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On some difference equations with exponential nonlinearity
Department of Civil Engineering, University of Patras, 26500 Patras, Greece |
The problem of the existence of complex $\ell_1$ solutions of two difference equations with exponential nonlinearity is studied, one of which is nonautonomous. As a consequence, several information are obtained regarding the asymptotic stability of their equilibrium points, as well as the corresponding generating function and $z-$ transform of their solutions. The results, which are obtained using a general theorem based on a functional-analytic technique, provide also a rough estimate of the region of attraction of each equilibrium point for the autonomous case. When restricted to real solutions, the results are compared with other recently published results.
References:
[1] |
A. S. Ackleh and P. L. Salceanu,
Competitive exclusion and coexistence in an $n-$species Ricker model, J. Biol. Dynamics, 9 (2015), 321-331.
doi: 10.1080/17513758.2015.1020576. |
[2] |
D. Aruğaslan and L. Güzel, Stability of the logistic population model with generalized piecewise constant delays, Adv. Difference Equations, 2015 (2015). Google Scholar |
[3] |
I. Györi and L. Horváth,
A new view of the $\ell^p$
-theory for a system of higher order difference equations, Comput. Math. Appl., 59 (2010), 2918-2932.
doi: 10.1016/j.camwa.2010.02.010. |
[4] |
I. Györi and L. Horváth,
$\ell^p$
-solutions and stability analysis of difference equations using the Kummer's test, Appl. Math. Comput., 217 (2011), 10129-10145.
doi: 10.1016/j.amc.2011.05.008. |
[5] |
T. Hüls and C. Pötzsche,
Qualitative analysis of a nonautonomous Beverton-Holt Ricker model, SIAM J. Appl. Dyn. Syst., 13 (2014), 1442-1488.
doi: 10.1137/140955434. |
[6] |
E. K. Ifantis,
On the convergence of power series whose coefficients satisfy a Poincaré-type linear and nonlinear difference equation, Complex Variables Theory Appl., 9 (1987), 63-80.
doi: 10.1080/17476938708814250. |
[7] |
Y. Kang and H. Smith,
Global dynamics of a discrete two-species Lottery-Ricker competition model, J. Biol. Dynamics, 6 (2012), 358-376.
doi: 10.1080/17513758.2011.586064. |
[8] |
R. May,
Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
doi: 10.1007/978-0-387-21830-4_7. |
[9] |
G. Papaschinopoulos, N. Fotiades and C. J. Schinas,
On a system of difference equations including negative exponential terms, J. Differ. Equations Appl., 20 (2014), 717-732.
doi: 10.1080/10236198.2013.814647. |
[10] |
G. Papaschinopoulos, M. A. Radin and C. J. Schinas,
On the system of two difference equations of exponential form: $x_{n+1}=a+bx_{n-1}e^{-y_{n}}$, $y_{n+1}=c+dy_{n-1}e^{-x_{n}}$, Math. Comp. Mod., 54 (2011), 2969-2977.
doi: 10.1016/j.mcm.2011.07.019. |
[11] |
E. N. Petropoulou and P. D. Siafarikas, Functional analysis and partial difference equations, in Some Recent Advances in Partial Difference Equations (ed. E. N. Petropoulou), Bentham eBooks (2010), 49–76. Google Scholar |
[12] |
W. E. Ricker, Stock and recruitmnet, J. Fish. Res. Board Canada, 11 (1954), 559-623. Google Scholar |
[13] |
G. Stefanidou, G. Papaschinopoulos and C. J. Schinas,
On a system of two exponential type difference equations, Comm. Appl. Nonlinear Anal., 17 (2010), 1-13.
|
[14] |
S. Stevic, On a discrete epidemic model Discrete Dynam. Nat. Soc. , 2007 (2007), Article ID 87519, 10pp.
doi: 10.1155/2007/87519. |
show all references
References:
[1] |
A. S. Ackleh and P. L. Salceanu,
Competitive exclusion and coexistence in an $n-$species Ricker model, J. Biol. Dynamics, 9 (2015), 321-331.
doi: 10.1080/17513758.2015.1020576. |
[2] |
D. Aruğaslan and L. Güzel, Stability of the logistic population model with generalized piecewise constant delays, Adv. Difference Equations, 2015 (2015). Google Scholar |
[3] |
I. Györi and L. Horváth,
A new view of the $\ell^p$
-theory for a system of higher order difference equations, Comput. Math. Appl., 59 (2010), 2918-2932.
doi: 10.1016/j.camwa.2010.02.010. |
[4] |
I. Györi and L. Horváth,
$\ell^p$
-solutions and stability analysis of difference equations using the Kummer's test, Appl. Math. Comput., 217 (2011), 10129-10145.
doi: 10.1016/j.amc.2011.05.008. |
[5] |
T. Hüls and C. Pötzsche,
Qualitative analysis of a nonautonomous Beverton-Holt Ricker model, SIAM J. Appl. Dyn. Syst., 13 (2014), 1442-1488.
doi: 10.1137/140955434. |
[6] |
E. K. Ifantis,
On the convergence of power series whose coefficients satisfy a Poincaré-type linear and nonlinear difference equation, Complex Variables Theory Appl., 9 (1987), 63-80.
doi: 10.1080/17476938708814250. |
[7] |
Y. Kang and H. Smith,
Global dynamics of a discrete two-species Lottery-Ricker competition model, J. Biol. Dynamics, 6 (2012), 358-376.
doi: 10.1080/17513758.2011.586064. |
[8] |
R. May,
Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.
doi: 10.1007/978-0-387-21830-4_7. |
[9] |
G. Papaschinopoulos, N. Fotiades and C. J. Schinas,
On a system of difference equations including negative exponential terms, J. Differ. Equations Appl., 20 (2014), 717-732.
doi: 10.1080/10236198.2013.814647. |
[10] |
G. Papaschinopoulos, M. A. Radin and C. J. Schinas,
On the system of two difference equations of exponential form: $x_{n+1}=a+bx_{n-1}e^{-y_{n}}$, $y_{n+1}=c+dy_{n-1}e^{-x_{n}}$, Math. Comp. Mod., 54 (2011), 2969-2977.
doi: 10.1016/j.mcm.2011.07.019. |
[11] |
E. N. Petropoulou and P. D. Siafarikas, Functional analysis and partial difference equations, in Some Recent Advances in Partial Difference Equations (ed. E. N. Petropoulou), Bentham eBooks (2010), 49–76. Google Scholar |
[12] |
W. E. Ricker, Stock and recruitmnet, J. Fish. Res. Board Canada, 11 (1954), 559-623. Google Scholar |
[13] |
G. Stefanidou, G. Papaschinopoulos and C. J. Schinas,
On a system of two exponential type difference equations, Comm. Appl. Nonlinear Anal., 17 (2010), 1-13.
|
[14] |
S. Stevic, On a discrete epidemic model Discrete Dynam. Nat. Soc. , 2007 (2007), Article ID 87519, 10pp.
doi: 10.1155/2007/87519. |
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