CONVERGENCE OF ISHIKAWA TYPE ITERATIVE SEQUENCE WITH ERRORS FOR QUASI-CONTRACTIVE MAPPINGS IN CONVEX METRIC SPACES

Some convergence theorems of Ishikawa type iterative sequence with errors for nonlinear general quasi-contractive mapping in convex metric spaces are proved. The results not only extend and improve the corresponding results of L. B. Ciric, Q. H. Liu, H. E. Rhoades and H. K. Xu, et al., but also give an affirmative answer to the open question of Rhoades-Naimpally- Singh in convex metric spaces.

ISHIKAWA TYPE ITERATIVE SEQUENCES WITH ERRORS FOR LIPSCHITZIAN φ-STRONGLY ACCRETIVE OPERATOR EQUATIONS IN ARBITRARY BANACH SPACES

In this paper, we investigate the problem of approximating solutions of the equations of Lipschitzian ψ-strongly accretive operators and fixed points of Lipschitzian ψ-hemicontractive operators by lshikawa type iterative sequences with errors. Our results unify, improve and extend the results obtained previously by several authors including Li and Liu (Acta Math. Sinica 41 (4)(1998), 845-850), and Osilike (Nonlinear Anal. TMA, 36(1)(1999), 1-9), and also answer completely the open problems mentioned by Chidume (J. Math. Anal. Appl. 151 (2)(1990), 453-461).

ON THE ITERATIVE APPROXIMATION PROBLEM OF FIXED OINTS FOR ASYMPTOTICALLY NON-EXPANSIVE TYPE MAPPINGS IN BANACH SPACES

Some iterative approximation theorems of fixed points for asymptotically nonexpansive type mappings in Banach spaces are obtained.

CONVERGENT PROBLEM OF ITERATIVE SEQUENCES FOR NONLINEAR MAPPINGS WITH ERROR MEMBERS IN BANACH SPACES

In this paper,the approximation problems of Ishikawa iteration with errors of fixed points for asymptotically nonexpansive mappings and asymptotically pseudocontractive mappings in arbitrary real Banach spaces are investigated.Some necessary condition and sufficient condition for the convergence of iterative sequences are given respectively.The results thus extend and improve some recent corresponding results.

INTERATIVE APPROXIMATION OF FIXED POINTS FOR ALMOST ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES

A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced,which includes a number of known classes of nonlinear Lipschitzian mappings and non_Lipschitzian mappings in Banach spaces as special cases; for example,the known classes of nonexpansive mappings,asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings.The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered.Not only S.S.Chang's inequality but also H.K.Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one.Moreover,Zhang Shi_sheng's method(Applied Mathematics and Mechanics(English Edition),2001,22(1):25-34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also,the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria.The above results unify,improve and generalize Zhang Shi_sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.

EXISTENCE AND APPROXIMATION OF SOLUTIONS TO VARIATIONAL INCLUSIONS WITH ACCRETIVE MAPPINGS IN BANACH SPACES

The purpose of this paper is to study the existence and approximation problem of solutions for a class of variational inclusions with accretive mappings in Banach spaces. The results extend and improve some recent results.

Chasing Sequencing Perfection:Marching Toward Higher Accuracy and Lower Costs

Next-generation sequencing(NGS),represented by Illumina platforms,has been an essential cornerstone of basic and applied research.However,the sequencing error rate of 1 per 1000 bp(10^(−3))represents a serious hurdle for research areas focusing on rare mutations,such as somatic mosaicism or microbe heterogeneity.By examining the high-fidelity sequencing methods developed in the past decade,we summarized three major factors underlying errors and the corresponding 12 strategies mitigating these errors.We then proposed a novel framework to classify 11 preexisting representative methods according to the corresponding combinatory strategies and identified three trends that emerged during methodological developments.We further extended this analysis to eight long-read sequencing methods,emphasizing error reduction strategies.Finally,we suggest two promising future directions that could achieve comparable or even higher accuracy with lower costs in both NGS and long-read sequencing.

Modified Ishikawa Iterative Process with Errors in Normed Linear Spaces

Ishikawa iterative sequences with errors different from the iterative sequences introduced by Liu and Xu are given. Moreover, the problem of approximating the fixed points of （ψ）-hemicontractive mapping in normed linear spaces by the modified Ishikawa iterative sequences with errors is investigated. The results presented in this paper improve and extend the results of the others.

BIGpre: A Quality Assessment Package for Next-Generation Sequencing Data

The emergence of next-generation sequencing （NGS） technologies has significantly improved sequencing throughput and reduced costs. However, the short read length, duplicate reads and massive volume of data make the data processing much more difficult and complicated than the first-generation sequencing technology. Al- though there are some software packages developed to assess the data quality, those packages either are not easily available to users or require bioinformatics skills and computer resources. Moreover, almost all the quality assessment software currently available didn＇t taken into account the sequencing errors when dealing with the du- plicate assessment in NGS data. Here, we present a new user-friendly quality assessment software package called BIGpre, which works for both Illumina and 454 platforms. BIGpre contains all the functions of other quality assessment software, such as the correlation between forward and reverse reads, read GC-content distribution, and base Ns quality. More importantly, BIGpre incorporates associated programs to detect and remove duplicate reads after taking sequencing errors into account and trimming low quality reads from raw data as well. BIGpre is primarily written in Perl and integrates graphical capability from the statistics package R. This package produces both tabular and graphical summaries of data quality for sequencing datasets from Illumina and 454 platforms. Processing hundreds of millions reads within minutes, this package provides immediate diagnostic information for user to manipulate sequencing data for downstream analyses. BIGpre is freely available at http：//bigpre.sourceforge.net/.

A survey on de novo assembly methods for single-molecular sequencing

Background:The single-molecular sequencing(SMS)is under rapid development and generating increasingly long and accurate sequences.De novo assembly of genomes from SMS sequences is a critical step for many genomic studies.To scale well with the developing trends of SMS,many de novo assemblers for SMS have been released.These assembly workflows can be categorized into two different kinds:the correction-and-assembly strategy and the assembly-and-correction strategy,both of which are gaining more and more attentions.Results:In this article we make a discussion on the characteristics of errors in SMS sequences・We then review the currently widely applied de novo assemblers for SMS sequences.We also describe computational methods relevant to de novo assembly,including the alignment methods and the error correction methods.Benchmarks are provided to analyze their performance on different datasets and to provide use guides on applying the computation methods.Conclusion:We make a detailed review on the latest development of de novo assembly and some relevant algorithms for SMS,including their rationales,solutions and results.Besides,we provide use guides on the algorithms based on their benchmark results.Finally we conclude the review by giving some developing trends of third generation sequencing(TGS).

Weak and Strong Convergence for Fixed Points of Asymptotically Non-expansive Mappings

A few weak and strong convergence theorems of the modified three-step iterative sequence with errors and the modified Ishikawa iterative sequence with errors for asymptotically non-expansive mappings in any non-empty closed convex subsets of uniformly convex Banach spaces are established. The results presented in this paper substantially extend the results due to Chang (2001), Osilike and Aniagbosor (2000), Rhoades (1994) and Schu (1991).