Several iterative algorithms for solving the split common fixed point problem of directed operators with applications

In this paper, we propose a new simultaneous iterative algorithm for solving the split common fixed point problem of directed operators. Inspired by the idea of cyclic iterative algorithm, we also introduce two iterative algorithms which combine the process of cyclic and simultaneous together. Under mild assumptions, we prove convergence of the proposed iterative sequences. As applications, we obtain several iteration schemes to solve the inverse problem of multiple-sets split feasibility problem. Numerical experiments are presented to confirm the efficiency of the proposed iterative algorithms.     

半序度量空间中公共二元随机重合点定理及其应用

在完备可分的半序度量空间中,引入了随机映射对(F,G)关于g随机半序弱增以及(F,G)随机半序弱增的定义,研究了在满足一定非线性压缩条件下的随机映射列F_k:Ω×X×X→X,k=1,2…,g:Ω×X→X和h:Ω×X→X的公共二元随机重合点与公共二元随机不动点问题,所得结果推广了已有文献中的一些不动点定理.     

一些新的数论函数及其均值公式

对于给定的自然数m,我们利用最大公约数和最小公倍数定义数论函数(m,n)和[m,n]/m.本文主要目的是研究这两个新的函数的渐近性质,利用解析方法得到这两个函数的几个渐近公式.     

Novel evolutionary models and applications to sequence alignment problems

In this paper, we present a novel graph-theoretical approach for representing a wide variety of sequence analysis problems within a single model. The model allows incorporation of the operations “insertion”, “deletion”, and “substitution”, and various parameters such as relative distances and weights. Conceptually, we refer the problem as the minimum weight common mutated sequence (MWCMS) problem. The MWCMS model has many applications including multiple sequence alignment problem, the phylogenetic analysis, the DNA sequencing problem, and sequence comparison problem, which encompass a core set of very difficult problems in computational biology. Thus the model presented in this paper lays out a mathematical modeling framework that allows one to investigate theoretical and computational issues, and to forge new advances for these distinct, but related problems. Through the introduction of supernodes, and the multi-layer supergraph, we proved that MWCMS is -complete. Furthermore, it was shown that a conflict graph derived from the multi-layer supergraph has the property that a solution to the associated node-packing problem of the conflict graph corresponds to a solution of the MWCMS problem. In this case, we proved that when the number of input sequences is a constant, MWCMS is polynomial-time solvable. We also demonstrated that some well-known combinatorial problems can be viewed as special cases of the MWCMS problem. In particular, we presented theoretical results implied by the MWCMS theory for the minimum weight supersequence problem, the minimum weight superstring problem, and the longest common subsequence problem. Two integer programming formulations were presented and a simple yet elegant decomposition heuristic was introduced. The integer programming instances have proven to be computationally intensive. Consequently, research involving simultaneous column and row generation and parallel computing will be explored. The heuristic algorithm, introduced herein for multiple sequence alignment, overcomes the order-dependent drawbacks of many of the existing algorithms, and is capable of returning good sequence alignments within reasonable computational time. It is able to return the optimal alignment for multiple sequences of length less than 1500 base pairs within 30 minutes. Its algorithmic decomposition nature lends itself naturally for parallel distributed computing, and we continue to explore its flexibility and scalability in a massive parallel environment.     

Coupled common fixed point theorems for w-compatible mappings in ordered cone metric spaces

We establish coupled coincidence point results for mixed g-monotone mappings under general contractive conditions in partially ordered cone metric spaces over solid cones. We also present results on existence and uniqueness of coupled common fixed points. Our results generalize, extend and unify several well known comparable results in the literature. To illustrate our results and to distinguish them from the earlier ones, we equip the paper with examples.     

赋范线性空间中渐近拟伪压缩型映象不动点的修改的广义Ishikawa迭代逼近

本文在去掉lim inf n→∞||xn||＜∞,∞∑n=0(kn-1)＜∞条件下,并用αn→o(n→∞)取代∑α2n＜∞,使用新的分析技巧,在赋范线性空间中建立了一致Lipschitz的渐近拟伪压缩型映象公共不动点的修改的广义Ishikawa迭代序列的强收敛定理,从而本质改进和推广了唐玉超,刘理蔚新近的结果.     

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258??271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140??152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400??411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372??79 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336??46 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334??345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835??842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77??8 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766??76 (2007)].     

Approximating common fixed points of nearly asymptotically nonexpansive mappings

We use an iteration scheme to approximate common fixed points of nearly asymptotically nonexpansive mappings.We generalize corresponding theorems of [1] to the case of two nearly asymptotically nonexpansive mappings and those of [9] not only to a larger class of mappings but also with better rate of convergence.     

数学分析中若干具有共性理论的问题

探讨数学分析中若干具有共性理论的问题,并重点阐述数学分析中若干关于一致问题与Cauchy定理的教学探索和实践.     

一元多项式学习中易犯的几个错误

结合一元多项式中的一些重要概念,如多项式的最大公因式、多项式的重根及不可约多项式等,分析一元多项式学习中易犯的错误,并强调运用定理时要注意其适用的条件和前提．