摘要
我們主要討論方程式 f(λ,x)=0，其中 f 是一個從 Banach Space 映到 Banach Space 的算子， λ 是它的參數。這個方程式在數學、應用數學甚至理論物理的領域都是很常見的方程式。舉例來說，方程式 f(λ,x)=0 可以代表一個受λ參數控制的積分方程或微分方程系統。首先，我們有興趣的是 f 的解集合並且假設 (λ,0) 爲在其中的一條解曲線。實際上，我們就是在研究討論當ｆ在何種情況或具備哪種條件之下， (λ,0) 這條解曲線上的某一定點（非端點）會同時是ｆ的另外一條解曲線上的點，也就是 f 的解在那一定點上產生分歧。 我們會從有限維空間的分歧理論開始介紹，接著應用度理論證明在有限維空間以及無限維空間的分歧理論，最後我們給個例子作爲結束。
關鍵字
分歧並列摘要
Many problems in mathematics, and its applications to theoretical physics, lead to a problem of the form f(λ,x)=0, (1) where f is an operator on R×X into Y , and X and Y are Banach spaces. For example, (1) could represent a system of differential or integral equations, depending on a parameter λ. We are interested in the structure of the solution set; namely, the set f^(-1) (0)={ (λ,x)∈R×X∶f(λ,x)=0 }. (2) Since we are interested in bifurcation from trivial solutions, we may assume that (λ,0) is a solution curve of (1). In particular, we seek conditions on f to see if a solution (λ,0) of (1) whether or not lies on the other solution curves of (1). In this paper, we start with introducing the bifurcation theory in finite dimensional space case. The degree theory is used in both finite and the infinite dimensional space cases. We conclude the article with some examples.
並列關鍵字
bifurcation參考文獻
[3]. Tian Ma and Shouhong Wang : Bifurcation Theory and Applications, World Scienti c 2005.
[4]. Donal O'Regan, Yeol Je Cho and Yu-Qing Chen : Topological Degree Theory and Applications, Chapman and Hall/CRC 2006.
[5]. Joel smoller : Shock Waves and Reaction-Diffusion Equations, Second Edition, Springer-Verlag
[6]. Micheal G. Crandall and Paul H. Rabinnowitz : Bifurcation from Simple Eigenvalues, Journal of Functional Analysis 8, 321-340(1971).
[1]. Wislaw Krawcewitz and Jianhong Wu : Theory of Degree with Applications to Bifurcations and Differential Equations, John Wiley and Sons, Inc. 1997.