| Abstract: | 在本論文中，我們將探討以下型態的非線性邊界值問題正解之存在性與非存在性： (*) u'(t)+f(t,u(t))=0, 0 <t <1; u屬於B，其中B為適當的邊界條件。給予f(t,.)適當的條件，利用 Krasnoselskii 的固定點定理，我們將給出在幾種不同邊界值條件下的微分方程式多重正解的存在或非存在性。 經由(*) 問題的探討，我們將一般的常微分方程式推廣至延遲的微分方程式 u'(t)+f(t,u(t+s))=0, 0<t <1 , -r < s < a, 來討論其解的存在性。更經由上述的延遲方程式的研究， 我們發現在時標(time scale)所定義的測度鏈(measure chain)上的微分方程式， (**)u'(t)+f(t,u(g(t)))=0, 0<t<1. 除了隱含方程式上的延遲性外，更可將一般的微分與差分方程做一個連結。 因此我們進一步討論(**)問題的正解存在性。 In this article, we concerned with the existence andxistence of positive solutions of the following nonlinear boundary value problem of the form: (*) u'(t)+f(t,u(t))=0, 0 <t <1. Under the suitable condition f(t,.), by using Krasnoselskii's fixed point theorem, we will give the existence andxistence of multiple positive solutions under several different boundary value conditions for the differential equations. It follows from the boundary value problem (*), we can extend general ordinary diferential equation to the delay differential equations u'(t)+f(t,u(t+s))=0, 0<t <1 , -r < s < a, and consider the existence of positive solutions. Moreover, it follows from above delay differential equations, we find that the differential equation on a measure chain defined on time scale of the form: (**) u'(t)+f(t,u(g(t)))=0, 0<t<1; combine the difference and differential equations. So we deal with the existence of positive solutions of the problem (**). |
