FIXED POINTS AND DUALITY OF CLOSED CONVEX SETS IN BANACH SPACES
Roxana-Irina Popescu · D-Scholarship@Pitt (University of Pittsburgh) · 2018
In the first chapter we construct a new example of an affine norm continuous mapping on a closed, convex, non-weakly compact set $C$ that cannot be extended to a continuous linear map on the entire space $X$. Although often used in the field of the fixed point theory, most of the examples known in the literature are restrictions of continuous, linear mappings from $X$ to $C$. The second chapter continues with a main focus on the notion of the affine dual of a closed, convex, bounded set. Using a theorem of M. Krein and D. Milman from Studia Mathematica 1940, one can show that certain spaces like $c_0$ and $L^1{[0,1]}$ are not dual spaces. However, it turns out that we can see them as affine dual spaces. In the third part of this thesis we provide a new proof that compactness in $\ell_1$ for closed, bounded, convex sets is equivalent with the fixed point property for cascading nonexpansive mappings. We also prove an analogue of this result in $L^1{[0,1]}$. The last part is dedicated to the study of the stability constant of the weak$^*$-fixed point property for the dual of separable Lindenstrauss spaces. Initiated in 1980 and 1982 by P. Soardi and T.C. Lim for the space $c_0$, we will now find a precise formula in the general case of an arbitrary predual of $\ell_1$ that depends only on a geometrical property of the unit ball of $\ell_1$ with respect to the predual considered. Therefore, this formula establishes a quantitative result in terms of geometric properties.