American Institute of Mathematical Sciences

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January  2020, 16(1): 37-53. doi: 10.3934/jimo.2018139

A new class of global fractional-order projective dynamical system with an application

Zeng-bao Wu 1, , Yun-zhi Zou 2, and  Nan-jing Huang 2,,

1. 

Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471934, China
2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Nan-jing Huang

Received  October 2016 Revised  September 2017 Published  January 2020 Early access  September 2018

Fund Project: This work was supported by the National Natural Science Foundation of China (11471230, 11671282) and the Program for Science Technology Innovation Research Team in Universities of Henan Province (18IRTSHN014).

In this article, some existence and uniqueness of solutions for a new class of global fractional-order projective dynamical system with delay and perturbation are proved by employing the Krasnoselskii fixed point theorem and the Banach fixed point theorem. Moreover, an approximating algorithm is also provided to find a solution of the global fractional-order projective dynamical system. Finally, an application to the idealized traveler information systems for day-to-day adjustments processes and a numerical example are given.

Keywords: Fractional-order projective dynamical system, delay, perturbation, Krasnoselskii fixed point theorem, approximating algorithm.
Mathematics Subject Classification: 49J40, 47J05, 34A08, 34K09.
Citation: Zeng-bao Wu, Yun-zhi Zou, Nan-jing Huang. A new class of global fractional-order projective dynamical system with an application. Journal of Industrial & Management Optimization, 2020, 16 (1) : 37-53. doi: 10.3934/jimo.2018139
References:
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K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341.  Google Scholar

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K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be.  Google Scholar

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K. Ding and N. J. Huang, A new interval projection neural networks for solving interval quadratic program, Chaos Solitons Fractals, 35 (2008), 718-725.  doi: 10.1016/j.chaos.2006.05.037.  Google Scholar

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T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin and S. Ganjlizadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems, Oper. Res., 42 (1994), 1120-1136.  doi: 10.1287/opre.42.6.1120.  Google Scholar

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M. L. Morgado, N. J. Ford and P. M. Lima, Analysis and numerical methods for fractional differential equations with delay, J. Comput. Appl. Math., 252 (2013), 159-168.  doi: 10.1016/j.cam.2012.06.034.  Google Scholar

[23]

B. P. Moghaddam and Z. S. Mostaghim, A numerical method based on finite difference for solving fractional delay differential equations, J. Taibah Univ. Sci., 7 (2013), 120-127.  doi: 10.1016/j.jtusci.2013.07.002.  Google Scholar

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N. Ozalp and I. Koca, A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Difference Equat., 2012 (2012), 7pp.  doi: 10.1186/1687-1847-2012-189.  Google Scholar

[26] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.   Google Scholar
[27]

S. B. Skaar, A. N. Michel and R. K. Miller, Stability of viscoelastic control systems, IEEE Trans. Automat. Control, 33 (1988), 348-357.  doi: 10.1109/9.192189.  Google Scholar

[28]

W. Y. Szeto and H. K. Lo, The impact of advanced traveler information services on travel time and schedule delay costs, J. Intel. Transportation Sys., 9 (2007), 47-55.  doi: 10.1080/15472450590916840.  Google Scholar

[29]

L. Song, S. Y. Xu and J. Y. Yang, Dynamical models of happiness with fractional order, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 616-628.  doi: 10.1016/j.cnsns.2009.04.029.  Google Scholar

[30]

P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.  doi: 10.1115/1.3167615.  Google Scholar

[31]

Z. Wang, A numerical method for delayed fractional-order differential equations, J. Appl. Math. , 2013 (2013), Article ID 256071, 7 pages.  Google Scholar

[32]

X. K. Wu, Z. B. Wu and Y. Z. Zou, Existence, uniqueness and stability for a class of interval projective dynamical systems, Comm. Appl. Nonlinear Anal., 20 (2013), 81-94.   Google Scholar

[33]

Z. B. Wu and Y. Z. Zou, Stability analysis of two related projective dynamical systems in Hilbert spaces, Nonlinear Anal. Forum, 19 (2014), 37-51.   Google Scholar

