. 60, p. 1We are concerned here with second-order forced nonlinear dynamic equations of the formx∆∆(t) + a(t)f(x(q(t))) = e(t) (1.1)on a time scale T, where a(t), q(t) and e(t) are real-valued rd-continuous functions defined on T,the function q(t) also satisfies q : T → T, q(t) → ∞ as t → ∞, and f ∈ C(R, R), xf(x) > 0whenever x 6= 0.Agarwal and Grace [4], Ou and Wong [27], Wong [28] studied the n-order forced functionaldifferential equations of the formx(n)(t) + a(t)f(x(q(t))) = e(t).Çakmak and Tiryaki [11] studied (1.1) in the case when T = R. Sun and Saker [26] considered then-order forced delay difference equations of the form∆mx(n) + q(n)f(x(n − τ)) = e(n).In recent years, there has been much research activity concerning the oscillation of solutionsof various second order dynamic equations on time scales, we refer the reader to the articles [2, 3,5, 6, 9, 10, 12–17, 20, 21, 23–25] and references cited therein. Agarwal et al. [3] investigated thesecond order forced dynamic equations with mixed nonlinearities(r(t)φα(x∆))∆ + f(t, xσ) = e(t), t ∈ T,with f(t, x) = q(t)φα(x) +∑ni=1 qi(t)φβi(x), φ∗(u) = |u|∗−1u. Anderson [5] considered the oscilla-tions of the forced dynamic equations(rx∆)∆(t) + p(t)|x(τ(t))|α−1x(τ(t)) + q(t)|x(θ(t))|β−1x(θ(t)) = f(t), t ∈ T,and Anderson [6] studied the second order dynamic equations(rx∆)∆(t) + f(t, xσ(t), x∆(t)) = 0, t ∈ T.Bohner and Tisdell [10] examined oscillation and nonoscillation for(rx∆)∆(t) + p(t)xσ(t) = e(t), t ∈ T.Huang and Feng [20] considered the following second-order forced nonlinear dynamic equationsx∆∆(t) + p(t)f(xσ(t)) = e(t), t ∈ T,and in [21] the authors studied the oscillation of the forced dynamic equationsx∆∆(t) + p(t)f(x(t)) = e(t), t ∈ T.Oscillatory criteria for the forced dynamic equations(a(t)x∆(t))∆ + p(t)f(x(σ(t))) = r(t), t ∈ T,where∞∫t0|r(s)|∆s < ∞ are analyzed in [24].Philos-type functions [22] are used extensively in the theory of oscillations, for example, Erbeet al. [12], Han et al. [17], Saker et al. [25] established some Philos-type oscillation criteriafor the second-order delay dynamic equations on time scales. In [12], the authors utilized theclass of functions as follows: Assume H is defined for t0 ≤ s ≤ σ(t), t, s ∈ [t0,∞)T, H(t, s) ≥0, H(σ(t), t) = 0, H∆s(t, s) ≤ 0 for t ≥ s ≥ t0, and for each fixed t, H∆s(t, s) is delta integrablewith respect to s.As we are interested in oscillatory behavior, we assume throughout this paper that the giventime scale T is unbounded above. We assume t0 ∈ T and it is convenient to assume t0 > 0. Wedefine the time scale interval of the form [t0,∞)T by [t0,∞)T = [t0,∞) ∩ T.EJQTDE, 2009 No. 60, p. 22 Main ResultsIn this section, we give some new oscillation criteria for (1.1). Motivated by [22], let us introducethe class of functions H which will be extensively used in the sequel.Let D0 ≡ {(t, s) ∈ T2 : t > s ≥ t0} and D ≡ {(t, s) ∈ T2 : t ≥ s ≥ t0}. We say that the functionH ∈ Crd(D, R) belongs to the class H, if(i) H(t, t) = 0, t ≥ t0, H(t, s) > 0 on D0,(ii) H has a nonpositive continuous ∆-partial derivative H∆s(t, s) and a nonnegative continuoussecond-order ∆-partial derivative H∆2s2 (t, s) with respect to the second variable,(iii) H∆s(σ(t), σ(t)) = 0,(iv) limt→∞H∆s (σ(t),t0)H(σ(t),t0)= O(1).Theorem 2.1 Let a(t) ≥ 0. If there exists a function