Home On the Mittag–Leffler Stability of Impulsive Fractional Solow-Type Models
Article
Licensed
Unlicensed Requires Authentication

On the Mittag–Leffler Stability of Impulsive Fractional Solow-Type Models

  • Ivanka M. Stamova EMAIL logo and Gani Tr. Stamov
Published/Copyright: July 14, 2017

Abstract

In this article, we introduce fractional-order Solow-type models as a new tool for modeling and analysis in mathematical finance. Sufficient conditions for the Mittag–Leffler stability of their states are derived. The main advantages of the proposed approach are using of fractional-order derivatives, whose nonlocal property makes the fractional calculus a suitable tool for modeling actual financial systems as well as using of impulsive perturbations which give an opportunity to control the dynamic behavior of the model. The modeling approach proposed in this article can be applied to investigate macroeconomic systems.

MSC 2010: 34K37; 34K45; 34K20; 34K60

References

[1] Solow R., A contribution to the theory of economic growth, Q. J. Econ. 70 (1956), 65–94.10.2307/1884513Search in Google Scholar

[2] Accinelli E. and Brida J. G., Population growth and the Solow–Swan model, Int. J. Ecol. Econ. Statistics 8 (2007), 54–63.Search in Google Scholar

[3] Acemoglu D., Introduction to modern economic growth, Princeton University Press, Princeton, 2009.Search in Google Scholar

[4] Capasso V., Engbers R. and La Torre D., On a spatial Solow model with technological diffusion and nonconcave production function, Nonlinear Anal. Real World Appl. 11 (2010), 3858–3876.10.1016/j.nonrwa.2010.01.016Search in Google Scholar

[5] Dohtani A., Growth-cycle model of Solow–Swan type, Int. J. Econ. Behav. Organ. 76 (2010), 428–444.10.1016/j.jebo.2010.07.006Search in Google Scholar

[6] Fanti L. and Manfredi P., The Solow’s model with endogenous population: a neoclassical growth cycle model, J. Econ. Dev. 28 (2003), 103–115.Search in Google Scholar

[7] Ferrara M., Guerrini L. and Sodini M., Nonlinear dynamics in a Solow model with delay and non-convex technology, Appl. Math. Comput. 228 (2014), 1–12.10.1016/j.amc.2013.11.082Search in Google Scholar

[8] Gandolfo G., Dynamics Economic, Springer, Berlin, 2009.10.1007/978-3-642-03871-6Search in Google Scholar

[9] Stamova I. M. and Stamov A. G., Impulsive control on the asymptotic stability of the solutions of a Solow model with endogenous labor growth, J. Franklin Inst. 349 (2012), 2704–2716.10.1016/j.jfranklin.2012.07.001Search in Google Scholar

[10] Stamova I. M. and Stamov A. G., On the stability of the solutions of an impulsive Solow model with endogenous population, Econ. Change Restruct. 46 (2013), 203–217.10.1007/s10644-012-9124-5Search in Google Scholar

[11] Guerrini L., The Solow–Swan model with a bounded population growth rate, J. Math. Econ. 42 (2006), 14–21.10.1016/j.jmateco.2005.05.001Search in Google Scholar

[12] Benchohra M., Henderson J. and Ntouyas S. K., Impulsive differential equations and inclusions, Hindawi Publishing Corporation, New York, 2006.10.1155/9789775945501Search in Google Scholar

[13] Stamova I. M., Stability analysis of impulsive functional differential equations, Walter de Gruyter, Berlin, New York, 2009.10.1515/9783110221824Search in Google Scholar

[14] Stamova I. M. and Stamov G. T., Applied impulsive mathematical models, Springer, Berlin, 2016.10.1007/978-3-319-28061-5Search in Google Scholar

[15] Basin M. V. and Pinsky M. A., Impulsive control in Kalman-like filtering problems, J. Appl. Math. Stoch. Anal. 11 (1988), 1–8.10.1155/S104895339800001XSearch in Google Scholar

[16] Korn R., Optimal impulse control when control consequences are random, Math. Oper. Res. 22 (1997), 639–667.10.1287/moor.22.3.639Search in Google Scholar

[17] Sun J., Qiao F. and Wu Q., Impulsive control of a financial model, Phys. Lett. A 335 (2005), 282–288.10.1016/j.physleta.2004.12.030Search in Google Scholar

[18] Sayevand K., Analytical treatment of Volterra integro-differential equations of fractional order, Appl. Math. Model. 39 (2015), 4330–4336.10.1016/j.apm.2014.12.024Search in Google Scholar

