Approximation methods for nonlinear operator equations

Let be a real normed linear space and be a uniformly quasi-accretive map. For arbitrary define the sequence by where is a positve real sequence satisfying the following conditions: (i) ; (ii) . For , assume that 0$"> and that , where (the set of all positive integers): and is a strictly increasing function with . It is proved that a Mann-type iteration process converges strongly to . Furthermore if, in addition, is a uniformly continuous map, it is proved, without the condition on , that the Mann-type iteration process converges strongly to . As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

     

A reconsideration on convergence of three-step iterations for asymptotically nonexpansive mappings

We illustrate that the control conditions of the main convergence theorems of Yao and Noor [Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput. in press] are incorrect. We also provide new control conditions which are complementary to Nilsrakoo and Saejung’s results [W. Nilsrakoo, S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput. 181 (2006) 1026–1034].     

Three-step Iterations with Errors for Nonlinear Strongly Accretive Operator Equations

In this paper, we suggest and analyse a three-step iterative scheme with errors for solving nonlinear strongly accretive operator equation Tx = f without the Lipshitz condition. The results presented in this paper improve and extend current results in the more general setting.     

On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings

In this note, we speed up the convergence of the Picard sequence of iterations for strongly accretive and strongly pseudo-contractive mappings. Our results improve the results of Chidume [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint no. IC2000098; C.E. Chidume, Iterative Algorithms for Non-expansive Mappings and Some of Their Generalizations, in: Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, vol. 1, 2, Kluwer Acad. Publ, Dordrecht, 2003, pp. 383–429], and Liu [L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Amer. Math. Soc. 125 (2) (1997) 1363–1366], and some other known results. The technique of the proof, presented in this paper, is different from the technique used by Chidume.     

AN ITERATIVE PROCESS FOR FINDING APPROXIMATE SOLUTIONS TO NONLINEAR EQUATIONS OF STRONGLY ACCRETIVE OPERATORS

In this paper, we investigate the Ishikawa iteration process in a p-uniformly smooth Banach space X. We prove that the Ishikawa iteration process converges strongly to the unique solution of the equation Tx=f when T is a Lipschitzian and strongly accretive operator frow X to X, or to the unique fixed point of T when T is a Lipschitzian and strictly pseudocontractive mapping from a nonempty closed convex subset K of X into itself. Our results are the extension and improvements of the earlier and recent results in this field.     

Viscosity approximation method for accretive operator in Banach space

Let X$X$ be a uniformly smooth Banach space, C$C$ be a closed convex subset of X$X$, and A$A$ an m-accretive operator with a zero. Consider the iterative method that generates the sequence {_xn}$\left\{x_n\right\}$ by the algorithm

_xn+1=_αnf(_xn)+(1−_αn)_Jrnxn,$x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)J_{r_n}{x}_n\text{,}$

一致平滑Banach空间中一类非线性算子的Ishikawa迭代序列的收敛定理

使用新的技巧，我们得到了连续强增生算子或连续强伪压缩算子的Ｉｓｈｉｋａｗａ迭代序列的强收敛定理，推广或改进了近期相关的结果．     

Parallel methods for regularizing systems of equations involving accretive operators

In this paper, two parallel methods for solving systems of accretive operator equations in Banach spaces are studied. The convergence analysis of the methods in both free-noise and noisy data cases is provided.     

Convergence of the steepest descent method for accretive operators

Let be a uniformly smooth Banach space and let be a bounded demicontinuous mapping, which is also -strongly accretive on . Let and let be an arbitrary initial value in . Then the approximating scheme

converges strongly to the unique solution of the equation , provided that the sequence fulfills suitable conditions.

     

On iterations methods for zeros of accretive operators in Banach spaces

In this paper, three iterations are designed to approach zeros of set-valued accretive operators in Banach spaces. The first one is the continuous Picard type iteration involving the resolvent, the second one is the approximate Picard type iteration involving the resolvent and the third one is the Halpern type iteration involving the resolvent. Some strong convergence theorems for three iterations are proved.     

一致光滑Banach空间中一类非线性映象的迭代过程

本文引入了广义Lipschitz的概念，研究了广义Lipschitz强增生映象的Mann型迭代和Ishikawa型迭代过程的收敛性，所得结果统一和扩展了近期相关结果．     

一类非线性算子方程解的迭代逼近

在一般的Banach空间中讨论强增生算子方程解和严格伪压缩算子不动点的迭代逼近问题,我们的结果改进并推广了文[1,2,5,7]中的相应结果。     

Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type

In this paper we study hybrid iterative scheme for finding a common element of set of solutions of generalized mixed equilibrium problem, set of common fixed points of finite family of weak relatively nonexpansive mapping and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. Our results extend, improve and generalize the results of several authors which are announced recently. Application of our theorem to solution of equations of Hammerstein-type is of independent interest.     

