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April  2020, 25(4): 1193-1212. doi: 10.3934/dcdsb.2019216

Existence of a global attractor for fractional differential hemivariational inequalities

Yirong Jiang 1, , Nanjing Huang 2, and  Zhouchao Wei 3,,

1. 

College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China
2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
3. 

School of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan, Hubei 430074, China

* Corresponding author: Zhouchao Wei, weizhouchao@163.com

Received  December 2018 Revised  May 2019 Published  April 2020 Early access  September 2019

Fund Project: The first author is supported by the Natural Science Foundation of Guangxi Province (2018GXNSFAA281021) and the Foundation of Guilin University of Technology (GUTQDJJ2017062). The second author is supported by the National Natural Science Foundation of China (11471230, 11671282). The third author is supported by the National Natural Science Foundation of China (11772306), Graduates education Teaching Research and Reform Project of China University of Geosciences (YJS2018311), and the Fundamental Research Funds for the Central Universities, China University of Geosciences (CUGGC05).

In this paper, we introduce and study a new class of fractional differential hemivariational inequalities ((FDHVIs), for short) formulated by an initial-value fractional evolution inclusion and a hemivariational inequality in infinite Banach spaces. First, by applying measure of noncompactness, a fixed point theorem of a condensing multivalued map, we obtain the nonemptiness and compactness of the mild solution set for (FDHVIs). Further, we apply the obtained results to establish an existence theorem of the mild solution of a global attractor for the semiflow governed by a fractional differential hemivariational inequality ((FDHVI), for short). Finally, we provide an example to demonstrate the main results.

Keywords: Fractional differential hemivariational inequalities, properties of the mild solution set, global attractor, measure of noncompactness.
Mathematics Subject Classification: Primary: 34D45, 35R11, 35R70; Secondary: 49J53.
Citation: Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216
References:
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N. T. V. Anh and T. D. Ke, On the differential variational inequalities of parabolic-elliptic type, Math. Methods Appl. Sci., 40 (2017), 4683-4695. 

[2]

D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Isr. J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044.

[3]

X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program. Ser. A, 146 (2014), 379-408.  doi: 10.1007/s10107-013-0689-1.

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F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

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J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Am. Math. Soc., 118 (1993), 447-453.  doi: 10.2307/2160321.

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J. Gwinner, On a new class of differential variational inequalities and a stability result, Math. Program. Ser. B, 139 (2013), 205-221.  doi: 10.1007/s10107-013-0669-5.

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A. Halanay, Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York and London, 1966.

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M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivaluedmaps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.

[9]

T. D. Ke and D. Lan, Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 17 (2014), 96-121.  doi: 10.2478/s13540-014-0157-5.

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T. D. Ke and D. Lan, Fixed point approach for weakly asymptotic stability of fractional differential inclusions involving impulsive effects, J. Fixed Point Theory Appl., 19 (2017), 2185-2208.  doi: 10.1007/s11784-017-0412-6.

[11]

T. D. Ke, N. V. Loi and V. Obukhovskii, Decay solutions for a class of fractional differential variational inequalities, Fract. Calc. Appl. Anal., 18 (2015), 531-553.  doi: 10.1515/fca-2015-0033.

[12]

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

[13]

X. S. Li, N. J. Huang and D. O'Regan, Differential mixed variational inqualities in finite dimensional spaces, Nonlinear Anal., 72 (2010), 3875-3886.  doi: 10.1016/j.na.2010.01.025.

[14]

X. W. Li and Z. H. Liu, Sensitivity analysis of optimal control problems described by differential hemivariational inequalities, SIAM J. Control Optim., 56 (2018), 3569-3597.  doi: 10.1137/17M1162275.

[15]

Z. H. Liu, S. Migorski and S. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations, 263 (2017), 3989-4006.  doi: 10.1016/j.jde.2017.05.010.

[16]

Z. H. Liu, S. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Analysis, 7 (2018), 571-586.  doi: 10.1515/anona-2016-0102.

[17]

N. V. Loi, T. D. Ke, V. Obukhovskii and P. Zecca, Topological methods for some classes of differential variational inequalities, J. Nonlinear Convex Anal., 17 (2016), 403-419. 

[18]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability Distributions Generated by Fractional Diffusion Equations, in: J. Kertesz, I. Kondor (Eds.), Econophysics: An Emerging Science, Kluwer Academic Publisher, Dordrecht Boston, London, 2000.

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V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.

[20]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.

[21]

S. Migórski, On existence of solutions for parabolic hemivariational inequalities, J. Comput. Appl. Math., 129 (2001), 77-87.  doi: 10.1016/S0377-0427(00)00543-4.

[22]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Springer-Verlag, New York, 2013.

[23]

J. S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424.  doi: 10.1007/s10107-006-0052-x.

[24]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.

[25]

A. U. Raghunathan, J. R. Pérez-Correa, E. Agosin and L. T. Biegler, Parameter estimation in metabolic flux balance models for batch fermentation-formulation and solution using differential variational inequalities, Ann. Oper. Res., 148 (2006), 251-270.  doi: 10.1007/s10479-006-0086-8.

