Search

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 $.

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.

Abstract In this paper, we study the eventual periodicity of the fuzzy max-type difference equation x n = max { C , x n − m − k x n − m } , n ∈ { 0 , 1 , … } $x_{n} =\max \{C , \frac{x_{n-m-k}}{x_{n-m} }\}, n\in \{0,1,\ldots \} $ , where m and k are positive integers, C and the initial values are positive fuzzy numbers. Let the support supp C = { t : C ( t ) > 0 } ‾ = [ C 1 , C 2 ] $\operatorname{supp} C=\overline{\{t : C(t) > 0\}}=[C_{1},C_{2}]$ of C. We show that: (1) if C 1 > 1 $C_{1}>1$ , then every positive solution of this equation equals C eventually; (2) there exists a positive fuzzy number C with C 1 = 1 $C_{1}=1$ such that this equation has a positive solution which is not eventually periodic; (3) if C 2 ≤ 1 $C_{2}\leq 1$ , then this equation has a positive solution which is not eventually periodic; (4) if C 1 < 1 < C 2 $C_{1}

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.

Abstract In this paper, we study the following max-type system of difference equations of higher order: { x n = max { A , y n − t x n − s } , y n = max { B , x n − t y n − s } , n ∈ { 0 , 1 , 2 , … } , $$ \textstyle\begin{cases} x_{n} = \max \{A ,\frac{y_{n-t}}{x_{n-s}} \}, \\ y_{n} = \max \{B ,\frac{x_{n-t}}{y_{n-s}} \},\end{cases}\displaystyle \quad n\in \{0,1,2,\ldots \}, $$ where A , B ∈ ( 0 , + ∞ ) $A,B\in (0, +\infty )$ , t , s ∈ { 1 , 2 , … } $t,s\in \{1,2,\ldots \}$ with gcd ( s , t ) = 1 $\gcd (s,t)=1$ , the initial values x − d , y − d , x − d + 1 , y − d + 1 , … , x − 1 , y − 1 ∈ ( 0 , + ∞ ) $x_{-d},y_{-d},x_{-d+1},y_{-d+1}, \ldots , x_{-1}, y_{-1}\in (0,+ \infty )$ and d = max { t , s } $d=\max \{t,s\}$ .

Abstract In this paper, we investigate hybrid adaptive synchronization issue for a class of perturbed fractional-order chaotic systems with nondeterministic nonlinear terms. On the basis of fractional-order extended version of Lyapunov stability criterion, a novel fuzzy adaptive synchronization control protocol coupled with backstepping-based method is constructed, ensuring that the synchronization errors converge to a sufficiently small region of the origin. In order to avert the occurrence of “explosion of complexity”, we take advantage of a fuzzy logic system to estimate the unknown systematic term approximately in every backstepping step. Finally, some numerical simulations are given to exemplify the effectiveness of the proposed approach.

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.

In this paper, we address the model of global attractor formulated in the form of evolution differential inclusions with second order in Banach spaces. Firstly, based on the fixed point theorem, the existence result of mild solutions is deduced. Then, by implementing the measure of noncompactness, the existence of global attractor associated with m-semiflow is validated. Finally, a concrete application of the main result is demonstrated to enhance the practical signification.

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

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.

Abstract In this paper, we study the following max-type system of difference equations: {xn=max{An,yn−1xn−2},yn=max{Bn,xn−1yn−2},n∈{0,1,2,…}, $$\textstyle\begin{cases}x_{n} = \max \{A_{n},\frac{y_{n-1}}{x_{n-2}} \},\\ y_{n} = \max \{B_{n} ,\frac{x_{n-1}}{y_{n-2}} \}, \end{cases}\displaystyle n\in \{0,1,2,\ldots\}, $$ where An,Bn∈(0,+∞) $A_{n},B_{n}\in(0, +\infty)$ are periodic sequences with period 2 and the initial values x−1,y−1,x−2,y−2∈(0,+∞) $x_{-1},y_{-1},x_{-2},y_{-2}\in (0,+\infty)$. We show that every solution of the above system is eventually periodic.

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

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 } .

Our aim in this paper is to investigate the convergence behavior of the positive solutions of a higher order fuzzy difference equation and show that all positive solutions of this equation converge to its unique positive equilibrium under appropriate assumptions. Furthermore, we give two examples to account for the applicability of the main result of this paper.

Let G a graph and f:G→G be a continuous map. Denote by h(f), R(f), and SA(f) the topological entropy, the set of recurrent points, and the set of special α-limit points of f, respectively. In this paper, we show that h(f)>0 if and only if SA(f)-R(f)≠∅.

