The periodic points of ε-contractive maps in fuzzy metric spaces

[EN] In this paper, we introduce the notion of ε-contractive maps in fuzzy metric space (X, M, ∗) and study the periodicities of ε-contractive maps. We show that if (X, M, ∗) is compact and f : X −→ X is ε-contractive, then P(f) = ∩ ∞n=1f n (X) and each connected component of X contains at most one periodic point of f, where P(f) is the set of periodic points of f. Furthermore, we present two examples to illustrate the applicability of the obtained results.Project supported by NNSF of China (11761011) and NSF of Guangxi (2020GXNSFAA297010) and PYMRBAP for Guangxi CU(2021KY0651)Sun, T.; Han, C.; Su, G.; Qin, B.; Li, L. (2021). The periodic points of ε-contractive maps in fuzzy metric spaces. Applied General Topology. 22(2):311-319. https://doi.org/10.4995/agt.2021.14449OJS311319222M. Abbas, M. Imdad and D. Gopal, ψ-weak contractions in fuzzy metric spaces, Iranian J. Fuzzy Syst. 8 (2011), 141-148.I. Beg, C. Vetro, D, Gopal and M. Imdad, (Φ, ψ)-weak contractions in intuitionistic fuzzy metric spaces, J. Intel. Fuzzy Syst. 26 (2014), 2497-2504. https://doi.org/10.3233/IFS-130920A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (1989), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4V. Gregori and J. J. Miñana, Some remarks on fuzzy contractive mappings, Fuzzy Sets Syst. 251 (2014), 101-103. https://doi.org/10.1016/j.fss.2014.01.002V. Gregori and J. J. Miñana, On fuzzy $Psi$-contractive sequences and fixed point theorems, Fuzzy Sets Syst. 300 (2016), 93-101. https://doi.org/10.1016/j.fss.2015.12.010V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets Syst. 125 (2002), 245-252. https://doi.org/10.1016/S0165-0114(00)00088-9J. Harjani, B. López and K. Sadarangani, Fixed point theorems for cyclic weak contractions in compact metric spaces, J. Nonl. Sci. Appl. 6 (2013), 279-284. https://doi.org/10.22436/jnsa.006.04.05X. Hu, Z. Mo and Y. Zhen, On compactnesses of fuzzy metric spaces (Chinese), J. Sichuan Norm. Univer. (Natur. Sei.) 32 (2009), 184-187.I. Kramosil and J. Michàlek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975), 336-344.D. Mihet, Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Sys. 159 (2008), 739-744. https://doi.org/10.1016/j.fss.2007.07.006D. Mihet, A note on fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Syst. 251 (2014), 83-91. https://doi.org/10.1016/j.fss.2014.04.010B. Schweizer and A. Sklar, Statistical metrics paces, Pacif. J. Math. 10 (1960), 385-389. https://doi.org/10.2140/pjm.1960.10.313Y. Shen, D. Qiu and W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Letters 25 (2012), 138-141. https://doi.org/10.1016/j.aml.2011.08.002S. Shukla, D. Gopal and A. F. Roldán-López-de-Hierro, Some fixed point theorems in 1-M-complete fuzzy metric-like spaces, Inter. J. General Syst. 45 (2016), 815-829. https://doi.org/10.1080/03081079.2016.1153084S. Shukla, D. Gopal and W. Sintunavarat, A new class of fuzzy contractive mappings and fixed point theorems, Fuzzy Sets Syst. 359 (2018), 85-94. https://doi.org/10.1016/j.fss.2018.02.010D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 222 (2013), 108-114. https://doi.org/10.1016/j.fss.2013.01.012D. Zheng and P. Wang, On probabilistic Ψ-contractions in Menger probabilistic metric spaces, Fuzzy Sets Syst. 350 (2018), 107-110. https://doi.org/10.1016/j.fss.2018.02.011D. Zheng and P. Wang, Meir-Keeler theorems in fuzzy metric spaces, Fuzzy Sets Syst. 370 (2019), 120-128. https://doi.org/10.1016/j.fss.2018.08.01

The depth and the attracting centre for a continuous map on a fuzzy metric interval