Banach空间中渐近非扩张映射的不动点迭代

设D是一致凸Banach空间X的非空闭凸子集 ,T∶D→D是渐近非扩张映射且kn ≥ 1 ,∑ ∞n =1(kn- 1 ) <∞ .设T的不动点集F(T) ≠ ,T是全连续的 (X满足Opial条件 ) ,{xn},{yn},{zn}由定义 2给出 ,如果 ∑∞n =1cn <∞ ,∑ ∞n =1c′n <∞ ,∑ ∞n =1c″n <∞ ,且下列条件之一满足 :(i)b″n ∈ [a ,b] ( 0 ,1 ) ;b′n ∈ [0 ,β];bn ∈[0 ,α],αβ β <1 ;(ii)b′n ∈ [a ,b] ( 0 ,1 ) ;b″n ∈ [a ,1 ];bn ∈ [0 ,b];(iii)bn ∈[a ,b] ( 0 ,1 ) ;b′n ∈ [a ,1 ],则 {xn},{yn},{zn}强收敛于T的不动点 .( {xn}弱收敛于T的不动点 ) .     

值域有界的一类非线性算子不动点的带误差迭代逼近

设Ｘ为一致光滑实Ｂａｎａｃｈ空间·Ｔ：Ｘ→Ｘ为连续强增生算子·ｆ∈Ｘ·定义算子Ｓ：Ｘ→Ｘ为Ｓｘ＝ｆ－Ｔｘ＋ｘ,ｘ∈Ｘ·设αｎ｛｝∞ｎ＝０与βｎ｛｝∞ｎ＝０为两个给定的实数列在（０,１）中且满足条件：（ⅰ）αｎ→０,βｎ→０（ｎ→∞）·（ⅱ）∑∞ｎ＝０αｎ＝∞·假设ｕｎ｛｝∞ｎ＝０和ｖｎ｛｝∞ｎ＝０为Ｘ中两个序列且满足‖ｕｎ‖＝ｏ（αｎ）,‖ｖｎ‖→０（ｎ→∞）·ｘ０∈Ｘ,迭代序列ｘｎ｛｝定义为：（ＩＳ）ｘｎ＋１＝（１－αｎ）ｘｎ＋αｎＳｙｎ＋ｕｎｙｎ＝（１－βｎ）ｘｎ＋βｎＳｘｎ＋ｖｎ（ｎ≥０）｛若Ｓｘｎ｛｝,Ｓｙｎ｛｝有界,则ｘｎ｛｝强收敛于Ｓ的唯一不动点     

Weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings

To approximate a common fixed point of a countable family of continuous pseudocontractive mappings, we introduce an implicit iteration sequence. A necessary and sufficient condition for the convergence of a sequence of such iterates for countably many continuous pseudocontractive mappings is given. We also prove the convergence theorems of an implicit iteration sequence for a countable family of strictly pseudocontractive mappings.     

任意Banach空间Φ-强伪压缩映射不动点的Ishikawa迭代逼近

文章借助于对偶映射的定义,给出了任意Banach空间中LipschitzΦ-强伪压缩映射不动点的Ishikawa迭代收敛定理的简化证明,并且推广了目前相应的已知结果.     

Iterative Approximation of Fixed Points for Uniformly Continuous and Strongly Pseudocontractive Mappings in Smooth Banach Spaces

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＬｅｔＸｂｅａｒｅａｌＢａｎａｃｈｓｐａｃｅｗｉｔｈｎｏｒｍｙ·ｙａｎｄｄｕａｌＸ．ＴｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇｆｒｏｍＸｉｎｔｏ２ＸｉｓｄｅｆｉｎｅｂｙＪ（ｘ）＝｛ｆ∈Ｘ：〈ｘ,ｆ〉＝ｙｘｙ２＝ｙｆｙ２｝,ｗｈｅｒｅ〈·,·〉ｄｅｎｏｔｅｓｔｈｅｇｅｎｅｒａｌｉｚｅｄｄｕａｌｉｔｙｐａｉｒｉｎｇ．Ｉｔｉｓｗｅｌｌ－ｋｎｏｗｎｔｈａｔＸｉｓｕｎｉｆｏｒｍｌｙｓｍｏｏｔｈｉｆａｎｄｏｎｌｙｉｆＪｉｓｓｉ…     

Banach空间中一类非线性算子Ishikawa迭代序列收敛定理

在一般的Banach空间中,研究了Ishikawa迭代序列收敛问题,去掉了通常文献中关于空间X的一致光滑或S-一致光滑的严格要求,此外,还去掉或减弱了其它某些条件。因而本质地改进了近期文献的一系列相应定理。