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研究生:李貤溶
研究生(外文):Yi-Jung Lee
論文名稱:包含四次多項式邊界值問題正解的分支曲線
論文名稱(外文):Bifurcation Curves of Positive Solutions for A Boundary Value Problem with A Quartic Polynomial
指導教授:葉宗鑫葉宗鑫引用關係
指導教授(外文):Tzung-Shin Yeh
口試委員:王信華黃印良
口試委員(外文):Shin-Hwa WangYin-Liang Huang
口試日期:2014-05-28
學位類別:碩士
校院名稱:國立臺南大學
系所名稱:應用數學系碩士班
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:2014
畢業學年度:102
語文別:英文
論文頁數:20
中文關鍵詞:斷裂S 型曲線確切解之個數正解四次多項式時間映射
外文關鍵詞:Broken S-shaped bifurcation curveexact multiplicitypositive solutionsquartic polynomialTime map
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我們研究邊界值問題正解的確切個數及分支曲線

u′′+λ(-u(u-β₁)(u-β₂)(u-β₃))=0,-1<x<1,
u(-1)=u(1)=0,

在這裡λ>0 是一個分支參數且非線性四次多項式 f(u)=-u(u-β₁)(u-β₂)(u-β₃) 有三個正根, 0<β₁≤β₂≤β₃。則在 (λ,||u||∞) 平面上,我們得到兩種不同性質的分支曲線:一種是斷裂S型曲線,另一種是單調遞增曲線。
We study the exact multiplicity and the bifurcation curves of positive solutions of the boundary value problem

u′′+λ(-u(u-β₁)(u-β₂)(u-β₃))=0,-1<x<1,
u(-1)=u(1)=0,

where λ>0 is a bifurcation parameter and the quartic polynomial nonlinearity f(u)=-u(u-β₁)(u-β₂)(u-β₃) has three positive zeros, 0<β₁≤β₂≤β₃. Then on the (λ,||u||∞)-plane, we obtain two qualitatively different bifurcation curves: a broken S-shaped curve, and a monotone increasing curve.

中文摘要 i
英文摘要 ii
誌  謝 iii
目  錄 iv
1 Introduction 1
2 Main results 6
3 Lemmas 9
4 Proofs of Main results 12
Reference 19
[1] I. Addou, and S.-H. Wang, Exact multiplicity results for some p-Laplacian nonpositone problems with concave-convex-concave nonlinearities. Nonlinear Analysis 53 (2003) 111-137.
[2] K.-C. Hung, S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc. 365 (2013) 1933-1956.
[3] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J. 20 (1970) 1-13.
[4] J. Smoller, A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981) 269-290.
[5] C.-C. Tzeng, K.-C. Hung, S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations 252 (2012) 6250-6274.
[6] S.-H. Wang, A correction for a paper by J. Smoller and A. Wasserman, J. Differential Equations 77 (1989) 199-202.[7] S.-H. Wang, T.-S. Yeh, S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension, J. Differential Equations 255 (2013) 812-839.
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