Hilbert's projective metric and its applications to eigenvalue problems

• Main Authors: Chao-Ya Huang, 黃昭雅
• Other Authors: Min-Jei Huang
• Format: Others
• Language: en_US
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/28311592360214273532

博士 === 國立清華大學 === 數學系 === 95 === 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.