A~new class of fractional differential hemivariational inequalities

S. D. Zeng ; S. Migórski ; W. Han

Izvestiya. Mathematics , Tome 87 (2023) no. 2, pp. 326-361.

Résumé

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly–weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier–Stokes system supplemented by a fractional reaction–diffusion equation, which is studied as a fractional hemivariational inequality.

Détail

Keywords: fractional differential hemivariational inequality, Clarke subgradient, $C_0$-semigroup, Navier–Stokes system.
Mots-clés : existence

Bibliographie

[1] J. S. Pang and D. E. Stewart, “Differential variational inequalities”, Math. Program., 113:2, Ser. A (2008), 345–424 | DOI | MR | Zbl

[2] Z. H. Liu, D. Motreanu, and S. D. Zeng, “Nonlinear evolutionary systems driven by mixed variational inequalities and its applications”, Nonlinear Anal. Real World Appl., 42 (2018), 409–421 | DOI | MR | Zbl

[3] Z. H. Liu, D. Motreanu, and S. D. Zeng, “On the well-posedness of differential mixed quasi-variational-inequalities”, Topol. Methods Nonlinear Anal., 51:1 (2018), 135–150 | DOI | MR | Zbl

[4] Z. H. Liu, S. D. Zeng, and D. Motreanu, “Evolutionary problems driven by variational inequalities”, J. Differential Equations, 260:9 (2016), 6787–6799 | DOI | MR | Zbl

[5] Z. H. Liu, S. Migórski, and S. D. Zeng, “Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces”, J. Differential Equations, 263:7 (2017), 3989–4006 | DOI | MR | Zbl

[6] X. J. Chen and Z. Y. Wang, “Differential variational inequality approach to dynamic games with shared constraints”, Math. Program., 146:1-2, Ser. A (2014), 379–408 | DOI | MR | Zbl

[7] T. D. Ke, N. V. Loi, and V. Obukhovskii, “Decay solutions for a class of fractional differential variational inequalities”, Fract. Calc. Appl. Anal., 18:3 (2015), 531–553 | DOI | MR | Zbl

[8] J. Gwinner, “On a new class of differential variational inequalities and a stability result”, Math. Program., 139:1-2, Ser. B (2013), 205–221 | DOI | MR | Zbl

[9] Z. H. Liu, N. V. Loi, and V. Obukhovskii, “Existence and global bifurcation of periodic solutions to a class of differential variational inequalities”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23:7 (2013), 1350125 | DOI | MR | Zbl

[10] B. Brogliato and A. Tanwani, “Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness, and stability”, SIAM Rev., 62:1 (2020), 3–129 | DOI | MR | Zbl

[11] X. J. Chen and Z. Y. Wang, “Convergence of regularized time-stepping methods for differential variational inequalities”, SIAM J. Optim., 23:3 (2013), 1647–1671 | DOI | MR | Zbl

[12] L. S. Han, A. Tiwari, M. K. Camlibel, and J. S. Pang, “Convergence of time-stepping schemes for passive and extended linear complementarity systems”, SIAM J. Numer. Anal., 47:5 (2009), 3768–3796 | DOI | MR | Zbl

[13] L. S. Han and J. S. Pang, “Non-Zenoness of a class of differential quasi-variational inequalities”, Math. Program., 121:1, Ser. A (2010), 171–199 | DOI | MR | Zbl

[14] Y. R. Jiang and Z. C. Wei, “Weakly asymptotic stability for fractional delay differential mixed variational inequalities”, Appl. Math. Optim., 84:1 (2021), 273–297 | DOI | MR | Zbl

[15] Z. H. Liu and S. D. Zeng, “Differential variational inequalities in infinite Banach spaces”, Acta Math. Sci. Ser. B (Engl. Ed.), 37:1 (2017), 26–32 | DOI | MR | Zbl

[16] N. V. Loi, “On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities”, Nonlinear Anal., 122 (2015), 83–99 | DOI | MR | Zbl

[17] S. Migórski and S. D. Zeng, “A class of generalized evolutionary problems driven by variational inequalities and fractional operators”, Set-Valued Var. Anal., 27 (2019), 949–970 | DOI | MR | Zbl

