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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On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points

This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.The first author thanks the Spanish Ministry of Economy and Competitiveness for partial support of this work through Grant DPI2012-30651. He also thanks the Basque Government for its support through Grant IT378-10, and to the University of Basque Country by its support through Grant UFI 11/07

Best Proximity Points of Cyclic φ-Contractions on Reflexive Banach Spaces

We provide a positive answer to a question raised by Al-Thagafi and Shahzad (Nonlinear Analysis, 70 (2009), 3665-3671) about best proximity points of cyclic φ-contractions on reflexive Banach spaces

Effects of choline supplementation in mothers with hypothyroidism on the brain-derived neurotrophic factor gene expression changes in pre-pubertal offspring rats

Background Thyroid hormones play a vital function in the maturation in the course of mind development. Regarding the well-known effects of choline on brain-derived neurotrophic factor (BDNF), the study examined the effects of choline on hippocampal BDNF gene expression in maternal hypothyroidism rats. Methods and results To induce the hypothyroidism, 6-propyl-2-thiouracil was introduced to the ingesting water from the sixth day of gestation to twenty-first postnatal day (PND). Choline-treatment started twice a day on the first day of gestation until PND 21. On PND28, pups were sacrificed. The expression of BDNF gene was evaluated after the hippocampus was harvested. Our results demonstrated that both male and female pre-pubertal offspring rats' BDNF gene expression was decreased by developmental hypothyroidism. Choline increases the ratio of relative gene expression of BDNF in the hippocampus of males and females in the control/hypothyroidism group, especially in males. Conclusions It can be concluded that maternal choline supplementation on the first day of gestation until PND 21 improves brain development and cognitive function in pre-pubertal offspring rats regarding control/hypothyroidism groups