[EN] Let I be a fuzzy metric interval and f be a continuous map from I to I. Denote by R(f), Ω(f) and ω(x, f) the set of recurrent points of f, the set of non-wandering points of f and the set of ω- limit points of x under f, respectively. Write ω(f) = ∪x∈Iω(x, f), ωn+1(f) = ∪x∈ωn(f)ω(x, f) and Ωn+1(f) = Ω(f|Ωn(f)) for any positive integer n. In this paper, we show that Ω2(f) = R(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2.Project supported by NNSF of China (11761011, 71862003) and NSF of Guangxi (2018GXNSFAA294010) and SF of Guangxi University of Finance and Economics (2019QNB10).Sun, T.; Li, L.; Su, G.; Han, C.; Xia, G. (2020). The depth and the attracting centre for a continuous map on a fuzzy metric interval. Applied General Topology. 21(2):285-294. https://doi.org/10.4995/agt.2020.13126OJS285294212A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Sys. 64 (1994), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Sys. 27 (1989), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4V. Gregori and J. J. Miñana, Some remarks on fuzzy contractive mappings, Fuzzy Sets Sys. 251 (2014), 101-103. https://doi.org/10.1016/j.fss.2014.01.002V. Gregori and J. J. Miñana, On fuzzy Ψ-contractive sequences and fixed point theorems, Fuzzy Sets Sys. 300 (2016), 93-101. https://doi.org/10.1016/j.fss.2015.12.010V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets Sys. 125 (2002), 245-252. https://doi.org/10.1016/S0165-0114(00)00088-9X. Hu, Z. Mo and Y. Zhen, On compactnesses of fuzzy metric spaces (Chinese), J. Sichuan Norm. Univer. (Natur. Sei.) 32 (2009), 184-187.I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975), 336-344.C. Li and Y. Zhang, On connectedness of the Hausdorff fuzzy metric spaces, Italian J. Pure Appl. Math. 42 (2019), 458-466.D. Mihet, Fuzzy Ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Sys. 159 (2008), 739-744. https://doi.org/10.1016/j.fss.2007.07.006D. Mihet, A note on fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Sys. 251 (2014), 83-91. https://doi.org/10.1016/j.fss.2014.04.010J. Rodríguez-López and S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets Sys. 147 (2004), 273-283. https://doi.org/10.1016/j.fss.2003.09.007B. Schweizer and A. Sklar, Statistical metrics paces, Pacif. J. Math. 10 (1960), 385-389. https://doi.org/10.2140/pjm.1960.10.313Y. Shen, D. Qiu and W. Chen, Fixed point theorems in fuzzy metric spaces, Appl. Math. Letters 25 (2012), 138-141. https://doi.org/10.1016/j.aml.2011.08.002D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Sys. 222 (2013), 108-114. https://doi.org/10.1016/j.fss.2013.01.012D. Zheng and P. Wang, On probabilistic Ψ-contractions in Menger probabilistic metric spaces, Fuzzy Sets Sys. 350 (2018), 107-110. https://doi.org/10.1016/j.fss.2018.02.011D. Zheng and P. Wang, Meir-Keeler theorems in fuzzy metric spaces, Fuzzy Sets Sys. 370 (2019), 120-128. https://doi.org/10.1016/j.fss.2018.08.01

Reproductive Outcomes of In Vitro Fertilization and Fresh Embryo Transfer in Infertile Women With Adenomyosis: A Retrospective Cohort Study

BackgroundAdenomyosis is commonly encountered in infertile women; however, it is still unclear whether adenomyosis has a detrimental effect on in vitro fertilization and embryo transfer (IVF-ET) outcomes.MethodWe enrolled 1146 patients with adenomyosis and 1146 frequency-matched control women in a 1:1 ratio based on age, BMI, and basal follicle-stimulating hormone (FSH) level. After controlling for other factors, the rates of clinical pregnancy, miscarriage, live birth, and obstetric complications were compared between two groups.ResultsThere was no significant difference in clinical pregnancy rate between the two groups (38.1% vs. 41.6%; P=0.088). The implantation rate (25.6% versus 28.6%, P=0.027) and live birth rate (26% versus 31.5%, P=0.004) were significantly lower in the women with adenomyosis than in the controls. The miscarriage rate in the adenomyosis group was higher than that in the control group (29.1% versus 17.2%, P=0.001). After adjusting for confounding factors, multivariate analysis showed the clinical pregnancy rate was not statistically different between the two groups (OR: 0.852, P=0.070). In the adenomyosis group, the rate of miscarriage(OR: 1.877, P=0.000), placenta previa (OR: 2.996, P=0.042)and preeclampsia (OR: 2.287, P=0.042)were increased significantly, while live birth rate (OR: 0.541, P=0.000) was reduced significantly than control group.ConclusionAdenomyosis has negative effect on IVF-ET outcomes in which miscarriage risk increased, live birth rate reduced and obstetric complications increased

