A discrete analogue of Lyapunov-type inequalities for nonlinear systems

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1 Computers Mathematics with Applications A discrete analogue of Lyapunov-type inequalities for nonlinear systems Mehmet Ünal a,, Devrim Çakmak b, Aydın Tiryaki c a Department of Mathematics Computer Science, Bahçeşehir University, Beşiktaş, İstanbul, Turkey b Department of Mathematics Education, Faculty of Education, Gazi University, Teknikokullar-Ankara, Turkey c Department of Mathematics, Faculty of Arts Science, Gazi University, Teknikokullar-Ankara, Turkey Received 11 December 2006; received in revised form 10 September 2007; accepted 10 October 2007 Abstract In this paper, by using elementary analysis, we establish some new Lyapunov-type inequalities for nonlinear systems of difference equations when the coefficent β 2 t is not necessarily nonnegative valued when the end points are not necessarily usual zeros, but rather, generalized zeros. Applying these inequalities, we obtain a disconjugacy criterion boundedness for the solution of our system. Some special cases of our results contain recently developed Lyapunov inequalities for discrete linear Hamiltonian systems. The inequalities obtained here can be used as hy tools in the study of the qualitative behaviour of solutions of the associated equations. c 2007 Elsevier Ltd. All rights reserved. Keywords: Discrete Hamiltonian system; Lyapunov-type inequality; Discongugacy; Generalized zero; Weakly oscillatory 1. Introduction We shall be interested in obtaining Lyapunov-type inequalities for the discrete nonlinear systems of the form xt = α 1 txt β 1 t ut γ 2 ut ut = β 2 t xt + 1 β 2 xt + 1 α 1 tut, t Z, 1 where γ > 1 β > 1 are constants, the functions α 1 t, β 1 t, β 2 t are real valued with β 1 t > 0 1 α 1 t 0 for all t Z, denotes the forward difference operator, that is, xt := xt + 1 xt. We recall that a nontrivial solution x, u of the discrete nonlinear system 1 defined for t Z is said to be proper if sup { xs + us : s t} > 0 for t Z. In the discrete case, instead of the usual zero, the concept of a generalized zero is used. A function f : Z R is said to have a generalized zero at t 0 Z provided either f t 0 = 0 or f t 0 1 f t 0 < 0. A proper solution x, u of the discrete nonlinear system 1 is called weakly oscillatory if at least one component has a sequence of generalized zeros tending to +. This solution is said to be oscillatory if Corresponding author. addresses: munal@bahcesehir.edu.tr M. Ünal, dcakmak@gazi.edu.tr D. Çakmak, tiryaki@gazi.edu.tr A. Tiryaki /$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi: /j.camwa

2 2632 M. Ünal et al. / Computers Mathematics with Applications both components have sequences of generalized zeros tending to +. If both components at least one component are eventually positive or negative, then the solution x, u is called nonoscillatory weakly nonoscillatory. System 1 is said to be oscillatory if all solutions are oscillatory. Before we give the precise formulation of Lyapunov-type inequalities for the system 1, we mention a few background details which serve to motivate the results of this paper. The behaviour of solutions for equations of the type x t + qtxt = 0, t R 2 2 xt + qtxt + 1 = 0, t Z 3 have been studied extensively in the literature for continuous discrete