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半線性橢圓型偏微分方程式之解的漸近行為及不穩定性的研究
http://ir.ncue.edu.tw/ir/handle/987654321/16362
title: 半線性橢圓型偏微分方程式之解的漸近行為及不穩定性的研究 abstract: 我們將研究下列半線性橢圓型偏微分方程式之解的漸近行為及不穩定性. ? -.DELTA.u = .beta. f(u) in .OMEGA. P(.beta.) |? u = 0 on .PRTL..OMEGA.其中,.OMEGA. 為 n 域,且.OMEGA. 之邊界.PRTL..OMEGA.為平滑.當 f(u) 成長速度大於線性,則我們將證明,對任意緊緻子集 K ,當.beta.趨近於零時,恆有Lim Min {u(x,.beta.)| x in K} =+.inf.其中, u(x ,.beta.) 之選取受到下列限制:若 f(0)=0, 則 u(x ,.beta.) 可為 P(.beta. ) 之任意非零解.若 f(0)>0 , 則 u(x ,.beta.) 可為 P(.beta. ) 之任意非極小解. ( 關於 "非極小解" 之定義,請參閱論文第一節. )除此之外,我們也將證明 P(.beta.) 之非極小解 u(x ,.beta.)在 .OMEGA.內部之振動現象將逐漸擴大.當 f(u) 成長速度幾近乎線性時,則我們將證明,存在正數區間 I= (0,r),使得下列敘述成立: (1)當.beta. 在 I 內部時,則 P(.beta. )恰有一個解u(x,.beta.);而當 .beta. > r 時,P(.beta.) 無解. (2) 若 K 為 .OMEGA. 的任意一個緊緻子集,則當.beta.趨近於 r 時, Lim Min { u(x,.beta. ) | x inK } =.inf.一般而言,大家知道 r 的上限估計;在本文之中,我們有估計 r之下限的方法.尤其,當 f(u) 成長速度幾近乎線性時,我們可以得到 r的真正數值.在這篇論文的最後一節中,我們將研究半線性橢圓型偏微分方程式之解的不穩定性. 我們將証明,若適當地限制非線性項,則半線性橢圓型偏微分方程式之所有的正解均為不穩定解.另一方面,對一般的非線性項,我們將証明,若所討論的區域足夠小,則方程式之所有正解均為不穩定解。
<br>極小極大定理
http://ir.ncue.edu.tw/ir/handle/987654321/16361
title: 極小極大定理 abstract: 傳統上,數學方法從其所處理之物理問題衍生而來,因而伴隨了許多來自自然科學的觀念,如連續性、緊緻性等;數學分析的發展明白地顯現此一特性。時至今日,社會科學之迅速發展,仍求助於這源於自然科學的傳統數學。問題是:它們是否具有社會科學所需要的最佳模式呢?這篇文章選擇了數學經濟與最佳控制理論中著名的極小極大定理,嘗試給予其另一種模型。傳統的數學分析,均會先給定一個集合及其拓業性質再討論定義於此的函數之種種特性。我們將避開這「習慣」而給予集合與函數相等之地位,也就是,我們將同時賦予集合與函數某些性質,使極小極大定理仍能成立。
<br>Central Limit Theorems for Instantaneous Filters of Linear Random Fields on Z^2
http://ir.ncue.edu.tw/ir/handle/987654321/14080
title: Central Limit Theorems for Instantaneous Filters of Linear Random Fields on Z^2 abstract: This note considers the stationary sequence generated by applying an instantaneous filter to a linear random field in Z2. The class of filters under consideration includes polynomials and indicator functions. Instead of imposing certain mixing conditions on the random fields, it is assumed that the weights of the innovations satisfy a summability property. By building a martingale decomposition based on a suitable filtration, asymptotic normality is proved for the partial sums of the stationary sequence.
<br>Systems of Generalized Quasivariational Inclusion Problems with Weak Continuity and Weak Convexity and Variants of Set-valued Ekeland’s Variational Principle
http://ir.ncue.edu.tw/ir/handle/987654321/8750
title: Systems of Generalized Quasivariational Inclusion Problems with Weak Continuity and Weak Convexity and Variants of Set-valued Ekeland’s Variational Principle abstract: An existence theorm of systems of generalized quasivariational inclusion problems with very weak continuity and convexity assumptions is proven. From this result, recent existence theorems of other types of systems of generalized quasivariational inclusion problems with very weak continuity and convexity assumptions are established. We establish these results with a very simple ane elegant method. Our results improve and gengralize many recent results of these types of problems. As applications of our results, we study variants of set-valued vector Ekeland's variational principle with weak continuity assumprions on the maps we consider, and with a simple method. Our results on set-valued or vector-valued Ekeland's variational principle are different from known existence results of these types of problems.
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