摘要
在這篇論文中，我們研究某些類型的非線性算子，在Banach空間中的特徵值問題與定點問題。
我們令K是Banach空間中的closed cone，則對於K中一個非零正向量f，我們利用希氏正射距離，定義K之子集合K(f)為與f 距離有限的所有正向量所成的集合。我們考慮將K映射至K的非線性正算子T之特徵值問題，以及伴隨著算子T的定點問題。當算子T是遞增且將正向量f 映射至集合K(f)內，Bushell證明了，若T是p-齊次算子，其中0<p<1，則算子T在集合K(f)上有唯一的定點。在第一章中，我們推廣這結論到p-凹性算子，並且我們也明確估計了算子T之定點的長度。我們的方法是建立在遞增且p-凹性算子在希氏正射距離下是p-收縮映射。所以利用收縮映射定理，可以得到特徵值與特徵向量之存在性與唯一性的結果。
在第二章中，我們繼續探討某些緊緻算子之特徵值問題。我們考慮的算子是T(1)+T(p)算子，其中T(1)是遞增且1-凹性算子、T(p)是遞增且p-凹性算子，其中0<p<1。特別地，我們不需要假設K具有內點，並且我們考慮的算子T(1)+T(p)將所有非零正向量映射至集合K(f) 內。我們利用Zeidler的結論，證明特徵值與特徵向量的存在性。另一方面，我們使用Nussbaum的技巧，獲得特徵值與特徵向量的唯一性。並且我們也證明特徵值具有單調連續的性質，這些性質保證算子T有唯一的定點。我們這些結論可以應用到非線性代數方程、非線性邊界值問題、一階遞延方程以及矩陣方程的研究。
最後在第三章中，我們將古典的線性Krein-Rutman定理，推廣到齊次且 f-遞增的非線性算子上，我們也探討正齊次算子譜的性質，並且給了一些應用。

Abstract
In this thesis we study the eigenvalue problems for certain classes of positive nonlinear operators defined on a cone in a Banach space. To be precise we suppose that K is a closed cone in a real Banach space X. For a given f in K{+}= K\{0}, we set K(f) = {x in K{+} : d(x, f) is finite}, where d denotes Hilbert's projective metric. We consider the eigenvalue problem: Tx = lambda x, where x in K, lambda >0, and the associated fixed-point equation Tx = x for certain nonlinear cone mappings T. Assuming T is increasing and Tf in K(f) for some f in K{+}, Bushell proved that if T is homogeneous of degree p with 0<p<1, then T has a unique fixed point in K(f). In Chapter 1 we extend this result to p-concave operators. We also give explicit norm-estimates for the fixed point. Our approach is based on the fact that increasing p-concave operators are p-contractions in Hilbert's projective metric. Consequently certain existence and uniqueness results can be proved using the contraction mapping principle. In Chapter 2 we continue the study of such problems for cone mappings which are compact, and which are of the form T = T(1) + T(p), where T(j) is increasing j-concave for j = 1, p with 0<p<1. In particular, we do not assume that the cone is solid, and we consider the case of a self-mapping of a subset of the boundary of the cone. We shall apply a result of Zeidler to prove the existence of solutions to the eigenvalue problem. On the other hand, we shall use the techniques of Nussbaum to obtain uniqueness results. As in Chapter 1, we next show that the solutions have some monotonicity and continuity properties which will ensure that the fixed-point equation Tx = x has a unique solution. Applications to nonlinear systems of equations, to nonlinear boundary-value problems, to differential delay equations and to matrix equations are considered. Finally, in Chapter 3 we extend the linear Krein-Rutman theorem to the case when T is homogeneous and f-increasing. We also study the spectral properties of positively homogeneous operators. Some examples are given.

1. P.J. Bushell, Hilbert's metric and positive contraction mappings in a Banach space, Arch. Rat. Mech. Anal., 52(1973), 330-338.
2. P.J. Bushell, On the projective contraction ratio for positive linear mappings, J. London Math. Soc., (2), 6(1973), 256-258.
3. P.J. Bushell, On solutions of the matrix equation T′AT=A², Linear Algebra Appl., 8(1974), 465-469.
4. P.J. Bushell, The Cayley-Hilbert metric and positive operators, Linear Algebra Appl., 84(1986), 271-280.
5. L.H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120(1994), 743-748.
6. G. Fournier and M. Martelli, Eigenvectors for nonlinear maps. Topol. Methods Nonlin. Anal. 2(1993), 203-224.
7. M.-J. Huang and D.-Y. Chen, Existence and uniqueness of positive periodic solutions for a class of differential delay equations, preprint.
8. R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge 1994.
9. M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
10. M.G. Krein and M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk, 3(1), 23(1948); English transl. Amer. Math. Soc. Transl., No. 26.
11. X.-L. Liu and W.-T. Li, Existence and uniqueness of positive periodic solutions of functional differential equations, J. Math. Anal. Appl., 293(2004), 28-39.
12. C. Loewner, Über monotone Matrixfunktionen, Math. Z., 38(1934), 177-216.
13. A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
14. M. Martelli, Positive eigenvectors of wedge maps. Ann. Mat. Pura Appl. 145(1986), 1-32.
15. R.D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, in Fixed Point Theory, E. Fadell and G. Fournier, editors, Springer Lecture Notes in Math., 886, Springer Verlag, New York, 1981, pgs. 309-331.
16. R.D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Memoirs Amer. Math. Soc., 75(1988), No. 391.
17. A.J.B. Potter, Hilbert's projective metric applied to a class of positive operators, Ordinary and Partial Differential Equations, Lecture Notes in Mathematics, 564(1976), 377-382.
18. A.J.B. Potter, Applications of Hilbert's projective metric to certain classes of non-homogeneous operators, Quart. J. Math. Oxford, 28(1977), 93-99.
19. A.C. Thompson, On the eigenvectors of some not-necessarily-linear transformations, Proc. London Math. Soc.(3), 15(1965), 577-598.
20. E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, 1985.