settings, respectively. In fact, Lyapunov, in his celebrated paper [1], proved that if xt is a solution of 2 with xa = 0 = xba < b xt 0 for t a, b, then the so called Lyapunov inequality b a b a qs ds > 4 4 holds, the constant 4 can not be replaced by a larger number. This result has found applications in the study of various properties of solutions of 2 such as oscillation, disconjugacy eigenvalue problems. Since an exhaustive list of references is impossible due to the incredible number of papers devoted to this subject, we are forced to confine ourselves to those papers which have contributed the above result: Reid [2,3], Hartman [4], Hochstadt [5], Eliason [6], Singh [7], Kwong [8] Cheng [9]. In fact, Hartman in [4] has obtained an inequality which is more general than 4. Inequality 4 has been generalized to second order nonlinear differential equations by Eliason [10] Pachpatte [11], to delay differential equations of the second order by Eliason [12], by Dahiya Singh [13], to higher order differential equations by Pachpatte [14]. Lyapunov-type inequalities for the Emden Fowler type equations can be found in Pachpatte s paper [11]. Lyapunov-type inequalities for the half-linear equation were obtained independently by Lee et al. [15] by Pinasco [16]. The proof for the half-linear extension can be found in Došlý Rehák s recent book [17, p. 190]. A thorough literature review of continuous discrete Lyapunov inequalities their applications can be found in the survey paper [18] by S. S. Cheng the references quoted therein. Although there is an extensive literature on the Lyapunov-type inequalities for the above mentioned equations, there is not much done for the linear Hamiltonian system x t = α 1 txt + β 1 tut u t = β 2 txt α 1 tut 5 for the nonlinear systems of differential equations of the form x t = α 1 txt + β 1 t ut γ 2 ut u t = β 2 t xt β 2 xt α 1 tut. 6 We refer the reader to the introductory papers by Guseinov Kaymakçalan [19] for 5 by Tiryaki et al. [20] for 6, respectively. There has been much attention paid to the existence of proper solutions in both general special cases of the system 6. For a comprehensive treatment of the subject, we refer the reader to the books by Došlý Řehák [17], Kiguradze Chanturia [21] Mirzov [22], the paper by Kitano Kusano [23]. The purpose of this paper is to obtain a discrete analogue of Lyapunov-type inequalities for nonlinear systems 1. As far as we know, the discrete nonlinear systems 1 have never been the subject of investigation in this direction before. We remark that the discrete Hamiltonian system, in case of two scalar linear difference equations, has the form xt = atxt btut ut = ctxt + 1 atut, t Z, 7

3 M. Ünal et al. / Computers Mathematics with Applications where the coefficient at satisfying the condition 1 at 0 for all t Z. Clearly 7 is a special case of the nonlinear discrete system 1 with γ = β = 2. Also notice that the second order difference equation pt xt + qtxt + 1 = 0, t Z, 8 where pt > 0 for all t Z, can be written as an equivalent discrete Hamiltonian system of type 7. Indeed, let xt be a solution of 8 set ut = pt xt. Then we obtain xt = 1 ut, ut = qtxt + 1. pt So, 8 is equivalent to 7 with at 0, bt = 1 pt ct = qt. We also remark that the Emden Fowler type difference equation pt xt α 2 xt + qt xt + 1 β 2 xt + 1 = 0, 9 the half-linear difference equation