In chapter 2 , we present a family of iterative method with the convergence of order three . the family of iterative methods avoid evaluating the second frechet derivative 第二章，提出了一族具有三阶收敛迭代法，这族迭代法避免了求f ( x )的二阶导数。
2.
Some inaccurated definitions and properties in text [ 1 , 2 ] are introduced and reformed . the definition of infinite space , cover frechet ( v ) spaces , net cover spaces and their property are discussed 摘要指出了文献[ 1 , 2 ]对邻域空间等的定义存在的问题并进行了修正，给出了无限空间、盖邻空间、网盖空间的定义，并且讨论了它们的一些性质。
3.
In chapter 2 , we discuss lipschitz condition which is satisfied by the second frechet - derivative of operator through the use of recurrence relations , so as to make it meaningful in general and get the convergence theorem 第二章，通过运用递归技巧，对算子的二阶fr chet导数满足的lipschitz条件进行讨论，以使其在一般情况下有意义，并得到newton法的收敛性定理。
4.
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。
5.
Reich [ 2 ] proved the ergodic theorems to nonexpansive semigroups in hilbert spaces . takahashi and zhang [ 3 ] , tan and xu [ 4 ] extended baillon ' s theorem to asymptotically nonexpansive and asymptotically nonexpansive type semigroups in hilbert spaces . recently , reich [ 6 ] , bruck [ 5 ] , oka [ 7 ] gave the ergodic convergence theorems for nonexpansive , asymptotically nonexpansive mappings and semigroups in uniformly convex banach spaces with frechet differentiable norm . li and ma [ 13 ] obtained the ergodic convergence theorems for general commutative asymptotically nonexpansive type topological semigroups in reflexive banach space , which is a great breakthrough Baillon [ 1 ]首先在hilbert空间的非空凸闭子集上给出了非扩张映照的弱遍历收敛定理。 baillon的定理引起了很多数学家的兴趣， reich [ 2 ]在hilbert空间中证明了非扩张半群的遍历收敛定理。 takahashi和zhang [ 3 ] ， tan和xu [ 4 ]分别将baillon的定理推广到渐近非扩张半群及渐近非扩张型半群。
6.
By using bruck ' s lemma [ 10 ] , passty [ 31 ] extended the results of [ 1 , 16 ] to uniformly convex banach space with a frechet differentiable norm . however , there existed more or less limitations in their methods adopted . by using new techniques , chapter2 of this paper discussed the weak convergence theorem for right reversible semigroup of asymptotically nonexpansive type semigroup and the corresponding theorem for its almost - orbit in the reflexive banach space with a frechet differentiable norm or opial property Feattieranddotson 16 ]和bose [ l ]通过使用opial引理17 }在具弱连续对偶映照的一致凸b ~ h空间中证明了渐近非扩张映照的弱收敛定理， passty 31通过使用bruck引理10 ]把1 ， 16 ]的结果推广到具freehet可微范数的一致凸banach空间，然而，他们的证明存在着种种局限性。
7.
Chapter 2 of this paper , by using a new method of proof , we obtain the weak ergodic convergence theorem for general semigroups of asymptotically nonexpansive type semigroups in reflexive banach space . by theorem 2 . 1 of chapter 1 we get the weak ergodic convergence theorem of almost orbit for general semigroups of asymptotically nonexpansive type semigroups in reflexive banach space . by this method of proof , we give the weak ergodic convergence theorems for right reversible semigroups . by theorem 2 . 1 of chapter l , we generalize the result to almost orbit case . so we can remove a key supposition that almost orbit is almost asymptotically isometric . it includes all commutative semigroups cases . baillon [ 8 ] , hirano and takahashi [ 9 ] gave nonlinear retraction theorems for nonexpansive semigroups . recently mizoguchi and takahashi [ 10 ] proved a nonlinear ergodic retraction theorem for lipschitzian semigroups . hirano and kido and takahashi [ 11 ] , hirano [ 12 ] gave nonlinear retraction theorems for nonexpansive mappings in uniformly convex banach spaces with frechet differentiable norm . . in 1997 , li and ma [ 16 ] proved the ergodic retraction theorem for general semitopological semigroups in hilbert space without the conditions that the domain is closed and convex , which greatly extended the fields of applications of ergodic theory . chapter 2 of this paper , we obtain the ergodic retraction theorem for general semigroups and almost orbits of asymptotically nonexpansive type semigroups in reflexive banach spaces . and we give the ergodic retraction theorem for almost orbits of right reversible semitopological semigroups 近年来， bruck [ 5 ] ， reich [ 6 ] ， oka [ 7 ]等在具frechet可微范数的一致凸banach空间中给出了非扩张及渐近非扩张映射及半群的遍历收敛定理。 li和ma [ 13 ]在具frechet可微范数的自反banach空间中给出了一般交换渐近非扩张型拓扑半群的遍历收敛定理，这是一个重大突破。本文第二章用一种新的证明方法在自反banach空间中，研究了扬州大学硕士学位论文2一般半群上的( r )类渐近非扩张型半群的弱遍历收敛定理，即:定理3 . 1设x是具性质( f )的实自反banach空间， c是x的非空有界闭凸子集， g为含单位元的一般半群， s =仕工， 。
