Asymptotic behavior results for solutions to some nonlinear difference equations

Jamieson, William T.; Merino, Orlando

doi:10.1016/j.jmaa.2015.04.094

Cited by 5 publications

(6 citation statements)

References 18 publications

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“…By using the results in [ 13 ] we can give precise asymptotic behavior of solutions of Model ( 6 ) and so the rate of convergence as well. As it was proven in Theorem 1 of [ 13 ] and Theorem 3.1 in [ 19 ] this rate depends on the eigenvalues of the matrix of the linearized system at the given equilibrium, which is attractor. In other words the eigenvalues of Jacobian matrix evaluated at the given equilibrium determine the asymptotic behavior and so the rate of convergence.…”

Section: Global Disease Persistence and Invariant Rectanglesmentioning

confidence: 99%

“…In other words the eigenvalues of Jacobian matrix evaluated at the given equilibrium determine the asymptotic behavior and so the rate of convergence. See Corollary 5 in [ 13 ] and Theorem 3.1 in [ 19 ]. However in epidemic problems the emphasis is on determination of , and so we will skip the asymptotic of solutions.…”

Section: Global Disease Persistence and Invariant Rectanglesmentioning

confidence: 99%

“…Thus the convergence to the solution of such system depends on the numerical method used. The precise asymptotic behavior of solutions converging to the equilibrium solution of the model presented can be derived from formulas in [ 13 ] or [ 19 ] and is given in Theorem 5 of this paper. Our results correct the error in a related paper [ 11 ].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior of a discrete-time density-dependent SI epidemic model with constant recruitment

Kulenović

Nurkanović

Yakubu

2021

J. Appl. Math. Comput.

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We use the epidemic threshold parameter, R 0 , and invariant rectangles to investigate the global asymptotic behavior of solutions of the density-dependent discrete-time SI epidemic model where the variables S n and I n represent the populations of susceptibles and infectives at time n = 0, 1,. . ., respectively. The model features constant survival "probabilities" of susceptible and infective individuals and the constant recruitment per the unit time interval [n, n + 1] into the susceptible class. We compute the basic reproductive number, R 0 , and use it to prove that independent of positive initial population sizes, R 0 < 1 implies the unique disease-free equilibrium is globally stable and the infective population goes extinct. However, the unique endemic equilibrium is globally stable and the infective population persists whenever R 0 > 1 and the constant survival probability of susceptible is either less than or equal than 1/3 or the constant recruitment is large enough.

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Section: Global Disease Persistence and Invariant Rectanglesmentioning

confidence: 99%

Section: Global Disease Persistence and Invariant Rectanglesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic behavior of a discrete-time density-dependent SI epidemic model with constant recruitment

Kulenović

Nurkanović

Yakubu

2021

J. Appl. Math. Comput.

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“…With the continuous development of science and technology in the fields of economy, biology, computer science, and so on, the research of nonlinear difference equations has been rapidly pushed forward, (for example, see [1][2][3][4] and the relevant reference cited therein). In the recent years, because the max operator has a great importance in automatic control models (see [5,6]), max-type difference equations which are a special type of difference equations have aroused the concern and attention of many scholars.…”

Section: Introductionmentioning

confidence: 99%

Periodic Solution for a Max-Type Fuzzy Difference Equation

Wang

2020

Journal of Mathematics

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The paper is concerned with the dynamics behavior of positive solutions for the following max-type fuzzy difference equation system: xn+1=maxA/xn, A/xn−1, xn−2, n=0,1,2,…, where xn is a sequence of positive fuzzy numbers, and the parameter A and the initial conditions x−2, x−1, x0 are also positive fuzzy numbers. Firstly, the fuzzy set theory is used to transform the fuzzy difference equation into the corresponding ordinary difference equations with parameters. Then, the expression for the periodic solution of the max-type ordinary difference equations is obtained by the iteration, the inequality technique, and the mathematical induction. Moreover, we can obtain the expression for the periodic solution of the max-type fuzzy difference equation. In addition, the boundedness and persistence of solutions for the fuzzy difference equation is proved. Finally, the results of this paper are simulated and verified by using MATLAB 2016 software package.

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“…Although difference equations which appear naturally as discrete analogues in the numerical solutions of differential and delay differential equations have been predominantly studied so far (see, for example, [11,16,21] and the references therein), the study of nonlinear difference equations which are not discrete analogues of differential equations has been also of a great interest recently (see, for example, [7,9,13] and the references therein). Recently, so called, max-type difference equations have attracted some attention (see, for example, [6,17,20] and the references therein) because the max operator have great importance in automatic control models (see, [15,19]) and have wide applications in biology (see, [8]), ecology (see, [12,18]), and physics (see, [4,10]).…”

Section: Introductionmentioning

confidence: 99%

On the periodicity of a max-type rational difference equation

Wang¹,

Jing²,

Hu³

et al. 2017

J. Nonlinear Sci. Appl.

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This paper shows that every well-defined solution of the following max-type difference equationwhere A ∈ R and the initial conditions x −2 , x −1 , x 0 are arbitrary non-zero real numbers is eventually periodic with period three by using new iteration method for the more general nonlinear difference equations and inequality skills as well as the mathematical induction. Our main results considerably improve results appearing in the literature.

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Asymptotic behavior results for solutions to some nonlinear difference equations

Cited by 5 publications

References 18 publications

Asymptotic behavior of a discrete-time density-dependent SI epidemic model with constant recruitment

Asymptotic behavior of a discrete-time density-dependent SI epidemic model with constant recruitment

Periodic Solution for a Max-Type Fuzzy Difference Equation

On the periodicity of a max-type rational difference equation

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