不动点理论中的半序方法及其应用

The normality of cones is necessary for the recent researches on fixed point theory of discontinuous opeartors in Banach spaces, and there are rarely references involving the non-normal case. In this project, we are main concerned with fixed point theory in Banach spaces with non-normal cones which leads to that the Sandwich theorem is invalid in the sense of norm convergence. To fulfill this end, we need to introduce the concept of weak convergence base on the partial order of the cones, and then discuss the convergence of sequences, Sandwich theorem in the sense of weak convergence, the convergence of vector series, and so on. It is known that generalized metric does not satisfy all the three axioms, the lackness of either of the three axioms will certainly lead to the difficulties in the researches of fixed point theory. In this project, we shall further consider fixed point theory of nonlinear contractions，the existence of largest and least fixed points of Caristi-type mappings and Caristi's fixed point theory by eatablishing the existence of maximal and minimal elements and the sufficiency for uniqueness of limits. It seems that the researches of this project will enrich fixed point theory of discontinuous operators and enlarge its applications.

目前Banach空间中非连续算子不动理论研究大多要求锥是正规的，有关非正规锥情形的工作十分少见。正规性的缺乏使得范数收敛意义下的Sandwich定理未必成立，为了克服这一不足， 本项目拟引入基于锥半序的弱收敛概念，并通过讨论序列弱收敛性判别、弱收敛意义下的Sandwich定理及向量级数的弱收敛性判别等基本性质，来研究非正规锥情形Banach空间中非连续凹凸型及Lipschitz型算子和非正规锥度量空间中非线性压缩算子的不动点问题。广义度量空间不全满足度量的三条公理，任一公理的缺失都给其不动点研究带来相当困难，本项目拟通过考虑极大极小元存在性、收敛序列极限唯一的充分性条件等，对半序广义度量空间中非连续算子不动点理论进行更深入的研究，包括非线性压缩型算子及Caristi型算子的最大最小不动点和Caristi型算子不动点定理等。本项目研究有望进一步丰富非连续算子不动点理论研究并拓广其应用。

本研究通过引入基于锥半序的弱收敛概念，并讨论序列弱收敛性判别、弱收敛意义下的Sandwich定理及向量级数的弱收敛性判别等基本性质，研究了非正规锥情形Banach空间中非连续凹凸型及Lipschitz型算子和非正规锥度量空间中非线性压缩算子的不动点问题，针对广义度量空间不全满足度量的三条公理，某一公理的缺失都给其不动点研究带来相当困难，本研究通过考虑极大极小元存在性、收敛序列极限唯一的充分性条件等，对半序广义度量空间中非连续算子不动点理论进行更深入的研究，包括非线性压缩型算子及Caristi型算子的最大最小不动点和Caristi型算子不动点定理等。本研究有望进一步丰富非连续算子不动点理论研究并拓广其应用。