Study On Non-linear Operator In Semi-ordered Banach Space

Nonlinear functional analysis has become an important direction of research in modern mathematics,nonlinear operator theory is also an important part of nonlinear functional analysis.Since the early 1980s,Guo Dajun,Sun Jingxian,Du Yihong,Liu Zhaoli,Zhu Chuanxi and Li Fuyi have been researched the nonlinear problems which lack of compactness or lack of continuity,and obtained some new results.In this paper,we mainly study some problems in semi-ordered Banach space and obtain a few new results.This paper is divided into four parts.In chapter one,the backgrounds and current situation of operator theory in Banach space are introduced and the preliminaries of Banach space are given.In chapter two,we study the existence and uniqueness of fixed point for decreasing strict-set-contraction operator in product space.Under the weak continuous condition,we obtain the existence and uniqueness of fixed point for decreasing operator and we give an application of these results; In chapter three,we get several positive fixed point theorems of decreasing operator and the existence and uniqueness theorems of positive fixed point for operator C=A+B,C=D-A in real Banach space where the order is decided by a normal cone; In chapter four,we obtained some existence and uniqueness theorems of fixed point for some mixed monotone operators in semi-ordered Banach space.