Nonlinear operators and differential equations in abstract spaces - Chen, Yuqing (ProQuest Dissertations & Theses, 2000)
The aim of this dissertation is to study some open problems in nonlinear operator theory, and to prove the existence of solutions to evolution equations in abstract spaces. The main contributions of this dissertation are the following: We give a partial answer to Schauder's fixed point conjecture in Chapter 1 and we generalize the Crandall-Liggett exponential formula to locally convex spaces and show that a continuous accretive operator in a locally convex space needs not to be m-accretive, in Chapter 2. We establish a topological degree theory for multivalued mappings of class (S+) in Chapter 3. We prove that a Hausdorff continuous or upper semicontinuous accretive operator in a Banach space is m-accretive, or satisfies the range condition under additional conditions in Chapter 4. We generalize some variational inequalities and surjectivity results for monotone type operators in non-reflexive Banach spaces in Chapter 5. Finally, in Chapter 6, we prove that Massera's theorem still holds in the anti-periodic case and prove various existence results for periodic solutions to first and second order differential equations.