[34]

Z. B. Wu and Y. Z. Zou, Global fraction-order projective dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2811-2819.  doi: 10.1016/j.cnsns.2014.01.007.  Google Scholar

[35]

Z. B. Wu, Y. Z. Zou and N. J. Huang, A system of fractional-order interval projection neural networks, J. Comput. Appl. Math., 294 (2016), 389-402.  doi: 10.1016/j.cam.2015.09.007.  Google Scholar

[36]

Z. B. Wu, Y. Z. Zou and N. J. Huang, A class of global fractional-order projective dynamical systems involving set-valued perturbations, Appl. Math. Comput., 277 (2016), 23-33.  doi: 10.1016/j.amc.2015.12.033.  Google Scholar

[37]

Z. B. Wu, J. D. Li and N. J. Huang, A new system of global fractional-order interval implicit projection neural networks, Neurocomputing, 282 (2018), 111-121.   Google Scholar

[38]

Z. B. Wu, C. Min and N. J. Huang, On a system of fuzzy fractional differential inclusions with projection operators, Fuzzy Sets Syst., 347 (2018), 70-88.  doi: 10.1016/j.fss.2018.01.005.  Google Scholar

[39]

Y. S. Xia and T. L. Vincent, On the stability of global projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150.  doi: 10.1023/A:1004611224835.  Google Scholar

[40]

Y. S. Xia, Further results on global convergence and stability of global projected dynamical systems, J. Optim. Theory Appl., 122 (2004), 627-649.  doi: 10.1023/B:JOTA.0000042598.21226.af.  Google Scholar

[41]

Z. H. Yang and J. D. Cao, Initial value problems for arbitrary order fractional differential equations with delay, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2993-3005.  doi: 10.1016/j.cnsns.2013.03.006.  Google Scholar

[42]

D. Zhang and A. Nagurney, On the stability of projected dynamical systems, J. Optim. Theory Appl., 85 (1995), 97-124.  doi: 10.1007/BF02192301.  Google Scholar

[43]

X. M. Zhao and G. Orosz, Nonlinear day-to-day traffic dynamics with driver experience delay: Modeling, stability and bifurcation analysis, Phys. D, 275 (2014), 54-66.  doi: 10.1016/j.physd.2014.02.005.  Google Scholar

[44]

Y. Zhou, F. Jiao and J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal., 71 (2009), 3249-3256.  doi: 10.1016/j.na.2009.01.202.  Google Scholar

[45]

Y. Z. Zou, X. Li, N. J. Huang and C. Y. Sun, Global dynamical systems involving generalized $f$-projection operators and set-valued perturbation in Banach spaces, J. Appl. Math. , 2012 (2012), Article ID 682465, 12 pages.  Google Scholar

[46]

Y. Z. Zou and C. Y. Sun, Equilibrium points for two related projective dynamical systems, Comm. Appl. Nonlinear Anal., 19 (2012), 111-119.   Google Scholar

[47]

Y. Z. Zou, X. K. Wu, W. B. Zhang and C. Y. Sun, An iterative method for a class of generalized global dynamical system involving fuzzy mappings in Hilbert spaces, Lecture Notes in Commput. Sci., 7666 (2012), 44-51.  doi: 10.1007/978-3-642-34478-7_6.  Google Scholar

show all references

References:
[1]

W. M. Ahmad and R. El-Khazali, Fractional-order dynamical models of love, Chaos Solit. Fract., 33 (2007), 1367-1375.  doi: 10.1016/j.chaos.2006.01.098.  Google Scholar

[2]

R. P. Agarwal, Y. Zhou and Y. Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095-1100.  doi: 10.1016/j.camwa.2009.05.010.  Google Scholar

[3]

S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron J. Differential Equations, 2011 (2011), 1-11.   Google Scholar

[4]

S. Bhalekar and V. Daftardar-geji, A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order, J. Fract. Calc. Appl., 1 (2011), 1-9.   Google Scholar

[5]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 9-42.  doi: 10.1007/BF02073589.  Google Scholar

[6]

K. Diethelm, N. J. Ford and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341.  Google Scholar

[7]