[18] S. Migórski and S. D. Zeng, “Mixed variational inequalities driven by fractional evolution equations”, Acta Math. Sci. Ser. B (Engl. Ed.), 39:2 (2019), 461–468 | DOI | MR | Zbl

[19] J. S. Pang and D. E. Stewart, “Solution dependence on initial conditions in differential variational inequalities”, Math. Program., 116:1-2, Ser. B (2009), 429–460 | DOI | MR | Zbl

[20] J. L. Shen and J. S. Pang, “Linear complementarity systems: Zeno states”, SIAM J. Control Optim., 44:3 (2005), 1040–1066 | DOI | MR | Zbl

[21] N. T. V. Anh and T. D. Ke, “Asymptotic behavior of solutions to a class of differential variational inequalities”, Ann. Polon. Math., 114:2 (2015), 147–164 | DOI | MR | Zbl

[22] P. D. Panagiotopoulos, “Nonconvex energy functions. Hemivariational inequalities and substationarity principles”, Acta Mech., 48:3-4 (1983), 111–130 | DOI | MR | Zbl

[23] P. D. Panagiotopoulos, Hemivariational inequalities. Applications in mechanics and engineering, Springer–Verlag, Berlin, 1993 | DOI | MR | Zbl

[24] Z. H. Liu, S. D. Zeng, and D. Motreanu, “Partial differential hemivariational inequalities”, Adv. Nonlinear Anal., 7:4 (2018), 571–586 | DOI | MR | Zbl

[25] S. Migórski and S. D. Zeng, “Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model”, Nonlinear Anal. Real World Appl., 43 (2018), 121–143 | DOI | MR | Zbl

[26] S. D. Zeng, Z. H. Liu, and S. Migórski, “A class of fractional differential hemivariational inequalities with application to contact problem”, Z. Angew. Math. Phys., 69:2 (2018), 36 | DOI | MR | Zbl

[27] Y. R. Jiang, N. J. Huang, and Z. C. Wei, “Existence of a global attractor for fractional differential hemivariational inequalities”, Discrete Contin. Dyn. Syst. Ser. B, 25:4 (2020), 1193–1212 | DOI | MR | Zbl

[28] X. W. Li and Z. H. Liu, “Sensitivity analysis of optimal control problems described by differential hemivariational inequalities”, SIAM J. Control Optim., 56:5 (2018), 3569–3597 | DOI | MR | Zbl

[29] X. W. Li, Z. H. Liu, and M. Sofonea, “Unique solvability and exponential stability of differential hemivariational inequalities”, Appl. Anal., 99:14 (2020), 2489–2506 | DOI | MR | Zbl

[30] Z. H. Liu, D. Motreanu, and S. D. Zeng, “Nonlinear evolutionary systems driven by quasi-hemivariational inequalities”, Math. Methods Appl. Sci., 41:3 (2018), 1214–1229 | DOI | MR | Zbl

[31] Z. H. Liu, D. Motreanu, and S. D. Zeng, “Generalized penalty and regularization method for differential variational-hemivariational inequalities”, SIAM J. Optim., 31:2 (2021), 1158–1183 | DOI | MR | Zbl

[32] Z. H. Liu, V. T. Nguyen, J. C. Yao, and S. D. Zeng, “History-dependent differential variational-hemivariational inequalities with applications to contact mechanics”, Evol. Equ. Control Theory, 9:4 (2020), 1073–1087 | DOI | MR | Zbl

[33] S. Migórski and S. D. Zeng, “A class of differential hemivariational inequalities in Banach spaces”, J. Global Optim., 72:4 (2018), 761–779 | DOI | MR | Zbl

[34] G. J. Tang, J. X. Cen, V. T. Nguyen, and S. D. Zeng, “Differential variational-hemivariational inequalities: existence, uniqueness, stability, and convergence”, J. Fixed Point Theory Appl., 22:4 (2020), 83 | DOI | MR | Zbl

[35] S. D. Zeng, S. Migórski, and A. A. Khan, “Nonlinear quasi-hemivariational inequalities: existence and optimal control”, SIAM J. Control Optim., 59:2 (2021), 1246–1274 | DOI | MR | Zbl

[36] V. Barbu, Analysis and control of nonlinear infinite dimensional systems, Math. Sci. Engrg., 190, Academic Press, Inc., Boston, MA, 1993 | MR | Zbl