Global Behavior of the Difference Equation x

We study the following difference equation xn+1=(p+xn-1)/(qxn+xn-1), n=0,1,…, where p,q∈(0,+∞) and the initial conditions x-1,x0∈(0,+∞). We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by Kulenović and Ladas (2002) is true

Global behavior of a max-type system of difference equations of the second order with four variables and period-two parameters

In this paper, we study global behavior of the following max-type system of difference equations of the second order with four variables and period-two parameters $\left\{\begin{array}{ll}x_{n} = \max\Big\{A_n , \frac{z_{n-1}}{y_{n-2}}\Big\}, \ y_{n} = \max \Big\{B_n, \frac{w_{n-1}}{x_{n-2}}\Big\}, \ z_{n} = \max\Big\{C_n , \frac{x_{n-1}}{w_{n-2}}\Big\}, \ w_{n} = \max \Big\{D_n, \frac{y_{n-1}}{z_{n-2}}\Big\}, \ \end{array}\right. \ \ n\in \{0, 1, 2, \cdots\},$ where $A_n, B_n, C_n, D_n\in (0, +\infty)$ are periodic sequences with period 2 and the initial values $x_{-i}, y_{-i}, z_{-i}, w_{-i}\in (0, +\infty)\ (1\leq i\leq 2)$. We show that if \min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\} < 1 , then this system has unbounded solutions. Also, if $\min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\}\geq 1$, then every solution of this system is eventually periodic with period $4$.</p

Impaired large-scale cortico–hippocampal network connectivity, including the anterior temporal and posterior medial systems, and its associations with cognition in patients with first-episode schizophrenia

Background and objectiveThe cortico–hippocampal network is an emerging neural framework with striking evidence that it supports cognition in humans, especially memory; this network includes the anterior temporal (AT) system, the posterior medial (PM) system, the anterior hippocampus (aHIPPO), and the posterior hippocampus (pHIPPO). This study aimed to detect aberrant patterns of functional connectivity within and between large-scale cortico–hippocampal networks in first-episode schizophrenia patients compared with a healthy control group via resting-state functional magnetic resonance imaging (rs-fMRI) and to explore the correlations of these aberrant patterns with cognition.MethodsA total of 86 first-episode, drug-naïve schizophrenia patients and 102 healthy controls (HC) were recruited to undergo rs-fMRI examinations and clinical evaluations. We conducted large-scale edge-based network analysis to characterize the functional architecture of the cortico–hippocampus network and investigate between-group differences in within/between-network functional connectivity. Additionally, we explored the associations of functional connectivity (FC) abnormalities with clinical characteristics, including scores on the Positive and Negative Syndrome Scale (PANSS) and cognitive scores.ResultsCompared with the HC group, schizophrenia patients exhibited widespread alterations to within-network FC of the cortico–hippocampal network, with decreases in FC involving the precuneus (PREC), amygdala (AMYG), parahippocampal cortex (PHC), orbitofrontal cortex (OFC), perirhinal cortex (PRC), retrosplenial cortex (RSC), posterior cingulate cortex (PCC), angular gyrus (ANG), aHIPPO, and pHIPPO. Schizophrenia patients also showed abnormalities in large-scale between-network FC of the cortico–hippocampal network, in the form of significantly decreased FC between the AT and the PM, the AT and the aHIPPO, the PM and the aHIPPO, and the aHIPPO and the pHIPPO. A number of these signatures of aberrant FC were correlated with PANSS score (positive, negative, and total score) and with scores on cognitive test battery items, including attention/vigilance (AV), working memory (WM), verbal learning and memory (Verb_Lrng), visual learning and memory (Vis_Lrng), reasoning and problem-solving (RPS), and social cognition (SC).ConclusionSchizophrenia patients show distinct patterns of functional integration and separation both within and between large-scale cortico–hippocampal networks, reflecting a network imbalance of the hippocampal long axis with the AT and PM systems, which regulate cognitive domains (mainly Vis_Lrng, Verb_Lrng, WM, and RPS), and particularly involving alterations to FC of the AT system and the aHIPPO. These findings provide new insights into the neurofunctional markers of schizophrenia

Accuracy of triage strategies for human papillomavirus DNA-positive women in low-resource settings: A cross-sectional study in China

CareHPV is a human papillomavirus (HPV) DNA test for low-resource settings (LRS). This study assesses optimum triage strategies for careHPV-positive women in LRS