Geometrical Coefficients and Measures of Noncompactness

This thesis studies various measures of noncompactness and some geometrical coefficients in metric or Bauach spaces. These geometrical numbers are useful in the study of measures of noncompactness, some of which are interesting quantities in fixed point theory, In Chapter 1, we give some definitions and km.= au results for convenience and later use. Some of these results have been obtained very recently. In Chapter 2, we study a very useful geometrical coefficient in fixed point theory - the Lifschitz characteristic, which is also useful in the study of measures of noncompactness. We will compare this number with other interesting geometrical numbers, and estimate its values in certain spaces. Also some fixed point theorems which employ this quantity are given. This chapter contains the work of [WebZ-1] and part of the work of [WebZ-2]. In Chapter 3, the notion of normal structure and several normal structure coefficients in Banach space are studied. The notion of normal structure has proved to be a very useful one and various types of normal structure coefficients, such as N(X), WCS(X) and D(X), have been well studied in recent years. We will give several equivalent definitions of normal structure. Via these new characterizations of normal structure, some other normal structure coefficients are defined. We connect these geometrical numbers with N(X), WCS(X) and D(X), and use them to relate various measures of noncompactness. This chapter contains part of the work of [Z-2]. In Chapter 4, we give various connections between the set measure of noncompactness alpha(O), the ball measure of noncompactness beta(O) and the separation measure of noncompactness delta(O). In Chapter 5, we study alpha(T) and beta(T) when T is a linear operator. We relate beta(T) and beta(T*) by using a geometrical number