【作者】 张玲玲；

【导师】 梁展东；

【作者基本信息】 郑州大学 ， 基础数学， 2003， 博士

【摘要】 全文共分三章： 第一章，讨论了一类α—t型凹凸的混合单调算子，给出了其存在唯一不动点的充分必要条件。 通过运用锥理论，采用上下解方法、单调迭代技巧等讨论了一类α—t型凹凸的混合单调算子，给出了其存在唯一不动点的充分必要条件，含盖了相关文献的部分工作，所给的充要条件弥补了以往只给出充分条件的局限性，对文献中的相关工作做了本质性的改进；对于广义α—t型凹凸混合单调算子，通过引入“伴随列”的概念，也给出了其存在唯一不动点的充分必要条件；对于次线性混合单调算子，通过运用双下标排列的对角线方法，也给出了其存在唯一不动点的充分必要条件；通过适当的变换技巧，讨论了(α，β)型凹凸混合单调算子，在一定的条件下，给出了其存在唯一不动点的充分必要条件。 第二章，讨论了ε—α正齐次算子以及α正齐次算子对ε—α正齐次算子的逼近问题。 由于α—正齐次算子的讨论，结果比较完善，混合单调的α—正齐次算子作为α—t型凹凸混合单调算子的特殊情况，我们自然有其存在不动点的充要条件。本章将α正齐次算子推广为ε—α正齐次算子，并讨论了ε—α正齐次算子的连续性、范数性质、以及算子空间的完备性等。同时，探讨了α—正齐次算子对ε—α正齐次算子的逼近问题。 第三章，讨论了非线性混合单调算子在Banach空间中非线性脉冲微(积)分方程方面的应用。 作为混合单调算子理论的应用，本章讨论了非线性混合单调脉冲积微分方程和混合单调非线性椭圆方程方面的一些问题，不同程度地削弱了原有的条件，推广了已知的结果；还利用锥理论并结合算子半群的性质及其主要特征讨论了非线性脉冲发展方程初值问题、周期边值问题，给出了混合单调非线性脉冲发展方程的耦合周期解以及存在唯一解的充要条件。更多还原

【Abstract】 This paper includes three chapters.In chapter 1, a class of mixed monotone operators with a - t type convexity and concavity has been discussed , The sufficient and necessary conditions for existence and uniqueness of their fixed points are given.In this chapter,a class of mixed monotone operators with a - t type convexity and concavity has been discussed through cone theory and by using the method of lower-upper solutions,the skill of monotone iterative. The sufficient and necessary conditions for existence and uniqueness of their fixed points are given.The results in relevant literatures are generalized or included.These sufficient and necessary conditions are out of the limitation of only giving the sufficient conditions in previous papers,and improve the relevant results essentially;For a class of mixed monotone operators with general a - t type convexity and concavity ,the sufficient and necessary conditions for existence and uniqueness of their fixed points are given by introducing the concept of ?adjoint sequence ". For a class of sublinear mixed monotone operators ,the sufficient and necessary conditions for existence and uniqueness of their fixed points are also given by using diagonal process with double-lower ordinate arrangement.And a class of mixed monotone operators with (a, /3)type convexity and concavity is also studied through some proper transforms, the sufficient and necessary conditions for existence and uniqueness of their fixed points are also given under certain conditions.In chapter 2, e - a type positive homogeneous operators are studied,and the approximation to e - Q type positive homogeneous operators by a- type positive homogeneous operators is discussed.For a- type positive homogeneous operators,the research work has been complete. As a special case of mixed monotone operators with a - t type convexity and concavity , the mixed monotone operators with a- type positive homogeneity have similar results on the sufficient and necessary conditions for existence and uniqueness of their fixed points. In this chapter,a- type positive homogeneous operators are generalized as e -a type positive homogeneous operators. Moreover,the continuity,properties of norm,and completeness of operator space are investigated for e - a type positive homogeneous operators. The approximation to e -a type positive homogeneous operators by a- type positive homogeneous operators is preliminarily discussed.In chapter 3, the applications of mixed monotone operators to differential(-integro) equations in Banach space are given.As the applications of mixed monotone operators theory,some classes of equations are considered.Some initial value problems and boundary value problems for mixed monotone nonlinear impulsive integro-differential equations and nonlinear elliptic equations are discussed.Some known results are generalized under weak conditions. And initial value problems and periodic boundary value problems for mixed monotone nonlinear impulsive evolution equations are discussed by mixed monotone operators theory with considering the main properties of operator semigroup. The sufficient and necessary conditions for existence and uniqueness of their solution and coupled solution are obtained.更多还原