在本論文中，我們討論下列一維 p-拉普拉斯特徵值問題:
-((y’/s)^(p-1))’+(p-1)(q-λw)y^(p-1)=0 a.e. on (0,1)
且滿足
αy(0)+ α ’ (y’(0)/s(0))=0
βy(1)+ β’ (y’(1)/s(1))=0
其中定義f^(p-1)=|f|^p-2 f=|f|^p-1 sgnf; α, α’, β, β’都屬於實數且使得α^2+α’^2>0 和β^2+β’^2>0;且函數 s,q,w 必須滿足
(1) s,q,w∈L^1(0,1);
(2) 對於0≤x≤1,我們有s≥0,w≥0 a.e.;
(3) 對於任意x∈(0,1), ∫_0^1 s(t)dt>0, ∫_0^x w(t)dt>0,且∫_x^1 w(t)dt>0;
(4) 如果對於某些x_1<x_2,滿足∫_ x1^x2 w(t)dt=0,則∫_ x1^x2 |q(t)|dt=0;

(5) 對於所有n&#8712;N,在[0,1]間存在分割{ζ_i^(n)}_i=1 ^2n,使得對於任意0<k&#8804;n-1,
∫_ζ_2k^(n)^ ζ_2k+1^(n) w>0且 ∫_ζ_2k+1^(n)^ ζ_2k+2^(n) s>0.
我們稱上述的條件為Atkinson條件。此條件包含s,q,w&#8712;L^1(0,1)且s,w>0 a.e. 的情況。
我們利用廣義Prufer 型代入和 Caratheodory 定理去證明上述方程初始值問題解的存在性及唯一性。我們更將 Sturm 振盪定理推廣到帶 L^1 係數函數的 p-拉普拉斯算子, 從而說明此算子具有 Sturm-Liouville 性質。我們的結論填補了 Binding-Drabek [3] 論文中的一些漏洞。

In this thesis, we consider the following one dimensional p-Laplacian eigenvalue problem:
-((y’/s)^(p-1))’+(p-1)(q-λw)y^(p-1)=0 a.e. on (0,1) (0.1)
and satisfy
αy(0)+ α ’ (y’(0)/s(0))=0
βy(1)+β’ (y’(1)/s(1))=0 (0.2)
where f^(p-1)=|f|^p-2 f=|f|^p-1 sgnf; α, α’, β, β’ &#8712;R
such that α^2+α’^2>0 andβ^2+β’^2>0;
and the functions s,q,w are required to satisfy
(1) s,q,w&#8712;L^1(0,1);
(2) for 0&#8804;x&#8804;1, we have s&#8805;0,w&#8805;0 a.e.;
(3) for any x&#8712; (0,1), ∫_0^1 s(t)dt>0, ∫_0^x w(t)dt>0,and∫_x^1 w(t)dt>0;
(4) if for some x_1<x_2,we have∫_ x1^x2 w(t)dt=0,then∫_ x1^x2 |q(t)|dt=0;

(5) for all n&#8712;N, there is a partition {ζ_i^(n)}_i=1 ^2n of [0,1] such that for any 0<k&#8804;n-1, ∫_ζ_2k^(n)^ ζ_2k+1^(n) w>0 and ∫_ζ_2k+1^(n)^ ζ_2k+2^(n) s>0.
We call the above conditions Atkinson conditions, first introduce in [1].There conditions include the case when s,q,w&#8712;L^1(0,1) and s,w>0 a.e.
We use a generalized Prufer substitution and Caratheodory theorem to prove the existence and uniqueness for the solution of the initial value problem of (0.1) above. Then we generalize the Sturm oscillation theorem to one dimensional p-Laplacian and establish the Sturm-Liouville properties of the p-Laplacian operators with L^1 coefficient functions. Our results filled up some gaps in Binding-Drabek [3].

1 Introduction 1
2 Preliminaries 13
2.1 Properties of eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Generalized Prufer substitution . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Existence and uniqueness of solutions to Caratheodory problems . . . . 17
3 Sturm Liouville properties for Atkinson type problem 22
3.1 Θ as a function of λ. . . . . . . . 22
3.2 Proof of Sturm oscillation theorem . . . . . . . . . . . . . . . . . . . . 26
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Appendix 34
4.1 Sturm-Liouville properties when p= 2 . . . . . . . . . . . . . . . . . . 34
4.2 Proofs for existence and uniqueness theorems . . . . . . . . . . . . . . . 38
4.3 Sturm comparison theorem for p-Laplacian . . . . . . . . . . . . . . . . 41

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