Geometric Constants And Their Applications Of Orlicz Spaces Equipped With The P-Amemiya Norm

Geometric theory of Banach spaces is an important branch of modern func-tional analysis. In 1965, it is proved by W.Kirk that a reflexive Banach space withnormal structure has fixed point property, which leads to an extensive researchand rapid development on the fixed theory of single-valued or multi-valued non-expansive mappings by using the geometric properties of Banach spaces. It iswell known that fixed point theory arises in a large number of applications suchas differential equation, control theory, algebra, economy balance theory and soon. Some geometric constants are always introduced according to the space,such as the modular of convexity, Opial modulus, Jordan-von Neumann constan-t, Garc′?a-Falset coe?cient etc. Therefore, geometric constants of Banach spaceand their applications for the fixed point theory in Banach spaces play an impor-tant role in theoretical aspect, as well as from a practical point of view.In this dissertation, some modulus and constants relevant with fixed pointtheory in Orlicz sequence space endowed with the p-Amemiya norm and in Or-licz function space are discussed, as well as the geometric properties of Banachspace described by these parameters. The main contributions of the researchwork are presented as follows.Firstly, the Opial property of lΦ,p is discussed in Orlicz sequence space en-dowed with the p-Amemiya norm and the calculation formula with respect toOpial modulus of lΦ,p is presented. The criteria are investigated which guaranteethe lΦ,p possesses Opial property, uniform Opial property, local Opial propertyand positive Opial property. Further, the necessary and su?cient condition isobtained which proves that lΦ,p possesses(L) property.Secondly, the calculation formula of the weakly convergent sequence co-e?cient of Orlicz sequence space lΦ,pendowed with the p-Amemiya norm aregiven, and furthermore the necessary and su?cient condition is obtained for lΦ,pto get weakly uniform normal structure. In addition, the su?cient condition isproposed which guarantees the isometry and isostructuralism of the subspace oflΦ,p, c0 and lp, which leads to the necessary and su?cient condition for lΦ,ppos-sessing fixed point property.Thirdly, the packing spheres constant as well as Kottman constant in Orliczsequence space lΦ,p equipped with the p-Amemiya norm is studied, and the pack-ing spheres constant in lpspace is calculated according to the results developedin this paper. Moreover, the relationship between the packing spheres constantand reflexivity of lΦ,p is discussed. As a result, a new point of view is introducedto describe the reflexivity of the space.Finally, the monotonicity property is investigated with respect to Orliczsequence space lΦ,p equipped with the p-Amemiya norm and function spaceLΦ,p, and the necessary and su?cient condition is obtained to guarantee the u-niform monotonicity, locally uniform monotonicity and strict monotonicity forlΦ,p(LΦ,p). The coe?cient of monotonicity of lΦ,p(LΦ,p) is calculated, as well asthe upper(lower) local coe?cients of monotonicity with respect to the points lo-cated on the unit sphere. Further, the problem of best approximation is studiedand analyzed.