Geometric Constant Of Banach Space And Application In Fixed Point Theory

Geometric constant is an important tool for studying the geometric structure and the properties of fixed point, so it is a hot topics to study the relation between geometric structure and geometric constant. Recently a good deal of investigations have focused on finding the sufficient conditions with various geometrical constants for a Banach space to have normal structure.Several modulus and constants in Banach space and their applications in fixed point theory are studied in this thesis. Meanwhile, a new geometric constant is introduced, its properties and the relationship among uniform nonsquareness, normal structure and the constant is studied. The contents of this thesis are divided into three parts.First, the generalization of modulus of convexity, the modulus of U -convexity and modulus of *W -convexity are investigated. Some sufficient conditions for which a Banach space has normal structure by the modulus, weak orthogonal coefficient and Benavides coefficient are presented, which generalize some well-known results of Gao and Saejung, furthermore,some examples are given to show that the conclusions are strict.Second, lower bounds for the weakly convergent sequence coefficient in term of Gao constant and parameterized James constant is established. Some sufficient conditions which imply normal structure are obtained by means of these bounds. Our results not only generalize the results of Gao but also imply the existence of fixed point for multivalued nonexpansive mappings.At last, in order to generalize the modulus of smoothness and Gao constant, a new geometric constant is introduced. Some functional properties of this constant,such as monotonicity, continuity and convexity are discussed. The smoothness of Banach space and the q (1 < q≤2)-uniformly smooth can be characterized in term of the new constant. Some exact values of new constant for some classical Banach spaces are given. Finally, some sufficient conditions with the new constant which imply a Banach space has normal structure are presented.