Geometric Properties Related To Fixed Point Property In Banach Spaces

Fixed point theory is an important part of functional analysis.Polish mathematician Banach proved Banach contraction mapping principle in 1922,since the result is beau-tiful and successfully solve to the existence of the implicit function theorem,differential equation initial value of the existence and application for a series of significant problem,so that mathematicians carried out in-depth and extensive research on the fixed point the-ory.Many problems in mathematics,physics and engineering can be transformed into the fixed point problem of some operators,therefore,the fixed point theory becomes an ex-cellent research tool to solve practical problems.Geometric theory of Banach space is an important branch of modern functional analysis,which has important applications in fixed point theory and other fields,it has attracted a large number of mathematical researchers to study the fixed point property with it.Therefore,it has theoretical significance and ap-plication values to study the geometric properties related to fixed point property in Banach spaces.Based on the geometric properties of Banach spaces,this dissertation discusses the fixed point problems in Banach spaces.Firstly,the relationships between a generalized von Neumann-Jordan constant in Banach spaces and the fixed point property,normal structure,uniform normal structure are studied,as well as the application of the generalized von Neumann-Jordan constant in fixed point theory for multivalued nonexpansive mappings is shown.Some sufficien-t conditions for a Banach space has the fixed point property and normal structure are shown respectively by the generalized von Neumann-Jordan constant,coefficient R(a,X),coefficient R(X),coefficients ε0(X)and ρ’X(0).By the relationships of generalized von Neumann-Jordan constant,weakly convergent sequence coefficient WCS(X),coefficien-t R(a,X)and coefficient M(X),we obtain a criterion for a Banach space X has normal structure.Moreover,a sufficient condition for a Banach space has uniform normal struc-ture is discussed with the hyper-power techniques.In addition,the relationships between in the generalized von Neumann-Jordan constant,coefficient ω(X),R(a,X)and WCS(X)are shown.Then a sufficient condition for multivalued nonexpansive mappings in Banach spaces have the fixed point property is obtained.Secondly,a generalized Garcia-Falset coefficient and a generalized Dominguez-Benavides coefficient are introduced,and the relationships between the two coefficients and the fixed point property are studied.A sufficient and necessary condition for Ba-nach spaces have the fixed point property are shown.Some criteria for a Banach space has the fixed point property are shown by the generalized Garcia-Falset coefficient,von Neumann-Jordan constant and James constant.The formulae for these two coefficients in the Orlicz sequence space lΦ equipped with the Luxemburg norm are deduced respective-ly.Moreover,the criteria for the generalized Garcia-Falset coefficient is less than 2 and the generalized Dominguez-Benavides coefficient is less than 1 +a are shown respectively in the Orlicz sequence space lΦ equipped with the Luxemburg norm.Furthermore,these two coefficients are calculated for the space lp(1<p<∞)and the Orlicz sequence space lΦ equipped with the Luxemburg norm generated by Orlicz function Φ(x)=(?)x4+x2.·Finally,the definitions of nearly uniformly smooth R and modulus of nearly uniform smoothness R are introduced,and some of their basic properties are investigated.More-over,we prove that weak nearly uniformly smooth R implies the fixed point property.And a equivalent definition of modulus of nearly uniform smoothness R for reflexive Banach spaces is given,criteria for Banach spaces is nearly uniformly smooth R is obtained.The relationship between modulus of nearly uniform smoothness R and parameterized James constant is studied.