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September  2017, 22(7): 2587-2594. doi: 10.3934/dcdsb.2017098

On some difference equations with exponential nonlinearity

Eugenia N. Petropoulou

Department of Civil Engineering, University of Patras, 26500 Patras, Greece

In memory of Professor Evangelos K. Ifantis

Received  July 2016 Revised  December 2016 Published  March 2017

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.

Keywords: Nonautonomous difference equation, $\ell_1$ solution, bound, stability, Ricker equation.
Mathematics Subject Classification: Primary:39A30, 39A22;Secondary:39A45.
Citation: Eugenia N. Petropoulou. On some difference equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2587-2594. doi: 10.3934/dcdsb.2017098
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.  doi: 10.1007/978-0-387-21830-4_7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

S. Stevic, On a discrete epidemic model Discrete Dynam. Nat. Soc. , 2007 (2007), Article ID 87519, 10pp. doi: 10.1155/2007/87519.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459-467.  doi: 10.1007/978-0-387-21830-4_7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

S. Stevic, On a discrete epidemic model Discrete Dynam. Nat. Soc. , 2007 (2007), Article ID 87519, 10pp. doi: 10.1155/2007/87519.  Google Scholar

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