[26]

D. E. Stewart, Uniqueness for index-one differential variational inequalities, Nonlinear Anal., Hybrid Syst., 2 (2008), 812-818.  doi: 10.1016/j.nahs.2006.10.015.

[27]

A. Tasora, M. Anitescu, S. Negrini and D. Negrut, A compliant visco-plastic particle contact model based on differential variational inequalities, International Journal of Nonlinear Mechanics, 53 (2013), 2-12.  doi: 10.1016/j.ijnonlinmec.2013.01.010.

[28]

I. Vrabie, $C_{0}$-Semigroups and Applications, Elsevier, Amsterdam, 2003.

[29]

X. Wang and N. J. Huang, Differential vector variational inequalities in finite dimensional spaces, J. Optim. Theory Appl., 158 (2013), 109-129.  doi: 10.1007/s10957-012-0164-9.

[30]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.

show all references

References:
[1]

N. T. V. Anh and T. D. Ke, On the differential variational inequalities of parabolic-elliptic type, Math. Methods Appl. Sci., 40 (2017), 4683-4695. 

[2]

D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Isr. J. Math., 108 (1998), 109-138.  doi: 10.1007/BF02783044.

[3]

X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program. Ser. A, 146 (2014), 379-408.  doi: 10.1007/s10107-013-0689-1.

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[5]

J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Am. Math. Soc., 118 (1993), 447-453.  doi: 10.2307/2160321.

[6]

J. Gwinner, On a new class of differential variational inequalities and a stability result, Math. Program. Ser. B, 139 (2013), 205-221.  doi: 10.1007/s10107-013-0669-5.

[7]

A. Halanay, Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York and London, 1966.

[8]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivaluedmaps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.

[9]

T. D. Ke and D. Lan, Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces, Fract. Calc. Appl. Anal., 17 (2014), 96-121.  doi: 10.2478/s13540-014-0157-5.

[10]

T. D. Ke and D. Lan, Fixed point approach for weakly asymptotic stability of fractional differential inclusions involving impulsive effects, J. Fixed Point Theory Appl., 19 (2017), 2185-2208.  doi: 10.1007/s11784-017-0412-6.

[11]

T. D. Ke, N. V. Loi and V. Obukhovskii, Decay solutions for a class of fractional differential variational inequalities, Fract. Calc. Appl. Anal., 18 (2015), 531-553.  doi: 10.1515/fca-2015-0033.

[12]

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

[13]

X. S. Li, N. J. Huang and D. O'Regan, Differential mixed variational inqualities in finite dimensional spaces, Nonlinear Anal., 72 (2010), 3875-3886.  doi: 10.1016/j.na.2010.01.025.

[14]

X. W. Li and Z. H. Liu, Sensitivity analysis of optimal control problems described by differential hemivariational inequalities, SIAM J. Control Optim., 56 (2018), 3569-3597.  doi: 10.1137/17M1162275.

[15]

Z. H. Liu, S. Migorski and S. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations, 263 (2017), 3989-4006.  doi: 10.1016/j.jde.2017.05.010.

[16]

Z. H. Liu, S. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Analysis, 7 (2018), 571-586.  doi: 10.1515/anona-2016-0102.

[17]

N. V. Loi, T. D. Ke, V. Obukhovskii and P. Zecca, Topological methods for some classes of differential variational inequalities, J. Nonlinear Convex Anal., 17 (2016), 403-419. 

[18]

F. Mainardi, P. Paradisi and R. Gorenflo, Probability Distributions Generated by Fractional Diffusion Equations, in: J. Kertesz, I. Kondor (Eds.), Econophysics: An Emerging Science, Kluwer Academic Publisher, Dordrecht Boston, London, 2000.

[19]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.

[20]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403.  doi: 10.1023/A:1026514727329.

[21]

S. Migórski, On existence of solutions for parabolic hemivariational inequalities, J. Comput. Appl. Math., 129 (2001), 77-87.  doi: 10.1016/S0377-0427(00)00543-4.

[22]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Springer-Verlag, New York, 2013.

[23]

J. S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424.  doi: 10.1007/s10107-006-0052-x.

[24]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999.

[25]

A. U. Raghunathan, J. R. Pérez-Correa, E. Agosin and L. T. Biegler, Parameter estimation in metabolic flux balance models for batch fermentation-formulation and solution using differential variational inequalities, Ann. Oper. Res., 148 (2006), 251-270.  doi: 10.1007/s10479-006-0086-8.

[26]

D. E. Stewart, Uniqueness for index-one differential variational inequalities, Nonlinear Anal., Hybrid Syst., 2 (2008), 812-818.  doi: 10.1016/j.nahs.2006.10.015.

[27]

A. Tasora, M. Anitescu, S. Negrini and D. Negrut, A compliant visco-plastic particle contact model based on differential variational inequalities, International Journal of Nonlinear Mechanics, 53 (2013), 2-12.  doi: 10.1016/j.ijnonlinmec.2013.01.010.

[28]

I. Vrabie, $C_{0}$-Semigroups and Applications, Elsevier, Amsterdam, 2003.

[29]

X. Wang and N. J. Huang, Differential vector variational inequalities in finite dimensional spaces, J. Optim. Theory Appl., 158 (2013), 109-129.  doi: 10.1007/s10957-012-0164-9.

[30]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.

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