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

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

On ω-Limit Sets of Zadeh’s Extension of Nonautonomous Discrete Systems on an Interval

Let I = [ 0 , 1 ] and f n be a sequence of continuous self-maps on I which converge uniformly to a self-map f on I. Denote by F ( I ) the set of fuzzy numbers on I, and denote by ( F ( I ) , f ^ ) and ( F ( I ) , f ^ n ) the Zadeh ′ s extensions of ( I , f ) and ( I , f n ) , respectively. In this paper, we study the ω -limit sets of ( F ( I ) , f ^ n ) and show that, if all periodic points of f are fixed points, then ω ( A , f ^ n ) ⊂ F ( f ^ ) for any A ∈ F ( I ) , where ω ( A , f ^ n ) is the ω -limit set of A under ( F ( I ) , f ^ n ) and F ( f ^ ) = { A ∈ F ( I ) : f ^ ( A ) = A }

Eventually Periodic Solutions of a Max-Type System of Difference Equations of Higher Order

We study in this paper the following max-type system of difference equations of higher order: xn=max{A,yn-k/xn-1} and yn=max{B,xn-k/yn-1}, n∈{0,1,2,…}, where A≥B>0, k≥1, and the initial conditions x-k,y-k,x-k+1,y-k+1,…,x-1,y-1∈(0,+∞). We show that (1) if AB>1, then every solution of the above system is periodic with period 2 eventually. (2) If AB=1>B, then every solution of the above system is periodic with period 2k or 2 eventually. (3) If A=B=1 or AB<1, then the above system has a solution which is not periodic eventually

Decay Estimates for a Type of Fuzzy Viscoelastic Integro-Differential Model

We consider a type of fuzzy viscoelastic integro-differential model in this paper. With the aid of some appropriate hypotheses, a unified method and the multiplier technique are implemented to get priori estimates precisely without constructing any auxiliary function. By establishing the estimation of energy function, we derive the stability result of the global solution, and we calculate the estimations of energy attenuation in exponential and polynomial forms, respectively

On an Impulsive Food Web System with Mutual Interference and Distributed Time Delay

In this paper, we present a predator-prey system with mutual interference and distributed time delay and study its dynamical behavior. Based on the existence and universality of mutual interference among species, it is necessary to further study an impulsive food web system. By using stability theory, slight perturbation technique, and comparison theorem, we obtain some theoretical results of the system, such as boundedness and permanence. Moreover, numerical experiments are used to verify the theoretical results and to explore the dynamical behavior of the system, which exhibits rich dynamical behavior such as chaotic oscillation, periodic oscillation, symmetry-breaking bifurcations, chaotic crises, and period bifurcation. Finally, we give some practical guidelines for biological systems based on the theoretical results and numerical experiments of the system