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論文名稱 Title |
二階半線性微分方程的比較定理和振盪定理 Comparison and Oscillation Theorems for Second Order Half-Linear Differential Equations |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
64 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2012-04-17 |
繳交日期 Date of Submission |
2012-06-07 |
關鍵字 Keywords |
振盪型準則、Ricatti方程式、振盪定理、比較定理、半線性方程式 oscillation theorems, oscillation criterion, Ricatti equations, comparison theorems, Half-linear equations |
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統計 Statistics |
本論文已被瀏覽 5745 次,被下載 1019 次 The thesis/dissertation has been browsed 5745 times, has been downloaded 1019 times. |
中文摘要 |
這篇論文是對於 [c(x)u'^{(p-1)}]'+a(x)u^{(p-1)}=0, 其中 u^{(p-1)}=|u|^{p-2}u 的二階半線性微分方程的比較定理和振盪定理的一個概述。文中也包含一些範例。 針對上述的方程式,假如存在一個在 (0,∞) 的區間上具有無限多個零點的非零解,則稱之為振盪型方程;否則稱之為非振盪型方程。振盪定理能幫助我們決定一個方程式為振盪型或非振盪型。這些比較定理和振盪定理,對於上述方程式在 (0,∞) 的非零解,提供數量和位置的信息。本論文取材於 Li-Yeh 的幾篇論文,以及 Dosly-Rehak 的專著。但文中 Reid 類型的比較定理是新的。 |
Abstract |
This thesis is a short survey for the comparison theorems and oscillation theorems for the second order half-linear equation [c(x)u'^{(p-1)}]'+a(x)u^{(p-1)}=0, where u^{(p-1)}=|u|^{p-2}u. Some examples are also given. The above equation is said to be oscillatory ( O ) if there exists a nontrivial solution having an infinite number of zeros in (0,∞); and non-oscillatory ( NO ) if otherwise. Oscillation theorems help to determine whether an equation is ( O ) or ( NO ). These comparison theorem and oscillation theorems give information for the number and position of zeros in (0,∞) for a nontrivial solution of the above equation. Materials in this thesis originate from the papers of Li-Yeh and the monograph of Dosly and Rehak. But Reid type comparison theorem is new. |
目次 Table of Contents |
Contents 1 Introduction 1 2 Comparison Theorems for Half-Linear Equations 8 2.1 Leighton and Sturm-Picone type comparison theorems . . . . . . . . . 8 2.2 Picone type identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Levin type comparison theorem . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Reid type comparison theorem . . . . . . . . . . . . . . . . . . . . . . . 24 3 Oscillation Theorems for Half-Linear Equations 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Hille-Kneser type oscillation theorem . . . . . . . . . . . . . . . . . . . 28 3.3 Oscillation criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Hille-Wintner type comparison theorem . . . . . . . . . . . . . . . . . . 44 4 Appendix 49 |
參考文獻 References |
Bibliography [1] R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations, Dordrecht; Boston: Kluwer Academic Publishers, 2002. [2] W. Allegretto and Y.X. Huang, A Picone's identity for the p-Laplacian and applications, Nonlinear Analysis TMA, Vol. 32, No. 7 (1998), 819-830. [3] P. Binding, P.J. Browne and I.M. Karabash, Sturm-Liouville problems for the p-Laplacian on a half- line, Proc. Edinburgh Math. Soc., 53 (2010), 271-291. [4] C.Z. Chen, C.K. Law, W.C. Lian, and W.C. Wang, Optimal upper bounds for the eigenvalue ratios of one-dimensional p-Laplacian, Proc. Amer. Math. Soc., to appear. [5] O. Dosly and P. Rehak, Half-linear Dierential Equations, Amsterdam; Boston: Elsevier, 2005. [6] J. Jaros and T. Kusano, A Picone type identity for second order half-linear dierential equations, Acta Math. Univ. Comenianae, Vol. 68, 1(1999), 137-151. [7] T. Kusano and N. Yoshida, Nonoscillation theorems for a class of quasilinear dierential equations of second order, J. of Math. Anal. and Appl., 189 (1995), 115-127. [8] H.J. Li and C.C. Yeh, Sturmian comparison theorem for half-linear second-order dierential equations, Proc. Royal Soc. Edinburgh, 125A (1995), 1193-1204. [9] H.J. Li and C.C. Yeh, Nonoscillation criteria for second-order half-linear dierential equations, Appl. Math. Lett., Vol. 8, No. 5 (1995), 63-70. [10] H.J. Li and C.C. Yeh, An oscillation criterion of almost-periodic Sturm-Liouville equations, Rocky Mountain J. Math., Vol. 25, No. 4 (1995), 1417-1429. [11] S.A. Morris, The Schauder-Tychono xed point theorem and applications, Math. slov., Vol. 25, No. 2 (1975), 165-172. [12] J. Sugie and N. Yamaoka, Comparison theorem for oscillation of second-order half-linear dierential equations, Acta Math. Hungar., 111 (1-2) (2006), 165-179. [13] J. Sugie and M. Onitsuka, A non-oscillation theorem for nonlinear dierential equations with p-Laplacian, Proc. Royal Soc. Edinburgh, 136A (2006), 633-647. [14] C.A. Swanson, Comparison and Oscillation Theory of Linear Dierential Equations, Academic Press, New York, 1968. [15] W.Z. Wang, p-Laplacian operators with L1 coecient functions, Unpublished Master Thesis, National Sun Yat-sen University, Kaohsiung, Taiwan, 2011. [16] W.I Yen, Comparison and oscillation theorems for second order linear dierential equations, Unpublished Master Thesis, National Sun Yat-sen University, Kaohsiung, Taiwan, 2012. |
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