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研究生:李中芬
研究生(外文):Chung-Fen Lee
論文名稱:非線性常微分方程式之邊界值問題的討論
論文名稱(外文):Nonlinear Boundary Value Problems for Some Ordinary Differential Equations
指導教授:葉哲志
指導教授(外文):Cheh-Chih Yeh
學位類別:博士
校院名稱:國立中央大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:64
中文關鍵詞:非線性常微分方程式邊界值問題Laplacian 邊界值問題定點定理存在性與唯一性
外文關鍵詞:ordinary differential equationsnonlinear boundary value problemsexistence and uniquenessfixed point theoremLaplacian boundary value problems
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在本論文中,我們將討論非線性常微分方程式之邊界值問題。在第一、二章中,我們探討下列廣義的Laplacian 邊界值問題:
(g(u''))''=f(t,u(t),u''(t)), 0<t<1,
u 屬於 B
其中B是一個適當的邊界條件。考慮不同的函數g與f,我們給定適當的充分條件而得到正解的存在性與唯一性。
在第三、四章中,我們考慮下列四階微分方程式之邊界
值問題:
u''+rf(t,u(t))=0, 0<t<1,
u 屬於 B''
其中B’是一個適當的邊界條件。利用定點定理,給定函數 的條件,我們將討論解的存在性與多重性。
In this dissertation, we will study nonlinear boundary value problems for some ordinary differential equations. In chapters 1 and 2, we study the following generalized Laplacian boundary value problems :
(g(u''))''=f(t,u(t),u''(t)), 0<t<1.
We establish the solution existence and uniqueness for the problem under different conditions concerning f(t,u(t),u''(t)).
In chapters 3 and 4, we consider the following nonlinear fourth order boundary value problems :
u''+rf(t,u(t))=0, 0<t<1.
Consider different conditions concerning f(t,u(t)), we study the existence and multiplicity of solutions by using fixed point theorem.
Chapter 1. Existence and Uniqueness of Positive Solutions for Generalized Laplacian BVP''s with Sturm-Liouville Conditions ...........……….….(1)
Chapter 2. Existence of Positive Solutions for Generalized Laplacian Boundary
Value Problems ……………………………………………………...(26)
Chapter 3. Multiple Positive Solutions of Fourth Order Boundary Value
Problems ...........................................................................………......(39)
Chapter 4. Existence of Solutions for Singular Fourth Order Nonlinear
Boundary Value Problems ……………………………………….…(50)
1. R. B. Agarwal and F.H. Wong, Existence of positive solutions for higher order boundary value problems, Nonlinear Studies 5, (1998) 15-24.2. K. Deimling, Nonlinear Functional Analysis, Springer, New York, (1985).3. Dajun Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, 1988.4. Vasile I. Istratescu, Fixed Point Theory, An Introduction, Dordrecht, Holland, 1981.5. M. A. Krasnoselskii, Positive Solutions of Operator quations, Noordhoff, Groningen, 1964.6. Ma Ruyun, Wang Haiyan, On the existence of positive solutions of fourth order ordinary differential equations, Appl. Anal., 59 (1995), 225-2317. P. J. Y. Wong and R. P. Agarwal, Eigenvalues of boundary value problems for higher order differential equations, Mathematical Problems in Engineering, 2 (1996), 401-434.8. Yao Qingliu, Bai Zhanbing, Existence of positive solutions of BVP for u''(t)-r h(t)f(u(t))=0, Chinese Annals of Mathematics 20A: 5 (1999), 575-578. (In Chinese)9. Zhang Bingger, Kong Lingju, Existence of positive solutions of fourth order singular boundary value problems, Chinese Annals of Mathematics 22A: 4(2001), 397-402.(In Chinese)10. Zhang Xiaomei, Multiple nonnegative solutions of fourth order ordinary differential equations, Ann. of Diff. Eqs., 16:4 (2000), 407-413.
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