Positive Fixed Points Of Nonlinear Operators And Distribution Of Polynomial Zeros

This thesis is concerned with the following problems:(1) Positive fixed points of nonlinear operators and their applications;(2) Distribution of polynomial zeros, including the stability of polynomials and the annular bound of polynomial zeros.It is divided into five chapters. We now briefly discuss the contents of each chapter.Chapter 1 provides a brief introduction and an overview of background materials and recent developments in nonlinear functional analysis (especially, the partial ordering methods) and the distribution of polynomial zeros.Chapter 2 contains some elementary concepts and basic results which will be used in this thesis, including cone, solid cone, normal cone in real Banach spaces and several classes of nonlinear operators. In addition, we introduce briefly the theory of semigroups of bounded linear operators and positive semigroups.Chapters 3 and 4 constitute a survey of positive fixed points of nonlinear operators.In Chapter 3, combining the partial ordering methods of nonlinear functional analysis and the theory of positive semigroups, we obtain two positive fixed point theorems for decreasing operators with nonlinear perturbation via iterative technique. The perturbation operators are sublinear and affine, respectively. To our knowledge, results of this type are scare in the literature and the theorems in this chapter are definitely new. As applications we give examples to nonlinear integral equations, and show the existence and uniqueness of positive solutions.In Chapter 4, we further investigate positive fixed points of mixed monotone operators. §4.1 is centered around two classes of mixed monotone operators with convexity and concavity, and the convergence of the iterative sequences to positive fixed points is proved. In §4.2, we get a new existence and uniqueness theorem of positive fixed points for mixed monotone operators with affine perturbation. These theorems extend the existing corresponding ones and are applied to nonlinear inte-...