system of two second order linear differential equations

... solutions of second- order linear difference equations II: the second case is dedicated to the case when k<0. 1.1. The Result in [3] When k  1 When k  1, from 3 we know that 1.2 has two linearly ... completed the estimate of the error bounds for asymptotic solutions to second order linear difference equations in the first case. For the second case, we leave it to the second part of this paper: Error ... second- order linear difference equations, ” Journal of Computational and Applied Mathematics, vol. 41, no. 1-2, pp. 65–94, 1992. 3 R. Wong and H. Li, “Asymptotic expansions for second- order linear...

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... Difference Equations Volume 2010, Article ID 143298, 8 pages doi:10.1155/2010/143298 Research Article On Connection between Second- Order Delay Differential Equations and Integrodifferential Equations ... obtained by transforming second- order delay dierential equations ăxtat xgt btxht  0 to first -order differential equations. 1. Introduction and Preliminaries The second order delay dierential equation ăx t ... defined by 2.5. Proof. In the proof we will use notation defined in the proof of Theorem 2.1. The existence and uniqueness of a solution of 1.11, 2.1 is a consequence of Theorem 2.1. Thus,...

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... theorem of solution for second order nonlinear differential equations, ” Annals of Differential Equations, vol. 19, no. 3, pp. 480–486, 2003. 7 R K. Zhuang and H W. Wu, “Sturm comparison theorem of ... theorem of second- order differential equations see Section 1. Remark 3.6. In the discrete case: μt ≡ 1. This result is the same as Sturm comparison theorem of second- order difference equations ... of second- order linear equations on time scales. We first establish Picone identity on time scales and obtain our main result by using it. Also, our result unifies the existing ones of second- order...

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... Article Limit Properties of Solutions of Singular Second- Order Differential Equations Irena Rach ˚ unkov ´ a, 1 Svatoslav Stan ek, 1 Ewa Weinm ă uller, 2 and Michael Zenz 2 1 Department of Mathematical ... the solvability of nonlinear singular equations subject to different types of boundary conditions. In this section, we utilize Theorem 4.1 to show the existence of solutions for periodic problems. The rest of ... sufficient conditions for the existence of at least one solution of problem 5.1a and 5.1b with a>0. In the proof of this theorem, we work also with auxiliary two- point boundary conditions: u  0  ...

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... Article Global Behavior of Solutions to Two Classes of Second- Order Rational Difference Equations Sukanya Basu and Orlando Merino Department of Mathematics, University of Rhode Island, Kingston, ... Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002. 2 G. Ladas, “Invariants for generalized Lyness equations, ” Journal of Difference Equations ... about the global dynamics of 1.11 and 1.13, such as the character of the prime period -two solutions to either equation, or even for more general rational second- order equations, when such solutions...

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... translates into Z L =(r a,b )G L . A proof of Lemma 1.29 finishes the proof of the theorem. ✷ Proof of lemma 1.29. The proof is rather similar to the one of lemma 1.23. The only thing that we have ... as a group of matrices and has the structure of a linear algebraic group, that is, it is a group of invertible matrices defined by the vanishing of a set of polynomials on the entries of these matrices. ... O-submodules N 1 ,N 2 of N with N = N 1 ⊕N 2 and N i = F i for i =1, 2. Proof. The proof is similar to the proof of Proposition 3.17. Let S 1 and S 2 be the set of eigenvalues of E acting on F 1 and...

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... Hu, and Y. Wu, “Positive solutions of two- point boundary value problems for systems of nonlinear second- order singular and impulsive differential equations, ” Nonlinear Analysis: Theory, Methods ... S. G. Hristova and D. D. Ba ˘ ınov, “Existence of periodic solutions of nonlinear systems of differential equations with impulse effect,” Journal of Mathematical Analysis and Applications, vol. ... 3-4, pp. 307– 328, 2004. 4 B. Liu and J. Yu, “Existence of solution of m-point boundary value problems of second- order differential systems with impulses,” Applied Mathematics and Computation,...

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... Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1161–1173, 2007. 9 J. Wu and Z. Wang, Two periodic solutions of second- order neutral functional differential equations, ” ... Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 677–689, 2007. 10 M. R. Zhang, “Periodic solutions of linear and quasilinear neutral functional-differential equations, ” Journal ... and Lyapunov Stability of Periodic Solutions for Generalized Higher -Order Neutral Differential Equations Jingli Ren, 1 Wing-Sum Cheung, 2 and Zhibo Cheng 1 1 Department of Mathematics, Zhengzhou...

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... introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. 1; for an overview of existing results and of recent research areas of impulsive differential equations, ... Differential Equations, vol. 14, World Scientific, Singapore, 1995. 6 J. Yan, “Existence of positive periodic solutions of impulsive functional differential equations with two parameters,” Journal of Mathematical ... olution yt,t∈ J with property 3.17. Proof. The proof is similar to that of Theorem 3. 2of 6. Theorem 3.5. Assume that H 1 , H 2 , 3.1 of H 3  and 3.14 and 3.15 of H 4  hold. In addition,...

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Ngày tải lên: 23/06/2014, 00:20

... is just the repetition of the proof of Lemma 3.2, we omit the details of its proof. With the aid of above preparations, now we will give the proof of Theorem 1.2. Proof of Theorem 1.2 By(1.8), ... is in some way a type of the best expressive way of the structure of the solution space for recurrence relations occurring in the study of second- order linear differential equations. So, (1.3) ... similar to the proof of Lemma 3.1, so we omit the details of its proof. Lemma 3.4 Under the conditions of Theorem 1.2, F (x) satisfies the PS condition. Proof From the proof of Lemma 3.2, we see...

Ngày tải lên: 18/06/2014, 22:20

... is just the repetition of the proof of Lemma 3.2, we omit the details of its proof. With the aid of above preparations, now we will give the proof of Theorem 1.2. Proof of Theorem 1.2 By(1.8), ... similar to the proof of Lemma 3.1, so we omit the details of its proof. Lemma 3.4 Under the conditions of Theorem 1.2, F (x) satisfies the PS condition. Proof From the proof of Lemma 3.2, we see ... is in some way a type of the best expressive way of the structure of the solution space for recurrence relations occurring in the study of second- order linear differential equations. So, (1.3)...

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