pt xt α 2 xt + qt xt + 1 α 2 xt + 1 = 0, 10 where α > 1 β > 1 are constants, p q are real functions pt > 0 for all t Z, can clearly be written as special cases of the discrete nonlinear systems 1. Being motivated by the recent papers by Pachpatte [11], Lee et al. [15], Guseinov Kaymakçalan [19], Tiryaki et al. [20] Patula [24], we will set up prove our main theorems for the discrete nonlinear systems 1 in Section 2. The inequalities that we propose here can be used as a hy tool in the study of the qualitative nature of solutions. We give some applications to show the importance of our results in Section Main results Since our attention is restricted to the Lyapunov-type inequalities for the discrete nonlinear systems, we shall assume the existence of a nontrivial real solution x, u of the system 1. The main results of this paper are the following theorems. Theorem 1. Suppose β 1 t > 0 for all t Z. Let n, m Z with n m 2. Assume 1 has a real solution x, u such that xn = xm = 0 x is not identically zero on [n, m]. Then the inequality m 1 1/γ m 2 1/α α 1 t + M β α 1 β 1 t β + 2 t 2 11 holds, where α is the conjugate number to γ, i.e., β + 2 t = max{β 2t, 0}. 1 γ + 1 α = 1, M = xτ = max n+1 t m 1 xt Proof. Let xt, ut be nontrivial real solution of system 1 such that xn = xm = 0 xt is not identically zero on [n, m]. Then multiplying the first equation of 1 by ut the second one by xt + 1, then adding them up yields xtut = β 1 t ut γ β 2 t xt + 1 β. 12 Summing the last equation from n to m 1 taking into account that xn = xm = 0, we get 0 = β 1 t ut γ β 2 t xt + 1 β. 13

4 2634 M. Ünal et al. / Computers Mathematics with Applications Since xm = 0, we have β 1 t ut γ = β 2 t xt + 1 β β + 2 t xt + 1 β. 14 Choose τ [n + 1, m 1] such that M = xτ = max xt. n+1 t m 1 Hence xτ > 0. Summing the first equation of 1, first from n to τ 1, then from τ to m 1, we obtain, respectively, xτ = α 1 txt β 1 t ut γ 2 ut t=τ xτ = α 1 txt β 1 t ut γ 2 ut. t=τ Taking absolute values of above two equalities yield xτ α 1 t xt β 1 t ut γ 1 xτ = xτ α 1 t xt β 1 t ut γ 1, t=τ t=τ respectively. Adding the last two inequalities, we obtain 2 xτ α 1 t xt β 1 t ut γ On the other h, applying Hölder s inequality to the second sum of the right h side of 15 with the indices α γ, we have β 1 t ut γ 1 = m 1 1 γ β + α 1 1 t ut γ 1 1/γ m 1 1/α β 1 t β 1 t ut γ 1α where 1 γ + 1 α = 1/γ m 1 1/α β 1 t β 1 t ut γ, 16 m 1 = 1. Hence using 14, we obtain from 16 that β 1 t ut γ 1 m 1 1/γ m 2 1/α β 1 t β + 2 t xt + 1 β. Substituting the last inequality into 15 yields m 1 1/γ m 2 1/α 2 xτ α 1 t xt β 1 t β + 2 t xt + 1 β

5 M. Ünal et al. / Computers Mathematics with Applications xτ α 1 t + xτ β α 1/γ m 2 1/α β 1 t β + 2 t. m 1 Dividing the latter inequality by xτ, we obtain inequality 11. Theorem 2. Suppose 1 α 1 t > 0 β 1 t > 0 for all t Z. Let n, m Z with n m 2. Assume 1 has a real solution x, u such that xn = 0 xm 1xm < 0. Then the inequality m 2 1/γ m 2 1/α α 1 t + M β α 1 β 1 t β + 2 t > 1 17 holds, where β, γ, α, M β + 2 t are defined as before. Proof. Choose τ [n + 1, m 1] such that M = xτ = max xt. n+1 t m 1 Hence xτ > 0. Summing the first equation of 1, at first from n to τ 1, taking into account that xn = 0, we obtain Hence, xτ = α 1 txt β 1 t ut γ 2 ut. xτ α 1 t xt β 1 t ut γ 1 α 1 t xt β 1 t ut γ 1 α 1 t xt = α 1 t xt m 2 1/γ m 2 1/α β 1 t β 1 t ut