This dissertation consists of two parts . in part one , the weighted approximation by the linear operators in classical spaces and approximation in orlicz spaces are studied ; in part two , the approximation of multivariate linear operators is discussed 本学位论文分为上下两篇，上篇主要为一元线性算子在经典空间的加权逼近和orlicz空间的逼近：下篇为多元线性算子在经典空间的逼近和加权逼近。
Then the relation of regular operators , bounded operators and linear operators on banach lattices are given , that is lr ( e , f ) lb ( e , f ) l ( e , f ) ; order dual , operator dual and algebra dual are related , i . e . e " c e * c e # 然后给出banach格空问上正则算子，有界算子和线性算子的关系： lve ； f ） of 。 ef ） clp ， f ） ；给出了序对偶，算子对偶和代数对偶的关系： e ’ ce ” ce个然后引入赋值映射人证明了j是保格运算的格同态
In the third chapter , we derive a new family of deformed halley methods without the evaluation of the second frechet - derivative to approximate the roots of nondifferentiable equations in banach space . we also provided a existence - uniqueness theorem and a new system of recurrence relations 在第三章中，构造了一族避免二阶fre chet导数的修正halley迭代，用其去逼近banach空间中非线性算子方程的解，同时给出了存在唯一性定理和一种新型的递归关系。
In this paper we use pointwise modulus of smoothness ( f , t ) to study approximation direct theorem and equivalent theorem for some linear operators and quasi - interpolant operators ; using pointwise modulus we discuss the strong converse inequality on k - functional ; and using a modified weighted k - functional and weighted modulus of smoothness we study approximation with jacobi weight on operator with non - zero first order moments 本文利用点态光滑模_ ( ~ ) ~ ( 2r ) ( f ， t )来研究某些线性算子及逆中插式逼近正定理和等价定理；利用点态光滑模讨论其关于k -泛函的强逆不等式；同时利用一种改变的带权k -泛函和带权光滑模研究一阶矩不为零的算子的点态带jacobi权逼近。
We also give out the notion of mild degenerate a - times integrated existence family , and prove that the wellposedness of the a - times abstract cauchy problems is equivalent to mild degenerate a - times integrated existence famliy generated by operator a where a satisfies with some conditions and degenerate a - times integrated semigroups mild generated by a . at last , we conclude the generation theorem of degenerate a - times integrated semigroups . and we prove that degenerate a - times integrated semigroups mild generated by a is equivalent to generated by a 我们也给出了mild退化-次积分存在族的概念。我们证明了， ( + 1 ) ( r ) -次抽象cauchy问题的适定性和闭线性算子a在一定条件下， a的mild退化-次积分存在族以及a次生成退化的-次积分半群是等价的。最后，我们也给出了退化-次积分半群的生成定理。
It is wekk known that the fractional integrals , the commutators for the macinkiewicz integrals and the multilinear operators play a profound and extensive role in harmonic analysis and the partial equations . in this thesis , we ' ll discuss the mapping properties of these kinds of operators . our thesis consists of five chapters . chapterl is the introductions 由于分数次积分算子， marcinkiewicz算子的交换子，多线性算子是调和分析的重要算子，他们不仅在调和分析中有着重要的地位，而且在偏微分方程具有着及其重要的作用。本文致力于这些算子的研究。
Using the index theory of fixed points for a - proper semilinear operators , we study the existence of positive solutions for the multiple point boundary value problems at rersonance . the existence theorems of positive solutions for three - point boundary value problems at rersonance for a kind of nonlinear second - order differential equation are obtained . 3 利用a - proper半线性算子不动点指标理论，我们研究了多点共振边值问题正解的存在性，得到了一类二阶非线性微分方程三点共振边值问题正解存在性定理。
Similarly , we prove that the c - wellposedness of the a - times abstract cauchy problems is equivalent to mild degenerate a - times integrated c - existence famliy mild generated by operator a where a satisfies with some conditions and degenerate a - times integrated semigroups mild generated by a . finally , we obtain the generation theorem of degenerate a - times integrated c - semigroups . and we prove that degenerate a - times integrated c - semigroups mild generated by a is equivalent to generated by a 我们同样证明了( + 1 ) ( r ) -次抽象cauchy问题的c -适定性和闭线性算子a在一定条件下，其mild退化-次积分c存在族以及a次生成退化-次积分正则半群是等价的。我们也证明了退化-次积分正则半群的生成定理。
In this thesis , the solution of kolmogorov backward differential equations in birth and death process theory has been proved to be well - posedness by using the theories and methods of linear operator co semigroup in functional analysis . and the existence of superior eigenvalue of the coefficient matrix of the equations has been studied by using the theories of positive operator and conjugate operator 论文主要用泛函分析中的线性算子c _ 0半群理论研究生灭过程理论中柯尔莫哥洛夫向后微分方程组解的适定性，及用正算子和共轭算子的理论和一些结论研究了该方程组系数矩阵算子的占优本征值的存在性问题。
