The Equivalence of Convergence Results between Mann and Multistep Iterations with Errors for Uniformly Continuous Generalized Weak Φ

We prove the equivalence of the convergence of the Mann and multistep iterations with errors for uniformly continuous generalized weak Φ-pseduocontractive mappings in Banach spaces. We also obtain the convergence results of Mann and multistep iterations with errors. Our results extend and improve the corresponding results

The Equivalence of Convergence Results between Ishikawa and Mann Iterations with Errors for Uniformly Continuous Generalized Φ-Pseudocontractive Mappings in Normed Linear Spaces

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007

Remarks of Fixed Point for (ψ,φ)-Weakly Contractive Mappings

In this paper, we obtain a new fixed point theorem for weak contraction mappings on complete generalized metric spaces. Our result is different from that of (ψ,φ)-weakly contractive mappings which had been obtained by Lakzian–Samet’s [Appl. Math. Lett. 25 (2012) 902–906]. Moreover, an example is given to illustrate the usability of the obtained result

Probiotic Gastrointestinal Transit and Colonization After Oral Administration : A Long Journey

Orally administered probiotics encounter various challenges on their journey through the mouth, stomach, intestine and colon. The health benefits of probiotics are diminished mainly due to the substantial reduction of viable probiotic bacteria under the harsh conditions in the gastrointestinal tract and the colonization resistance caused by commensal bacteria. In this review, we illustrate the factors affecting probiotic viability and their mucoadhesive properties through their journey in the gastrointestinal tract, including a discussion on various mucosadhesion-related proteins on the probiotic cell surface which facilitate colonization.Funding Agencies|National Key Research and Development Program of China [2018YFC2000500]; National Natural Science Foundation of ChinaNational Natural Science Foundation of China (NSFC) [32001683]</p