γ 1α m 2 1/γ m 2 1/α β 1 t β 1 t ut γ. 18 Now summing Eq. 12 from n to m 2 taking into account that xn = 0, we obtain xm 1um 1 = β 1 t ut γ β 2 t xt + 1 β. 19 In addition, from the first equation of 1, we have, for t = m 1, 1 α 1 m 1 xm = xm 1 + β 1 m 1 um 1 γ 2 um 1. Multiplying the last equality by xm 1 yields 1 α 1 m 1 xmxm 1 = x 2 m 1 + β 1 m 1 um 1 γ 2 um 1xm 1. Since 1 α 1 t > 0, β 1 t > 0 for all t Z xmxm 1 < 0, the above latter equality implies that um 1xm 1 < 0 must hold. Hence, it follows from 19 that the inequality β 1 t ut γ < β 2 t xt + 1 β β + 2 t xt + 1 β

6 2636 M. Ünal et al. / Computers Mathematics with Applications holds. Substituting this string of relations into 18 yields m 2 1/γ m 2 1/α xτ < α 1 t xt β 1 t β + 2 t xt + 1 β xτ α 1 t + xτ β α m 2 Hence, dividing by xτ completes the proof. 1/γ m 2 1/α β 1 t β + 2 t. The proof of the following theorem can be obtained easily by the same method used in above theorem, with a slight modification. Hence it is omitted. Theorem 3. Suppose 1 α 1 t > 0 β 1 t > 0 for all t Z. Let n, m Z with n m 1. Assume that 1 has a real solution x, u such that xn 1xn < 0, xm = 0. Then the inequality β m 1 1/γ m 2 1/α α α 1 t + M 1 1 β 1 t β + 2 t > holds, where β, γ, α β + 2 t are defined as before M 1 = xτ = max n t m 1 xt. Theorem 4. Suppose 1 α 1 t > 0, β 1 t > 0 β 2 t > 0 for all t Z. Let n, m Z with n m 1. Assume 1 has a real solution x, u such that xn 1xn < 0 xm 1xm < 0, xt 0 for all t [n, m 1]. Then the inequality 1 γ m 1 1/α m 2 1/β α α 1 t + M 1 2 β 1 t β 2 t > holds, where α is the conjugate number to β M 2 = uτ 0 = max n 1 τ 1 uτ. Proof. Suppose that xt 0 for all t [n, m 1]. Let m 0 denote the smallest integer in [n, m] such that m 0 n xm 0 1xm 0 < 0. Then x does not have any generalized zeroes in [n + 1, m 0 1], without loss of generality we may assume that xt > 0, for all t [n, m 0 1]. Hence, we must have xn 1 < 0 xm 0 < 0. Let τ [n 1, m 0 1]. Summing the second equation of 1, first from n 1 to τ 1, then from τ to m 0 2, we obtain uτ un 1 = 1 β 2 t xt + 1 β 2 xt α 1 tut 25 m m um 0 1 uτ = β 2 t xt + 1 β 2 xt + 1 α 1 tut, 26 t=τ respectively. Here, notice that for τ = n 1 we write solely 26, for τ = m 0 1 only 25 is written. Now we claim that un 1 > 0 um 0 1 < 0. t=τ 27

7 M. Ünal et al. / Computers Mathematics with Applications Indeed, from the first equation of 1, we have 1 α 1 t xt + 1 = xt + β 1 t ut γ 2 ut. First multiplying this last equation by xt then setting t = n 1 t = m 0 1 in the obtained equation, respectively, yield 1 α 1 n 1 xn 1xn = x 2 n 1 + β 1 n 1 un 1 γ 2 un 1xn 1 1 α 1 m 0 1 xm 0 1xm 0 = x 2 m β 1 m 0 1 um 0 1 γ 2 um 0 1xm 0 1. Using the inequalities xn 1xn < 0, xm 0 1xm 0 < 0, since 1 α 1 t > 0 β 1 t > 0 for all t Z, we get from the above latter inequality that un 1xn 1 < 0 um 0 1xm 0 1 < Hence, taking into account that xn 1 < 0 xm 0 1 > 0, we obtain 27. Employing 25 if uτ < 0 26 if uτ > 0, taking into account 27, we obtain uτ m 1 β 2 t xt + 1 β 1 + m 1 α 1 t ut. 29 By using Hölder s inequality on the first sum of the right h side of 29 with indices α β with α 1 + β 1 get = 1, we uτ 2 1 1/β 2 1/α β 2 t β 2 t xt + 1 β + 1 Next, summing Eq. 12 from n 1 to m 0 1 yields m 1 α 1 t ut. 30 or xm 0 um 0 xn 1un 1 = m β 1 t ut γ m β 2 t xt + 1 β xm 0 um 0 + β 2 m 0 1 xm 0 β xn 1un 1 = Now we claim that m β 1 t ut γ m 1 β 2 t xt + 1 β. 31 xm 0 um 0 + β 2 m 0 1 xm 0 β > Indeed, from the second equation of 1 we have, for t = m 0 1, um 0 um 0 1 = β 2 m 0 1 xm 0 β 2 xm 0 α 1 m 0 1um 0 1, which upon multiplication by xm 0 yields um 0 xm 0 + β 2 m 0 1 xm 0 β = 1 α 1 m 0 1 um 0 1xm On the other h, from the inequalities xm 0 1xm 0 < 0 xm 0 1um 0 1 < 0, it follows that um 0 1xm 0 > 0. Therefore, our claim follows from 33 since 1 α 1 t > 0 for all t Z. By virtue of 28 32, from 31 the inequality m 1 β 2 t xt + 1 β < m β 1 t ut γ

