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系統識別號 U0002-1806201215364200
中文論文名稱 廣義區間上動態方程的存在性與穩定性
英文論文名稱 Existence and Stability for Dynamic Equation on Time Scales
校院名稱 淡江大學
系所名稱(中) 數學學系博士班
系所名稱(英) Department of Mathematics
學年度 100
學期 2
出版年 101
研究生中文姓名 林尚文
研究生英文姓名 Shang-Wen Lin
學號 893150044
學位類別 博士
語文別 英文
口試日期 2012-06-01
論文頁數 52頁
口試委員 指導教授-錢傳仁
委員-王富祥
委員-林賜德
委員-張茂盛
委員-陳建隆
中文關鍵字 廣義區間  二階常微分方程  兩點邊界值問題  劣線性  超線性  Schauder固定點定理  Lyapunov穩定性  隱函數方程 
英文關鍵字 Time scales  Green's function  Arzela-Ascoli theorem  Schauder fixed point theorem  sublinear  suplinear  Lyapunov Stability  Implicit Dynamic Equations  Index 1 
學科別分類
中文摘要 本論文主要分兩個部分。首先,我們探討廣義區間上二階非線性常微分方程,搭配兩點邊界值條件之下正解的存在性,並進一步以劣線性及超線性的觀點提出關於外力項的限制條件。此外,將本文的結果應用於連續型的方程,也能將先前的研究結果加以推廣。
其次,我們探討類線性隱函數方程之零解的穩定性。我們先以指標的概念處理存在性,再進一步以Lyapunov函數討論其穩定性。
英文摘要 In this thesis,we give a criterion for the existence of positive solutions for nonlinear second order ordinary differential equations with two-point boundary value conditions on time scales.
Moreover, for some source terms which are in the sense of sublinear or superlinear, we also formulate corollaries and examples for applications and we improve previous results.
Second, we investigate the stability of the solution x=0 for a class of quasilinear implicit dynamic equations on time scales of the form $A_tx^{Delta}=f(t,x)$.
We deal with an index concept to study the solvability and use Lyapunov functions as a tool to approach the stability problem.
論文目次 1.Introduction.............................................1

2.Some Basic Notations of the Theory of the Analysis on Time Scales................................................3

3.Nonlinear two-point boundary value problems on time scales.....................................................7
3.1Introduction............................................7
3.2Some preliminaries and main result......................8
3.3Consequence and examples...............................11

4.Lyapunov Stability of Quasilinear Implicit Dynamic Equations on Time Scales..................................16
4.1Introduction...........................................16
4.2Nonlinear Implicit Dynamic Equations on Time Scales....................................................18
4.3Quasilinear Implicit Dynamic Equations.................27
4.4Stability Theorem of Implicit Dynamic Equations........31
4.5Conclusion.............................................48

Reference.................................................49
參考文獻 [1] R. P. Agarwal, V. Oteoro-Espainar, K. Perera, D. R. Vivero, Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods, J. Math. Anal. Appl. 331 (2007) 1263-1274.

[2] R. P. Agarwal, V. Oteoro-Espainar, K. Perera, D. R. Vivero, Multiple positive solutions of singular Dirichlet problems on time scales viva variational methods, Nonlinear Anal. 67 (2007) 361-381.

[3] D. R. Anderson, Nonlinear triple-point problems on time scales, Electron. J. Differential Equations 2004 (47) (2004) 1-12.

[4] P. K. Anh and D. S. Hoang, Stability of a class of singular difference equations, International Journal of Difference Equations, vol. 1, no. 2, pp.181-193, 2006.

[5]F. M. Atici, G. Sh. Guseinov, On Green's functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math. 141 (2002) 75-99.

[6] M. Bohner, A Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.

[7] M. Bohner, A Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.

[8] M. Bohner and S. Stevic, Linear perturbations of a nonoscillatory second-order dynamic equation, Journal of Difference Equations and Applications, vol. 15, no. 11-12, pp. 1211-1221, 2009.

[9] M. Bohner and S. Stevic, Trench's perturbation theorem for dynamic equations, Discrete Dynamics in Nature and Society, vol. 2007, Article ID 75672, 11 pages, 2007.

[10] A. Cabada, External solutions and Green's functions of higher order periodic boundary value problems on time scales, J. Math. Anal. Appl. 290 (2004) 35-54.

[11] S. L. Campbell, Singular Systems of Differential EquationsI, II, vol. 40 of Research Notes in Mathematics, Pitman, London, UK, 1980.

[12] L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1989.

[13] T. S. Doan, A. Kalauch, S. Siegmund, and F. R. Wirth, Stability radii for positive linear time-invariant systems on time scales, Systems and Control Letters, vol. 59, no. 3-4, pp. 173-179, 2010.

[14] N. H. Du and V. H. Linh, Stability radii for linear time-varying differential-algebra equations with respect to dynamic perturbations, Journal of Differential Equations, vol. 230, no. 2, pp. 579-599, 2006.

[15] E. Griepentrog and R. Marz, Differential-Algebraic Equations and Their Numerical Treatment, vol. 88, Teubner, Leipzig, Germany, 1986.

[16] J. Henderson, C. C. Tisdell, Dynamic boundary value problems of the second-order: Bernstein-Nagumo conditions and solvability, Nonlinear Anal. 67 (5) (2007) 1374-1386.

[17] S. Hilger, Ein maskettenkalkul mit anwend ung auf zentrumsmannigfaltigkeiten, Ph. D. thesis, University of Wurzburg, 1988.

[18] S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990.

[19] P. E. Kloeden and A. Zmorzynska, Lyapunov functions for linear nonautonomous dynamical equations on time scales, Advances in Difference equations, vol. 2006, Article ID 69106, pp. 1-10, 2006.

[20] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations, EMS Textbooks in Mathematics, European Mathematical Society House, Z"{u}rich, Switzerland, 2006.

[21] V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.

[22] Y. Li, On the existence and nonexistence of positive solutions for nonlinear Sturm-Liouville boundary value problems, J. Math. Anal. Appl. 304 (2005) 74-86.

[23] L. C. Loi, N. H. Du, and P. K. Anh, On linear implicit non-autonomous systems of difference equations, Journal of Difference Equations and Applications, vol. 8, no. 12, pp. 1085-1105, 2002.

[24] R. Marz, Criteria for the trivial solution of differential algebraic equations with small nonlinearities to be asymptotically stable, Journal of Mathematical Analysis and Applications, vol. 225, no. 2, pp. 587-607, 1998.

[25] A. C. Peterson, Y. N. Raffoul, C. C. Tisdell, Three point boundary value problems on time scales, J. Difference Equ. Appl. 10 (9) (2004) 843-849.

[26] C. Potzsche, Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients, Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 317-335, 2004.

[27] C. Potzsche, Chain rule and invariance principle on measure chains, Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 249-254, 2002.

[28] C. Potzsche, Analysis Auf Mabketten, Universitat Augsburg, 2002.

[29]C. Potzsche, S. Siegmund, and F. R. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete and Continuous Dynamic Systems, vol. 9, no. 5, pp. 1223-1241, 2003.

[30] Christopher C. Tisdell, Atiya Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modeling, Nonlinear Analysis, 68 (2008) 3504-3524.
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