Measure Theory Of Noncompactness And Its Applications

In this dissertation,we study measure theory of noncompactness of Banach spaces and its application.As a result,we show that every infinite dimensional Banach space admits a regular measure of noncompactness which is not equivalent to the Hausdorff measure of noncompactness.Thus,the 40-year "fundamental question" is completely resolved.Then,we show that every monotone sublinear measures of noncompactness on a Banach space can be expressed as a support function defined on a space.As their application,we show a characterization guaranteeing that a strong-to-weak continuous self-mapping defined on a nonempty closed bounded convex set of a Banach space has a fixed point in terms of general regular measure of noncompactness;and prove some classical results in terms of Kuratowski's measure,or,Hausdorff's measure holds again for general(inequivalent)regular measure of noncompactness.We also show a characterization for existence of global attrac-tors of strong-to-weak continuous operator semigroups in Lp-spaces.As an example,we prove an existence theorem for the semigroups generated by a class of reaction-diffusion equations.