K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional adams method, Numer. Algorithms, 36 (2004), 31-52.  doi: 10.1023/B:NUMA.0000027736.85078.be.  Google Scholar

[8]

K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[9]

W. H. Deng, C. P. Li and J. H. Lu, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynam., 48 (2007), 409-416.  doi: 10.1007/s11071-006-9094-0.  Google Scholar

[10]

H. Dia and S. Panwai, Modelling drivers' compliance and rout choice behaviour in response to travel information, Nonlinear Dynam., 49 (2007), 493-509.   Google Scholar

[11]

K. Ding and N. J. Huang, A new interval projection neural networks for solving interval quadratic program, Chaos Solitons Fractals, 35 (2008), 718-725.  doi: 10.1016/j.chaos.2006.05.037.  Google Scholar

[12]

T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin and S. Ganjlizadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems, Oper. Res., 42 (1994), 1120-1136.  doi: 10.1287/opre.42.6.1120.  Google Scholar

[13]

T. L. Friesz, Z. G. Suo and D. H. Bernstein, A dynamic disequilibrium interregional commodity flow model, Transport. Res. B, 32 (1998), 467-483.  doi: 10.1016/S0191-2615(98)00012-5.  Google Scholar

[14]

Y. Jalilian and R. Jalilian, Existence of solution for delay fractional differential equations, Mediterr. J. Math., 10 (2013), 1731-1747.  doi: 10.1007/s00009-013-0281-1.  Google Scholar

[15]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar

[16] D. Kinderlehrer and G. Stampcchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.   Google Scholar
[17]

M. A. Krasnoselskii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl., 10 (1958), 345-409.   Google Scholar

[18]

W. H. Lin, A. Kulkarni and P. Mirchandani, Short-time arterial travel time prediction for advanced traveler infromation systems, J. Intel. Transportation Sys., 8 (2004), 143-154.   Google Scholar

[19]

C. P. Li and F. R. Zhang, A survey on the stability of fractional differential equations, Eur. Phys. J. Special Topics, 193 (2011), 27-47.  doi: 10.1140/epjst/e2011-01379-1.  Google Scholar

[20]

T. Maraaba, F. Jarad and D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within caputo derivatives, Sci. China Ser. A, 51 (2008), 1775-1786.  doi: 10.1007/s11425-008-0068-1.  Google Scholar

[21]

T. Maraaba, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507, 11pp.  doi: 10.1063/1.2970709.  Google Scholar

[22]

M. L. Morgado, N. J. Ford and P. M. Lima, Analysis and numerical methods for fractional differential equations with delay, J. Comput. Appl. Math., 252 (2013), 159-168.  doi: 10.1016/j.cam.2012.06.034.  Google Scholar

[23]

B. P. Moghaddam and Z. S. Mostaghim, A numerical method based on finite difference for solving fractional delay differential equations, J. Taibah Univ. Sci., 7 (2013), 120-127.  doi: 10.1016/j.jtusci.2013.07.002.  Google Scholar

[24] A. Nagumey and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications, Springer, New York, 1996.  doi: 10.1007/978-1-4615-2301-7.  Google Scholar
[25]

N. Ozalp and I. Koca, A fractional order nonlinear dynamical model of interpersonal relationships, Adv. Difference Equat., 2012 (2012), 7pp.  doi: 10.1186/1687-1847-2012-189.  Google Scholar

[26] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.   Google Scholar
[27]

S. B. Skaar, A. N. Michel and R. K. Miller, Stability of viscoelastic control systems, IEEE Trans. Automat. Control, 33 (1988), 348-357.  doi: 10.1109/9.192189.  Google Scholar

[28]

W. Y. Szeto and H. K. Lo, The impact of advanced traveler information services on travel time and schedule delay costs, J. Intel. Transportation Sys., 9 (2007), 47-55.  doi: 10.1080/15472450590916840.  Google Scholar

[29]

L. Song, S. Y. Xu and J. Y. Yang, Dynamical models of happiness with fractional order, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 616-628.  doi: 10.1016/j.cnsns.2009.04.029.  Google Scholar

