中文摘要
在本文中，吾人主要在討論二階橢圓形的積分微分方程之存在解及數值解的問題，而其形式如下所示，
(1.1) u"=f(x,u)+∫_0^1▒k(x,t,u(t))dt，0<x<1,
u(0)=u(1)=0.
和
(1.2) ∆u=f(x,u)+∫_Ω▒〖k(x,t,u(t))dt,in Ω〗，u=0 on ∂Ω.
在此Ω是在一個R^n(n≧2)中的有界區域，而這區域具有平滑的邊界，且∆代表n維的Laplace運算子.
在第一章中，吾人用Leray-Schauder degree定理來證(1.1)的存在解，而後由變分法(Variational method)，可以把(1.1)變為一組非線性的積分微分方程式，然後藉著同倫法(Homotopy method)來解此非線性的方程.最後可得在某個norm下，當h趨近於零時，真實解與數值解的收斂速度是最好的.我們將給予幾個例子來印證理論的結果.
在第二章中，首先經由[11]的結果來證明(1.2)的存在解，然後如同在第一章中討論的，給予一個適當的在h趨近於零時，可得到真實解與數值解的誤差會趨近於零.而最後，我們用幾個例子來確認理論的結果.
在第三章中，吾人討論一個非線性的積分微分方程式如下所示，
(1.3) ∆u=∅(x,u,K(u)) in Ω.
在此K(u)是某個非線性的積分運算子.
蔡[12]討論(1.3)藉由Leray-Schauder degree定理.近來有一個 mixed monotony的方法，對於邊界值的問題且當函數沒有任何遞增或遞減的性質時，此方法可造出一個收斂的單調數列收斂到唯一解.在此，我們創出一個新的方法，不需要去造單調的數列，而是藉著fixed point定理和微分不等式的技巧來證其存在解.

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