8 2638 M. Ünal et al. / Computers Mathematics with Applications follows. Substituting the last inequality above into 30 yields uτ < 2 1 1/β 1 1/α β 2 t β 1 t ut γ + 1 m 1 α 1 t ut, 34 for all τ [n 1, m 0 1]. Choose τ 0 [n 1, m 0 1] such that M 2 = uτ 0 = max n 1 τ 1 uτ. Then uτ 0 > 0 from 34, we obtain uτ 0 < uτ 0 γ α 2 1 1/β 1 1/α β 2 t β 1 t + uτ 0 1 Hence, dividing by uτ 0 we have γ α 1 < M 1 2 1/β 1 1/α 2 β 2 t β 1 t m 1 Since m 0 m, from the latter inequality follows inequality 21. α 1 t. m 1 α 1 t. Remark 5. We should note that inequality 21 is valid only for xt 0 on [n, m 1]. If xt has a zero on [n + 1, m 2], i.e. xt 0 = 0 for some t 0 in [n + 1, m 2], then we have the following two cases: in the case when xn 1xn < 0 xt 0 = 0, the inequality 21 is replaced by the inequality 20, in the case when xt 0 = 0 xm 1xm < 0, the inequality 21 is replaced by the inequality 17. Remark 6. If we would have imposed γ β to be the conjugate numbers in the discrete nonlinear system 1 to begin with, then all of the inequalities 11, 17, would still be obtained without M s by using Hölder s inequalities with indices γ β in the proofs of above theorems. By virtue of Remark 6, the following corollary follows by combining Theorems 1 4. Corollary 7. Suppose 1 α 1 t > 0, β 1 t > 0 β 2 t > 0 for all t Z. Let n, m Z with n m 2. Assume 1 with γ 1 + β 1 = 1 has a real solution x, u such that x has generalized zeros at n m, x is not identically zero on [n, m]. Then the inequality m 1 1/γ m 2 1/β α 1 t + β 1 t β 2 t > holds, where γ > 1 β > 1 are constants. Remark 8. Taking β = γ = 2 in the discrete nonlinear system 1 yields the following discrete linear Hamiltonian system xt = α 1 txt β 1 tut ut = β 2 txt + 1 α 1 tut, t Z. 35 Hence, all of above results presented in this section for the system 1 are also valid for system 35. Thus, we should remark here that the discrete nonlinear system 1 may be viewed as a natural generalization of the discrete linear Hamiltonian system 35. When β = γ = 2 in system 1, it is easy to see that Theorems 1 4 Corollary 7 reduce to Theorems Corollary 1.6 of Guseinov Kaymakçalan [19], respectively. We close this section with a comment on some special cases of the above theorems. Consider the following two special case of system 1, which are equivalent systems for the Emden Fowler type difference equation 9 for the half linear difference equation 10 xt = β 1 t ut γ 2 ut ut = β 2 t xt + 1 β 2 xt + 1, t Z 36

