Nonconvex Retracts And Fixed Point Theorems For Discontinuous Operators

The fixed point theory produced in topological transformation theory,an abstract mathematical theory which has important applications to analysis.The research of fixed point theory in nonlinear functional analysis has caused a lot of people’s interest.At the same time,people have made great achievements,such as fixed point theorems about cone expansion and compression,fixed point theorems of increasing operators.In this thesis,we further study the fixed point theory,on the basis of existed achievements at home and abroad.Firstly,the developments and present situations of the non-convex retract in Banach space and fixed point theorems for non-compactness and discontinuity of monotone operator are summarized.Secondly,we use concave and convex functionals to structure non-convex retracts D1∩D2 and D’1∩D’2 in Banach space.The conditions of theorems in literature are weakened or changed and some new theorems are obtained.Finally,we give new proofs of fixed point theorem of non-compactness and discontinuity of operator in regular or full regular cone in Banach space independent of Zorn’s lemma.It makes previous theorem more absolute.