scholarly journals On a Discrete Epidemic Model

2007 ◽  
Vol 2007 ◽  
pp. 1-10 ◽  
Author(s):  
Stevo Stevic
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Epidemic Model ◽  
Asymptotic Behavior Of Solutions ◽  
Negative Numbers ◽  
Initial Values ◽  
Discrete Epidemic Model ◽  
The Difference

We investigate the global asymptotic behavior of solutions of the difference equationxn+1=(1−∑j=0k−1xn−j)(1−e−Axn),n∈ℕ0, whereA∈(0,∞),k∈{2,3,…}, and the initial valuesx−k+1,x−k+2,…,x0are arbitrary negative numbers. Asymptotics of some positive solutions of the equation are also found.

scholarly journals Global Behavior of the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

scholarly journals Asymptotic Behavior of Solutions to Half-Linearq-Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Pavel Řehák
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Special Limit ◽  
Bounded Functions ◽  
Necessary And Sufficient

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linearq-difference equationDq(Φ(Dqy(t)))+p(t)Φ(y(qt))=0,t∈{qk:k∈N0}withq>1,Φ(u)=|u|α−1sgn⁡uwithα>1, to behave likeq-regularly varying orq-rapidly varying orq-regularly bounded functions (that is, the functionsy, for which a special limit behavior ofy(qt)/y(t)ast→∞is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

scholarly journals On the Asymptotic Behavior of a Difference Equation with Maximum

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Fangkuan Sun
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
The Difference

We study the asymptotic behavior of positive solutions to the difference equationxn=max{A/xn-1α,B/xn−2β},n=0,1,…,where0<α, β<1, A,B>0. We prove that every positive solution to this equation converges tox∗=max{A1/(α+1),B1/(β+1)}.

scholarly journals On the Difference Equationxn+1=xnxn-k/(xn-k+1a+bxnxn-k)

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Stevo Stević ◽  
Josef Diblík ◽  
Bratislav Iričanin ◽  
Zdenĕk Šmarda
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Closed Form ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference

We show that the difference equationxn+1=xnxn-k/xn-k+1(a+bxnxn-k),n∈ℕ0, wherek∈ℕ, the parametersa,band initial valuesx-i,i=0,k̅are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation.

scholarly journals On the Global Attractivity of a Max-Type Difference Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-5 ◽  
Author(s):  
Ali Gelişken ◽  
Cengiz Çinar
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Global Attractivity ◽  
Period 2 ◽  
The Difference

We investigate asymptotic behavior and periodic nature of positive solutions of the difference equation , where and . We prove that every positive solution of this difference equation approaches or is eventually periodic with period 2.

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

2021 ◽  
Vol 19 (1) ◽  
pp. 259-267
Author(s):  
Liuyang Shao ◽  
Yingmin Wang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Asymptotic Behavior Of Solutions ◽  
Suitable Assumption ◽  
Choquard Equation ◽  
Existence Of Positive Solutions

Abstract In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .

scholarly journals On the Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

The long-term behavior of solutions of the following difference equation: , , where the initial values , , are real numbers, is investigated in the paper.

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