摘 要
在本論文中，研究一維p-拉普拉斯算子正、反問題中，解的節點性質。我們首先考慮邊界值問題定義在(0,1)區間中，
-(y^(p-1))’+(p-1)q(x)y^(p-1) =(p-1)λw(x)y^(p-1), (0.1)
其中 f(p-1)：=∣f∣^(p-1) sgn f。此問題為非線性且退化的，其行為與傳統Sturm-Liouville問題很類似，是p=2時的特例。當此問題考慮線性分離的邊界值條件時，其特徵值是離散的，且對應於第n個的特徵函數 yn恰有(n-1)個零點在(0,1)中。利用一個Prufer型代換及廣義sine函數Sp(x)的性質，我們解答了當w(x)≡1時Dirichlet、週期及反週期邊界值條件下，反節點問題的重構公式及穩定性問題。另外，對應的Ambarzumyan問題也得到解答。
我們也研究一組含有非齊次項(p-1)w(x)f(y(x))在式子(0.1)右項的邊界值問題，其中w是恆正且連續可微的，q是連續可微的，f是正函數且Lipschitz連續在(0,∞)中，並且是在R中的奇函數使得
f0, f∞
不相等。我們推廣了Kong的結果，由p=2到p＞1，結果為當有一個特徵值λn落在(f0 , f∞)或(f∞ , f0)時，以上的非齊次式方程式結合任意線性分離的邊界值條件，存在一個解u有(n-1)個零點在(0,1)中。
雖然已知有些不同於p=2時的狀況，我們的結果證明了，一維p-拉普拉斯算子是非常相似於Sturm-Liouville算子，特別是使用Prufer代換的技巧。

In this thesis, direct and inverse problems concerning nodal solutions associated with the one-dimensional p-Laplacian operators are studied. We first consider the eigenvalue
problem on (0, 1),
−(y0(p−1))0 + (p − 1)q(x)y(p−1) = (p − 1) λw(x)y(p−1) (0.1)
Here f(p−1) := |f|p−2f = |f|p−1 sgn f. This problem, though nonlinear and degenerate, behaves very similar to the classical Sturm-Liouville problem, which is the special case
p = 2. The spectrum {λk} of the problem coupled with linear separated boundary conditions are discrete and the eigenfunction yn corresponding toλn has exactly n−1 zeros in (0, 1). Using a Pr‥ufer-type substitution and properties of the generalized sine function, Sp(x), we solve the reconstruction and stablity issues of the inverse nodal problems for Dirichlet boundary conditions, as well as periodic/antiperiodic boundary conditions whenever w(x) λ 1. Corresponding Ambarzumyan problems are also solved.
We also study an associated boundary value problem with a nonlinear nonhomogeneous
term (p−1)w(x) f(y(x)) on the right hand side of (0.1), where w is continuously differentiable and positive, q is continuously differentiable and f is positive and Lipschitz
continuous on R+, and odd on R such that
f0 := lim
y!0+
f(y)
yp−1 , f1 := lim
y!1
f(y)
yp−1 .
are not equal. We extend Kong’s results for p = 2 to general p > 1, which states that whenever an eigenvalue _n 2 (f0, f1) or (f1, f0), there exists a nodal solution un
having exactly n − 1 zeros in (0, 1), for the above nonhomogeneous equation equipped
with any linear separated boundary conditions.
Although it is known that there are indeed some differences, Our results show that the one-dimensional p-Laplacian operator is still very similar to the Sturm-Liouville operator, in aspects involving Pr‥ufer substitution techniques.

Contents
1 Introduction 5
1.1 The p-Laplacian eigenvalue problem . . . . . . . . . . . . . . . . . . . . 5
1.2 Two inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 A class of nonlinear p-Laplacian boundary value problems . . . . . . . 13
1.4 Chapter summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Preliminaries 17
2.1 The generalized sine function Sp . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Pr‥ufer-type substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Eigenvalue estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Inverse nodal problems and Ambarzumyan problems 24
3.1 Inverse nodal problem for Dirichlet boundary conditions . . . . . . . . 24
3.2 Inverse nodal problem for periodic/anti-periodic boundary conditions . 30
3.3 The Ambarzumyan problems . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Nonlinear boundary value problems for p-Laplacian 35
4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Results on initial value problems . . . . . . . . . . . . . . . . . . . . . 38
4.3 Some technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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