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

[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.

[3]

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

[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. 

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

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

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.
[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.

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

[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.

[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.

[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.

[31]

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

[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. 

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[46]

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

[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.

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.

[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.

[3]

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

[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. 

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

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

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

[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. 

[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.

[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.

[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.

[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.

[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.

[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.
[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.

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

[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.

[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.

[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.

[31]

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

[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. 

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[46]

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

[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.

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