[37] F. H. Clarke, Optimization and nonsmooth analysis, Canad. Math. Soc. Ser. Monogr. Adv. Texts, A Wiley-Interscience Publication. John Wiley Sons, Inc., New York, 1983 ; Russian transl. Nauka, Moscow, 1988 | MR | Zbl | MR | Zbl

[38] Z. Denkowski, S. Migórski, and N. S. Papageorgiou, An introduction to nonlinear analysis: theory, Kluwer Acad. Publ., Boston, MA, 2003 | DOI | MR | Zbl

[39] Z. Denkowski, S. Migórski, and N. S. Papageorgiou, An introduction to nonlinear analysis: applications, Kluwer Acad. Publ., Boston, MA, 2003 | MR | Zbl

[40] L. Gasiński and N. S. Papageorgiou, Nonlinear analysis, Ser. Math. Anal. Appl., 9, Chapman Hall/CRC, Boca Raton, FL, 2006 | DOI | MR | Zbl

[41] S. Migórski, A. Ochal, and M. Sofonea, Nonlinear inclusions and hemivariational inequalities. Models and analysis of contact problems, Adv. Mech. Math., 26, Springer, New York, 2013 | DOI | MR | Zbl

[42] E. Zeidler, Nonlinear functional analysis and its applications, v. II/A, Springer-Verlag, New York, 1990 ; v. II/B | DOI | MR | Zbl | DOI | MR | Zbl

[43] J. R. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls”, Nonlinear Anal. Real World Appl., 12:1 (2011), 262–272 | DOI | MR | Zbl

[44] F. Mainardi, P. Paradisi, and R. Gorenflo, “Probability distributions generated by fractional diffusion equations”, Econophysics: an emerging science, Kluwer Acad. Publ., Dordrecht, 2000, 312–350

[45] Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations”, Nonlinear Anal. Real World Appl., 11:5 (2010), 4465–4475 | DOI | MR | Zbl

[46] P. Kalita, “Convergence of Rothe scheme for hemivariational inequalities of parabolic type”, Int. J. Numer. Anal. Model., 10:2 (2013), 445–465 | MR | Zbl

[47] B. Zeng and S. Migórski, “Evolutionary subgradient inclusions with nonlinear weakly continuous operators and applications”, Comput. Math. Appl., 75:1 (2018), 89–104 | DOI | MR | Zbl

[48] S. Migórski and A. Ochal, “Hemivariational inequalities for stationary Navier–Stokes equations”, J. Math. Anal. Appl., 306:1 (2005), 197–217 | DOI | MR | Zbl

[49] M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Ser. Nonlinear Anal. Appl., 7, Walter de Gruyter Co., Berlin, 2001 | DOI | MR | Zbl

[50] W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Stud. Adv. Math., 30, Amer. Math. Soc., Providence, RI; International Press, Somerville, MA, 2002 | DOI | MR | Zbl

[51] C. Carstensen and J. Gwinner, “A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems”, Ann. Mat. Pura Appl. (4), 177 (1999), 363–394 | DOI | MR | Zbl

[52] R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach”, Phys. Rep., 339:1 (2000), 1–77 | DOI | MR | Zbl

[53] X. J. Li and C. J. Xu, “A space-time spectral method for the time fractional diffusion equation”, SIAM J. Numer. Anal., 47:3 (2009), 2108–2131 | DOI | MR | Zbl

[54] Y. N. Zhang, Z. Z. Sun, and H. L.Liao, “Finite difference methods for the time fractional diffusion equation on non-uniform meshes”, J. Comput. Phys., 265 (2014), 195–210 | DOI | MR | Zbl

[55] I. I. Vrabie, $C_0$-semigroups and applications, North-Holland Math. Stud., 191, North-Holland Publishing Co., Amsterdam, 2003 | MR | Zbl

[56] J. Franců, “Weakly continuous operators. Applications to differential equations”, Appl. Math., 39:1 (1994), 45–56 | DOI | MR | Zbl

[57] N. U. Ahmed, “Optimal control of hydrodynamic flow with possible application to artificial heart”, Dynam. Systems Appl., 1:2 (1992), 103–119 | MR | Zbl