9 M. Ünal et al. / Computers Mathematics with Applications xt = β 1 t ut γ 2 ut ut = β 2 t xt + 1 α 2 xt + 1, t Z, 37 respectively, where β 1 t = p 1 γ t β 2 t = qt α 1 + γ 1 = 1. Needless to say, all of the above results with the special case α 1 t 0, are also valid for the discrete nonlinear systems 36 37, hence for the Emden Fowler type difference equation 9 for the half linear difference equation 10. Remark 9. We should also note here that Theorem 1 is the discrete analogue of Theorem 1 in Tiryaki et al. [20]. 3. Some applications Applying the inequalities derived in Section 2, we have established some results related to disconjugacy boundedness for the solution of system 1. Let n m be integers with n m 2. Consider the discrete nonlinear system xt = α 1 txt β 1 t ut γ 2 ut ut = β 2 t xt + 1 β 2 xt + 1 α 1 tut, t [n, m]. 38 We will assume that γ > 1 β > 1 are constants, the coefficients α 1 t, β 1 t β 2 t are real valued functions defined on [n, m], 1 α 1 t > 0, β 1 t > 0, for all t [n, m]. 39 We also note that each solution x, u of the nonlinear system 38 will consist of vector valued functions defined on [n, m + 1]. In the sequel, we introduce the concept of a relatively generalized zero for the component x of real solution x, u of nonlinear system 38 also the concept of the disconjugacy of the same system on [n, m + 1]. The definition is relative to the interval [n, m + 1] the left end point n is treated separately. Definition 10. The component x of the real solution x, u of system 38 has a relatively generalized zero at n if only if xn = 0, while x has a relatively generalized zero at t 0 > n provided either xt 0 = 0 or xt 0 1xt 0 < 0. Definition 11. The system 38 is said to be disconjugate on [n, m + 1] if no real solution x, u of this system with x 0 has two or more relatively generalized zeros in [n, m + 1]. We should remark here, as mentioned in [19], that under condition 39, the definitions given above are equivalent to those given in [25, p. 354] in [26]. We conclude our remark with related works on the subject [27 31], the references given therein. Theorem 12. Assume condition 39 holds. If the inequality m 1/γ m 1 1/α α 1 t + M β α 1 β 1 t β + 2 t 1 40 holds, where α is the conjugate number to γ M = xτ = max n+1 t m 1 xt, then 38 is disconjugate on [n, m + 1]. Proof. Suppose, to the contrary, that system 38 is not disconjugate on [n, m + 1]. Then, by definition, there exists a real solution x, u of 38 with x which is nontrivial such that xn = 0 x has a generalized zero m 0 in [n + 1, m + 1]. We will have m 0 > n + 1 either xm 0 = 0 or xm 0 1xm 0 < 0. Therefore, applying Theorems 1 2, we obtain m α 1 t + M β 1 α 1 which contradicts to the condition of the theorem. 1/γ 2 1/α β 1 t β + 2 t > 1,

10 2640 M. Ünal et al. / Computers Mathematics with Applications Remark 13. We should mention here that if we impose γ 1 + β 1 = 1 on the system 38, then we would still obtain the inequality 40 without M. In addition to this, when β = γ = 2 in system 38, it is easy to see that Theorem 12 reduces to Theorem 7.1 of Guseinov Kaymakçalan [19]. Theorem 14. If β1 t < β2 t <, 41 then every weakly oscillatory proper solution x, u of 36 is bounded on Z. Proof. Let n, m Z with n m 2 x, u be any nontrivial weakly oscillatory proper solution of the discrete nonlinear system 36 on Z such that x has a sequence of generalized zeros tending to +. Suppose to the contrary that lim sup xt = ; then given any positive number M 3, we can find a positive integer N = NM 3 such that xt > M 3 for all t > N. Since x has a sequence of generalized zeros tending to +, there exists an interval [n, m] with n N such that one of the following cases holds. i xn = 0 = xm, ii xn = 0 xm 1xm < 0, iii xn 1xn < 0 xm = 0, iv xn 1xn < 0 xm 1xm < 0, xt 0 on [n, m 1]. Clearly all theorems given in Section 2 for the discrete nonlinear systems 1 with α 1 t 0 are valid for the discrete nonlinear systems 36. Notice that each of the above cases corresponds one theorem given in Section 2. That is, each of the cases i, ii, iii iv corresponds Theorems 1 4, respectively. Here we only prove the case xn = 0 = xm. The proofs of other cases can be obtained similarly. Now choose τ in [n, m] such that M = xτ = max{ xt : n < τ < m} > M 3. Clearly the inequality in Theorem 1 is satisfied on [n, m]. Because of 41, one can choose N t 0 large enough so that every n N β 1 t < M β α/α 1 β 2 t < Note that the inequality in Theorem 1 with α 1 t 0 is m 1 1/γ m 2 1/α M β α 1 β 1 t β + 2 t 2, hence the αth power of this inequality yields m 1 α 1 2 α M β α β 1 t β + 2 t. 43 Substituting 42 into 43, we obtain m 1 α 1 2 α M β α β 1 t β + 2 t α 1 M β α β 1 t β 2 t < M β α M β+α = 1, where α > 1, we use the fact that β + 2 t β 2t. This contradiction shows that xt is bounded on Z. Hence there exists a positive constant K such that xt K for all t Z. To show that ut is bounded, we sum the second equation of 36 from τ to t 1 to obtain t 1 ut = c + β 2 s xs + 1 β 2 xs + 1 s=τ 44

11 where c is a constant, hence M. Ünal et al. / Computers Mathematics with Applications ut c + K β 2 s, β 1 t 1 s=τ which implies that ut is bounded on Z since β 2 t <. This completes the proof. 45 Theorem 15. Assume that β + 2 t lµ [t 0, := { f s s N : s=t0 f s µ <, 1 µ < }. If x, u is any weakly oscillatory proper solution of 36 with β 1 t 1 on Z satisfying x has a sequence of relatively generalized zeros tending to +, then the distance between consecutive zeros of x tends to infinity as t +. Proof. Let xt, ut be a weakly oscillatory proper solution of 36 with β 1 t 1 on Z such that xt has a sequence of generalized zeros tending to +. Suppose that the conclusion is not true. Then there exists a xt with its sequence of generalized zeros {t n }, which sequence has a subsequence {t nm } such that t nm+1 t nm M4 < + for all m. Let s nm be point in t nm, t nm+1 where xt is maximized. Then s nm t nm < M4 for all m. We will proceed through the rest of the theorem as in Theorem 14 in the case of i; other cases can be obtained similarly with slight modifications. Let M = xs nm µ 1 be the index conjugate with µ, i.e., µ + µ 1 = 1. Suppose β + 2 t lµ [t 0,, 1 µ <, for m large enough so that 1/µ β + 2 s µ M α β M α 1+1/µ t nm By using the inequality in 44 with β 1 t = 1, we obtain 2 α M β α snm α 1 1 s nm t nm β + 2 t. 47 t=t nm Using Hölder s inequality with indices µ µ on the right-h side of 47 yields 2 α M β α snm α 1 1 s nm t nm β + 2 t µ t=t nm M β α s nm t nm α 1+1/µ M β α s nm t nm α 1+1/µ snm 1 β + 2 t µ t=t nm 1/µ snm 1 1/µ t=t nm 1/µ t=t nm β + 2 t µ 1/µ. 48 Substituting the inequality 46 into 48 taking into account that s nm t nm < M4 α > 1, we get 2 α M β α s nm t nm α 1+1/µ M α β M α 1+1/µ 4 < 1. This contradiction completes the proof. Acknowledgements The authors would like to thank the Referees for their valuable suggestions. References [1] A.M. Liapunov, Probleme Général de la Stabilité du Mouvement, vol. 17, Princeton Univ. Press, Princeton, NJ, [2] T.W. Reid, A matrix equation related to a non-oscillation criterion Lyapunov stability, Quart. Appl. Math. Soc

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