[19] Diethelm K., The analysis of fractional differential equations, an application-oriented exposition using differential operators of Caputo type, Springer-Verlag, Berlin, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[20] Kilbas A., H. M. Srivastava and Trujillo J. J., Theory and applications of fractional differential equations, North-Holland math. study, Elsevier, Amsterdam, 2006.Search in Google Scholar

[21] Podlubny I., Fractional differential equations, Academic Press, San Diego, 1999.Search in Google Scholar

[22] Chang Y. K. and Nieto J. J., Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators, Funct. Anal. Optim. 30 (2009), 227–244.10.1080/01630560902841146Search in Google Scholar

[23] Rehman M. and Eloe P., Existence and uniqueness of solutions for impulsive fractional differential equations, Appl. Math. Comput. 224 (2013), 422–431.10.1016/j.amc.2013.08.088Search in Google Scholar

[24] Stamova I. M., Global stability of impulsive fractional differential equations, Appl. Math. Comput. 237 (2014), 605–612.10.1016/j.amc.2014.03.067Search in Google Scholar

[25] Stamova I. M. and Stamov G. T., Stability analysis of impulsive functional systems of fractional order, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 702–709.10.1016/j.cnsns.2013.07.005Search in Google Scholar

[26] Suganya S., M. Mallika Arjunan and J. J. Trujillo, Existence results for an impulsive fractional integro-differential equation with state-dependent delay, Appl. Math. Comput. 266 (2015), 54–69.10.1016/j.amc.2015.05.031Search in Google Scholar

[27] Wang H., Existence results for fractional functional differential equations with impulses, J. Appl. Math. Comput. 38 (2012), 85–101.10.1007/s12190-010-0465-9Search in Google Scholar

[28] Xie S., Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay, Fract. Calc. Appl. Anal. 17 (2014), 1158–1174.10.2478/s13540-014-0219-8Search in Google Scholar

[29] Baleanu D., K. Diethelm, Scalas E. and Trujillo J. J., Fractional calculus: Models and numerical methods, World Scientific, Singapore, 2012.10.1142/8180Search in Google Scholar

[30] Chen L., Chai Y. and Wu R., Control and synchronization of fractional-order financial system based on linear control, Discrete Dyn. Nat. Soc. 2011 (2011), 1–21.10.1155/2011/958393Search in Google Scholar

[31] Cont R., Long range dependence in financial markets, in: J. Levy-Vehel, E. Lutton (Eds.), Fractals in engineering, pp. 159–179, Springer, London, 2005.10.1007/1-84628-048-6_11Search in Google Scholar

[32] Cunado J., Gil-Alana L. A. and F. Pérez de Gracia, AK growth models: New evidence based on fractional integration and breaking trends, Rech. Econ. Louvain 75 (2009) 131–149.10.3917/rel.752.0131Search in Google Scholar

[33] Danca M.-F., Garrappa R., Tang W. K. S. and Chen G., Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching, Comput. Math. Appl. 66 (2013), 702–716.10.1016/j.camwa.2013.01.028Search in Google Scholar

[34] Laskin N., Fractional market dynamics, Physica A 287 (2000), 482–492.10.1016/S0378-4371(00)00387-3Search in Google Scholar

[35] Li Y., Chen Y. Q. and Podlubny I., Mittag–Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), 1965–1969.10.1016/j.automatica.2009.04.003Search in Google Scholar

[36] Chen J., Zeng Z., Jiang P., Global Mittag–Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural Netw. 51 (2014), 1–8.10.1016/j.neunet.2013.11.016Search in Google Scholar PubMed

[37] Li Y., Chen Y. Q. and Podlubny I., Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability, Comput. Math. Appl. 59 (2010), 1810–1821.10.1016/j.camwa.2009.08.019Search in Google Scholar

[38] Liu S., Li X., Jiang W. and Zhou X., Mittag–Leffler stability of nonlinear fractional neutral singular systems, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3961–3966.10.1016/j.cnsns.2012.02.012Search in Google Scholar

[39] Stamova I.M., Global Mittag–Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays, Nonlinear Dynam. 77 (2014), 1251–1260.10.1007/s11071-014-1375-4Search in Google Scholar

Received: 2016-2-12
Accepted: 2017-5-4
Published Online: 2017-7-14
Published in Print: 2017-7-26

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 23.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2016-0027/html
Scroll to top button