H ∈ H, such thatlim supt→∞1H(σ(t), t0)σ(t)∫t0H (σ(t), σ(s)) e(s)∆s = +∞ (2.1)andlim inft→∞1H(σ(t), t0)σ(t)∫t0H (σ(t), σ(s)) e(s)∆s = −∞, (2.2)then (1.1) is oscillatory.Proof. Let x(t) be a nonoscillatory solution of (1.1). Without loss of generality we assumex(t) > 0, x(q(t)) > 0 for t ≥ t0. Multiplying (1.1) by H (σ(t), σ(s)) , and integrating from t0 toσ(t), we haveσ(t)∫t0H (σ(t), σ(s)) x∆∆(s)∆s+σ(t)∫t0H (σ(t), σ(s)) a(s)f(x(q(s)))∆s =σ(t)∫t0H (σ(t), σ(s)) e(s)∆s. (2.3)Using the integration by parts formula two times, we getσ(t)∫t0H (σ(t), σ(s)) x∆∆(s)∆s = −H(σ(t), t0)x∆(t0) −σ(t)∫t0H∆s(σ(t), s)x∆(s)∆s= −H(σ(t), t0)x∆(t0) + H∆s(σ(t), t0)x(t0) +σ(t)∫t0H∆2s2 (σ(t), s)x(σ(s))∆s. (2.4)Substituting (2.4) into (2.3) and dividing through by H(σ(t), t0), we arrive at1H(σ(t), t0)σ(t)∫t0H (σ(t), σ(s)) e(s)∆s = −x∆(t0) +H∆s(σ(t), t0)H(σ(t), t0)x(t0)+1H(σ(t), t0)σ(t)∫t0[H∆2s2 (σ(t), s)xσ(s) + H (σ(t), σ(s)) a(s)f(x(q(s)))]∆s. (2.5)Taking lim inf as t → ∞, we derive a contradiction. The proof is complete.EJQTDE, 2009 No. 60, p. 3Remark 2.1 This theorem is true for (1.1), i.e., q(t) = t, and true for q(t) ≤ t and forq(t) ≥ t.When T = R, let H(t, s) = (t − s)β , β > 1. It is easy to see that H ∈ H. Therefore, we havethe following result.Corollary 2.1 Let a(t) ≥ 0 and T = R. Iflim supt→∞1(t − t0)βt∫t0(t − s)βe(s)ds = +∞andlim inft→∞1(t − t0)βt∫t0(t − s)βe(s)ds = −∞,then (1.1) is oscillatory.When T = Z, let H(t, s) = (t − s)(k) = (t − s)(t − s + 1) · · · (t − s + k − 1), k ≥ 2. It is easy tosee that H ∈ H. Therefore, we have the following result.Corollary 2.2 Let a(t) ≥ 0 and T = Z. Iflim supt→∞1(t + 1 − t0)(k)t∑s=t0(t − s)(k)e(s) = +∞andlim inft→∞1(t + 1 − t0)(k)t∑s=t0(t − s)(k)e(s) = −∞,then (1.1) is oscillatory.Theorem 2.2 Assume H ∈ H,(1) a(t) < 0 for t ≥ t0, q(t) = σ(t),(2) there exist two positive constants c and λ such that either|f(x)| ≥ c|x|λ, λ > 1, (2.6)or|f(x)| ≤ c|x|λ, 0 < λ < 1, (2.7)(3)lim supt→∞1H(σ(t), t0)σ(t)∫t0(H(σ(t), σ(s))e(s) − G(t, s)) ∆s = +∞, (2.8)(4)lim inft→∞1H(σ(t), t0)σ(t)∫t0(H(σ(t), σ(s))e(s) − G(t, s)) ∆s = −∞, (2.9)whereG(t, s) = (λ − 1)λλ1−λ (H∆2s2 (σ(t), s))λλ−1 (−cH (σ(t), σ(s)) a(s))11−λ .Then (1.1) is oscillatory.EJQTDE, 2009 No. 60, p. 4Proof. Let x(t) be a nonoscillatory solution of (1.1). Without loss of generality we assumex(t) > 0 for t ≥ t0. Assume first that (2.6) holds. Multiplying (1.1) by H (σ(t), σ(s)) , andintegrating from t0 to σ(t), we haveσ(t)∫t0H (σ(t), σ(s)) x∆∆(s)∆s+σ(t)∫t0H (σ(t), σ(s)) a(s)f(x(σ(s)))∆s =σ(t)∫t0H (σ(t), σ(s)) e(s)∆s. (2.10)Substituting (2.4) and (2.6) into (2.10) and note that a(t) < 0, we obtain1H(σ(t), t0)σ(t)∫t0H (σ(t), σ(s)) e(s)∆s ≤ −x∆(t0) +H∆s(σ(t), t0)H(σ(t), t0)x(t0)+1H(σ(t), t0)σ(t)∫t0[H∆2s2 (σ(t), s)x(σ(s)) + cH (σ(t), σ(s)) a(s)xλ(σ(s))]∆s. (2.11)Set F (x) = ax − bxλ for x > 0, a ≥ 0, b > 0. If λ > 1, then F (x) has the maximum Fmax =(λ − 1)λλ1−λ aλλ−1 b11−λ , (see [18]). Thus from (2.11), we have1H(σ(t), t0)σ(t)∫t0(H(σ(t), σ(s))e(s) − G(t, s))∆s ≤ −x∆(t0) +H∆s(σ(t), t0)H(σ(t), t0)x(t0).Taking lim sup as t → ∞ into the above inequality and applying (2.8), we obtain a desiredcontradiction. If condition (2.7) holds, substituting (2.4) and (2.6) into (2.10) and note thata(t) < 0, we obtain1H(σ(t), t0)σ(t)∫t0H (σ(t), σ(s)) e(s)∆s ≥ −x∆(t0) +H∆s(σ(t), t0)H(σ(t), t0)x(t0)+1H(σ(t), t0)σ(t)∫t0[H∆2s2 (σ(t), s)x(σ(s)) + cH (σ(t), σ(s)) a(s)xλ(σ(s))]∆s. (2.12)Set F (x) = ax − bxλ for x > 0, a ≥ 0, b > 0. If 0 < λ < 1, then F (x) has the minimumFmin = (λ − 1)λλ1−λ aλλ−1 b11−λ , (see [18]). Thus from (2.12), we have1H(σ(t), t0)σ(t)∫t0(H(σ(t), σ(s))e(s) − G(t, s))∆s ≥ −x∆(t0) +H∆s(σ(t), t0)H(σ(t), t0)x(t0).Taking lim inf as t → ∞ into the above inequality and applying (2.9), we obtain a desired contra-diction. The proof is complete.Theorem 2.3 Assume a(t) < 0, q(t) = σ(t), H ∈ H, conditions (2.1) and (2.2) hold. If|f(x)| ≥ |x| holds withH∆2s2 (σ(t), s) + H(σ(t), σ(s))a(s) ≤ 0, (2.13)or |f(x)| ≤ |x| holds withH∆2s2 (σ(t), s) + H(σ(t), σ(s))a(s) ≥ 0, (2.14)then (1.1) is oscillatory.EJQTDE, 2009 No. 60, p. 5Proof. Let x(t) be a nonoscillatory solution of (1.1). Without loss of generality we assumex(t) > 0 for t ≥ t0. Substituting (2.4) and |f(x)| ≥ |x| into (2.10) and note that a(t) < 0, we obtain1H(σ(t), t0)σ(t)∫t0H (σ(t), σ(s)) e(s)∆s ≤ −x∆(t0) +H∆s(σ(t), t0)H(σ(t), t0)x(t0)+1H(σ(t), t0)σ(t)∫t0[H∆2s2 (σ(t), s)x(σ(s)) + H (σ(t), σ(s)) a(s)x(σ(s))]∆s. (2.15)From (2.13), taking lim sup as t → ∞ into (2.15) and applying (2.1), we obtain a desired contra-diction. Substituting (2.4) and |f(x)| ≤ |x| into (2.10) and note that a(t) < 0, we obtain1H(σ(t), t0)σ(t)∫t0H (σ(t), σ(s)) e(s)∆s ≥ −x∆(t0) +H∆s(σ(t), t0)H(σ(t), t0)x(t0)+1H(σ(t), t0)σ(t)∫t0[H∆2s2 (σ(t), s)x(σ(s)) + H (σ(t), σ(s)) a(s)x(σ(s))]∆s. (2.16)Taking lim inf as t → ∞ into (2.16) and applying (2.2), we obtain a desired contradiction. Theproof is complete.Remark 2.2 The results in Saker [24] cannot be applied in (1.1) when∞∫t0|r(s)|∆s = ∞.3 ExampleAs an application, we consider the following example.Example Consider the equationx∆∆(t) + a(t)f(x(q(t))) = tα sin t, (3.1)where a(t) ≥ 0, α > 0. When T = R, by Corollary 2.1, (3.1) is oscillatory for α > 1. When T = Z,by Corollary 2.2, we have that (3.1) is oscillatory for α > 1.AcknowledgmentsThe authors sincerely thank the reviewers for their valuable suggestions and useful commentsthat have led to the present improved version of the original manuscript.References[1] R. P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: asurvey. J. Comput. Appl. Math., 141 (2002), 1–26.[2] R. P. Agarwal, M. Bohner, S. H. Saker, Oscillation of second order delay dynamic equations,Can. Appl. Math. Q. 13 (2005), 1–18.[3] R. P. Agarwal, A. 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O’Regan, Oscillation results for second-order nonlinear neutraldelay dynamic equations on time scales, Applicable Analysis, 86 (1) (2007), 1–17.[26] Y. G. Sun, S. H. Saker, Forced oscillation of higher order nonlinear difference equations, Appl.Math. Comput., 187 (2007), 868–872.[27] C. H. Ou, J. S. W. Wong, Forced oscillation of nth-order functional differential equations, J.Math. Anal. Appl., 262 (2001), 722–732.[28] J. S. W. Wong, Oscillation criteria for a forced second-order linear differential equation, J.Math. Anal. Appl., 231 (1999), 235–240.(Received August 26, 2009)EJQTDE, 2009 No. 60, p. 8

Forced oscillation of second-order nonlinear dynamic equations on time scales

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Forced oscillation of second-order nonlinear dynamic equations on time scales

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