The aim of this paper is to investigate eventual periodicity of the following max-type system of difference equations of higher order with four variables ... [ABSTRACT FROM AUTHOR]

In this paper, we establish a regularization method of a class of quasi-mixed equilibrium problems in Banach spaces. First, an existence result of Minty inequality is given by applying the Tychonov fixed point theorem. Second, we show that the existence of solutions of the quasi-mixed equilibrium problems. Finally, some assumptions are proposed to ensure the boundedness and the convergence of regularized solutions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the system formulated by abstract second order nonlinear evolution differential equations which are subject to a generalized mixed variational inequalities. Firstly, based on Ky Fan inequality theorem, we examine that the solution set of variational inequalities is bounded, closed and convex by getting rid of the rigid restriction of monotonicity. Afterwards, the existence of solutions for a class of nonlinear differential equation is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Ɛ-contractive maps in fuzzy metric space (X,M,) and study the periodicities of Ɛ-contractive maps. We show that if (X,M,) is compact and f: X -! X is Ɛ-contractive, then P(f) = \1 n=1fn(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. [ABSTRACT FROM AUTHOR]

In this note, we establish a new Lyapunov-type inequality for a class of higher order dynamic equations under anti-periodic boundary conditions. Applying the inequality, we obtain the lower bound of the eigenvalue for a related eigenvalue problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the eventual periodicity of the following fuzzy max-type difference equation zn = max {1/zn-m, αn/zn-r}, n = 0,1,..., where {αn}n ≥ 0 is a periodic sequence of positive fuzzy numbers and the initial values z - d, z-d + 1,..., z-1 are positive fuzzy numbers with d = max {m,r}. We show that if max (supp αn) < 1, then every positive solution of this equation is eventually periodic with period 2m. [ABSTRACT FROM AUTHOR]

The purpose of this work is to investigate the convergence of the solutions of the following max-type difference equation zn = max {1/zn-s,Pn/zαnn-t}, n=0,1,2, . . . , where s, t ϵ {1, 2, 3, ...} with s ≠ t, αn ϵ (0, 1) is an s-periodic sequence, {Pn}+∞n=0=0 is a constant sequence satisfying Pn ϵ (0, 1] for every n ≥ 0. We show that if {zn}+∞n=-r (r = max{s, t}) is a positive solution of the above equation with the initial conditions z-r, z-r+1, . . . , z-1 ϵ (0,+∞), then limn→∞ zn = 1 or {z2sn+k}+∞n=0 is eventually monotone for every 0 ≤ k ≤ 2s - 1. Further, we show that if Pn is a periodic sequence, s = 1 and t is even, then limn→∞ zn = 1 or {zn}+∞n=-t is eventually periodic with period 2. [ABSTRACT FROM AUTHOR]

In this paper, we obtain several new Lyapunov-type inequalities for the following quasi-Hamiltonian systems x∆(t) = -W(t)x(σ(t))-U(t)∣y(t)∣p-2y(t); y∆(t) = V (t)∣x(σ(t))∣q-2x(σ(t))+WT (t)y(t) on the time scale interval [a; b]T [a; b]∨T for some [a; b] T (σ(a) < b), where U and V are real n ×n symmetric matrix-valued functions on [a; b]T with U being positive definite, W is real n × n matrix-valued function on [a; b]T with I + μ(t)W being invertible, and x; y are real vector-valued functions on [a; b]T. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the eventual periodicity of the following max-type system of difference equations {xn = max{1/yn-m, min {1, α/xn-r}}, yn = max {1 xn-s, min {1, β/yn-t}}, n = 0, 1, 2, ... , where α, β ∊ (0,+∞), m, s, r, t ∞ {1, 2, ...} with r ≠ m and t ≠ s. We show that every solution of this system with the initial values x-d, y-d, x-d+1, y-d+1; ... , x-1, y-1 ∊ (0,+∞) is eventually periodic with period m + s, where d = max{m; s; r; t}. [ABSTRACT FROM AUTHOR]

In this paper, we will generalize the notion of separation index, which was introduced by G. Manjunath et al (2006), to tree maps, and use the notion to characterize the intra-orbit. separation for the orbits of continuous transitive maps on a tree. For a. transitive map f, we will show: (i) if the separation index γ of f is positive, then for every 0 < τ < 7γ, (xs,xt) is a Li-Yorke pair of modulus τ, where xs and Xt be any two distinct points on a dense orbit; (ii) there is a natural number n such that the separation index of fn is greater than zero. [ABSTRACT FROM AUTHOR]

Let C0(I) be the set of all continuous self-maps of the compact interval I and P(u, v) = {f ∈ C0(I) : f has a cycle with over-rotation pair (u, v)} for any positive integers u and v with 2 ≤ 2u ≤ v. In this paper, we prove that if (2mns, 2mnt) ≻ (γ, λ), then P(2mns, 2mnt) ⊂ int P(γ, λ), where m ≥ 0 is an integer, n ≤ 1 odd, 2 ≤ 2s ≤ t with s, t coprime and 2 ≤ 2γ ≤ λ. [ABSTRACT FROM AUTHOR]