# American Institute of Mathematical Sciences

• Previous Article
Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model
• DCDS-B Home
• This Issue
• Next Article
Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels
September  2017, 22(7): 2587-2594. doi: 10.3934/dcdsb.2017098

## On some difference equations with exponential nonlinearity

 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.

Citation: Eugenia N. Petropoulou. On some difference equations with exponential nonlinearity. Discrete and 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. [2] D. Aruğaslan and L. Güzel, Stability of the logistic population model with generalized piecewise constant delays, Adv. Difference Equations, 2015 (2015). [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. [12] W. E. Ricker, Stock and recruitmnet, J. Fish. Res. Board Canada, 11 (1954), 559-623. [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

In memory of Professor Evangelos K. Ifantis

##### 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). [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. [12] W. E. Ricker, Stock and recruitmnet, J. Fish. Res. Board Canada, 11 (1954), 559-623. [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.
 [1] Huining Qiu, Xiaoming Chen, Wanquan Liu, Guanglu Zhou, Yiju Wang, Jianhuang Lai. A fast $\ell_1$-solver and its applications to robust face recognition. Journal of Industrial and Management Optimization, 2012, 8 (1) : 163-178. doi: 10.3934/jimo.2012.8.163 [2] Brian Ryals, Robert J. Sacker. Global stability in the 2D Ricker equation revisited. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 585-604. doi: 10.3934/dcdsb.2017028 [3] Hui Huang, Eldad Haber, Lior Horesh. Optimal estimation of $\ell_1$-regularization prior from a regularized empirical Bayesian risk standpoint. Inverse Problems and Imaging, 2012, 6 (3) : 447-464. doi: 10.3934/ipi.2012.6.447 [4] Pengbo Geng, Wengu Chen. Unconstrained $\ell_1$-$\ell_2$ minimization for sparse recovery via mutual coherence. Mathematical Foundations of Computing, 2020, 3 (2) : 65-79. doi: 10.3934/mfc.2020006 [5] Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385 [6] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [7] Leonid Shaikhet. Behavior of solution of stochastic difference equation with continuous time under additive fading noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 301-310. doi: 10.3934/dcdsb.2021043 [8] Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic and Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151 [9] Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6359-6376. doi: 10.3934/dcdsb.2021022 [10] Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141 [11] Carmen Cortázar, Marta García-Huidobro, Pilar Herreros. On the uniqueness of bound state solutions of a semilinear equation with weights. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6761-6784. doi: 10.3934/dcds.2019294 [12] Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663 [13] Xiao Tang, Yingying Zeng, Weinian Zhang. Interval homeomorphic solutions of a functional equation of nonautonomous iterations. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6967-6984. doi: 10.3934/dcds.2020214 [14] Esha Chatterjee, Sk. Sarif Hassan. On the asymptotic character of a generalized rational difference equation. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1707-1718. doi: 10.3934/dcds.2018070 [15] Seiji Ukai, Tong Yang, Huijiang Zhao. Exterior Problem of Boltzmann Equation with Temperature Difference. Communications on Pure and Applied Analysis, 2009, 8 (1) : 473-491. doi: 10.3934/cpaa.2009.8.473 [16] Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks and Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002 [17] E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323 [18] Gang Meng. The optimal upper bound for the first eigenvalue of the fourth order equation. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6369-6382. doi: 10.3934/dcds.2017276 [19] S. E. Kuznetsov. An upper bound for positive solutions of the equation \Delta u=u^\alpha. Electronic Research Announcements, 2004, 10: 103-112. [20] Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4887-4919. doi: 10.3934/dcds.2021061

2020 Impact Factor: 1.327