[30]

P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.  doi: 10.1115/1.3167615.  Google Scholar

[31]

Z. Wang, A numerical method for delayed fractional-order differential equations, J. Appl. Math. , 2013 (2013), Article ID 256071, 7 pages.  Google Scholar

[32]

X. K. Wu, Z. B. Wu and Y. Z. Zou, Existence, uniqueness and stability for a class of interval projective dynamical systems, Comm. Appl. Nonlinear Anal., 20 (2013), 81-94.   Google Scholar

[33]

Z. B. Wu and Y. Z. Zou, Stability analysis of two related projective dynamical systems in Hilbert spaces, Nonlinear Anal. Forum, 19 (2014), 37-51.   Google Scholar

[34]

Z. B. Wu and Y. Z. Zou, Global fraction-order projective dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2811-2819.  doi: 10.1016/j.cnsns.2014.01.007.  Google Scholar

[35]

Z. B. Wu, Y. Z. Zou and N. J. Huang, A system of fractional-order interval projection neural networks, J. Comput. Appl. Math., 294 (2016), 389-402.  doi: 10.1016/j.cam.2015.09.007.  Google Scholar

[36]

Z. B. Wu, Y. Z. Zou and N. J. Huang, A class of global fractional-order projective dynamical systems involving set-valued perturbations, Appl. Math. Comput., 277 (2016), 23-33.  doi: 10.1016/j.amc.2015.12.033.  Google Scholar

[37]

Z. B. Wu, J. D. Li and N. J. Huang, A new system of global fractional-order interval implicit projection neural networks, Neurocomputing, 282 (2018), 111-121.   Google Scholar

[38]

Z. B. Wu, C. Min and N. J. Huang, On a system of fuzzy fractional differential inclusions with projection operators, Fuzzy Sets Syst., 347 (2018), 70-88.  doi: 10.1016/j.fss.2018.01.005.  Google Scholar

[39]

Y. S. Xia and T. L. Vincent, On the stability of global projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150.  doi: 10.1023/A:1004611224835.  Google Scholar

[40]

Y. S. Xia, Further results on global convergence and stability of global projected dynamical systems, J. Optim. Theory Appl., 122 (2004), 627-649.  doi: 10.1023/B:JOTA.0000042598.21226.af.  Google Scholar

[41]

Z. H. Yang and J. D. Cao, Initial value problems for arbitrary order fractional differential equations with delay, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2993-3005.  doi: 10.1016/j.cnsns.2013.03.006.  Google Scholar

[42]

D. Zhang and A. Nagurney, On the stability of projected dynamical systems, J. Optim. Theory Appl., 85 (1995), 97-124.  doi: 10.1007/BF02192301.  Google Scholar

[43]

X. M. Zhao and G. Orosz, Nonlinear day-to-day traffic dynamics with driver experience delay: Modeling, stability and bifurcation analysis, Phys. D, 275 (2014), 54-66.  doi: 10.1016/j.physd.2014.02.005.  Google Scholar

[44]

Y. Zhou, F. Jiao and J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal., 71 (2009), 3249-3256.  doi: 10.1016/j.na.2009.01.202.  Google Scholar

[45]

Y. Z. Zou, X. Li, N. J. Huang and C. Y. Sun, Global dynamical systems involving generalized $f$-projection operators and set-valued perturbation in Banach spaces, J. Appl. Math. , 2012 (2012), Article ID 682465, 12 pages.  Google Scholar

[46]

Y. Z. Zou and C. Y. Sun, Equilibrium points for two related projective dynamical systems, Comm. Appl. Nonlinear Anal., 19 (2012), 111-119.   Google Scholar

[47]

Y. Z. Zou, X. K. Wu, W. B. Zhang and C. Y. Sun, An iterative method for a class of generalized global dynamical system involving fuzzy mappings in Hilbert spaces, Lecture Notes in Commput. Sci., 7666 (2012), 44-51.  doi: 10.1007/978-3-642-34478-7_6.  Google Scholar

Figure 1.  Transient behavior of the system (21) on [0, 0.4]
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