Fixed point results for generalized cyclic contraction mappings in partial metric spaces

Rus (Approx. Convexity 3:171–178, 2005) introduced the concept of cyclic contraction mapping. P˘acurar and Rus (Nonlinear Anal. 72:1181–1187, 2010) proved some fixed point results for cyclic φ-contraction mappings on a metric space. Karapinar (Appl. Math. Lett. 24:822–825, 2011) obtained a unique fixed point of cyclic weak φ- contraction mappings and studied well-posedness problem for such mappings. On the other hand, Matthews (Ann. New York Acad. Sci. 728:183–197, 1994) introduced the concept of a partial metric as a part of the study of denotational semantics of dataflow networks. He gave a modified version of the Banach contraction principle, more suitable in this context. In this paper, we initiate the study of fixed points of generalized cyclic contraction in the framework of partial metric spaces. We also present some examples to validate our results.S. Romaguera acknowledges the support of the Ministry of Science and Innovation of Spain, grant MTM2009-12872-C02-01.Abbas, M.; Nazir, T.; Romaguera Bonilla, S. (2012). Fixed point results for generalized cyclic contraction mappings in partial metric spaces. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 106(2):287-297. https://doi.org/10.1007/s13398-011-0051-5S2872971062Abdeljawad T., Karapinar E., Tas K.: Existence and uniqueness of a common fixed point on partial metric spaces. Appl. Math. Lett. 24(11), 1894–1899 (2011). doi: 10.1016/j.aml.2011.5.014Altun, I., Erduran A.: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. article ID 508730 (2011). doi: 10.1155/2011/508730Altun I., Sadarangani K.: Corrigendum to “Generalized contractions on partial metric spaces” [Topology Appl. 157 (2010), 2778–2785]. Topol. 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Microfluidic droplet platform for ultrahigh-throughput single-cell screening of biodiversity

© 2017, National Academy of Sciences. All rights reserved.Ultrahigh-throughput screening (uHTS) techniques can identify unique functionality from millions of variants. To mimic the natural selection mechanisms that occur by compartmentalization in vivo, we developed a technique based on single-cell encapsulation in droplets of a monodisperse microfluidic double water-in-oil-in-water emulsion (MDE). Biocompatible MDE enables in-droplet cultivation of different living species. The combination of droplet-generating machinery with FACS followed by next-generation sequencing and liquid chromatography-mass spectrometry analysis of the secretomes of encapsulated organisms yielded detailed genotype/phenotype descriptions. This platform was probed with uHTS for biocatalysts anchored to yeast with enrichment close to the theoretically calculated limit and cell-to-cell interactions. MDE-FACS allowed the identification of human butyrylcholinesterase mutants that undergo self-reactivation after inhibition by the organophosphorus agent paraoxon. The versatility of the platform allowed the identification of bacteria, including slow-growing oral microbiota species that suppress the growth of a common pathogen, Staphylococcus aureus, and predicted which genera were associated with inhibitory activity

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