Decay mild solutions of fractional differential hemivariational inequalities
DOI:
https://doi.org/10.12775/TMNA.2021.032Keywords
Decay mild solutions, fractional differential hemivariational inequalities, fixed point theorem, measure of noncompactness, Mittag-Leffler functionAbstract
The goal of this paper is to consider fractional differential hemivariational inequalities (FDHVIs, for short) in the framework of Banach spaces. Our first aim is to investigate the existence of mild solutions to FDHVIs by means of a fixed point technique avoiding the hypothesis of compactness on the semigroup. The second step of the paper is to study the existence of decay mild solutions to FDHVIs via giving asymptotic behavior of Mittag-Leffler function.References
R. Agarwal, S. Hristova and D. O’Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal. 19 (2016), no. 2, 290–318.
R.R. Akhmerov, M.I. Kamenskiı̆, A.S. Potapov, A.E. Rodkina and B.N. Sadovskiı̆, Measures of Noncompactness and Condensing Operators, Birkhäuser, Boston, Basel, Berlin, 1992.
J.P. Aubin and A. Cellina, Differential Inclusions, Springer–Verlag, New York, 1984.
D. Baleanu and A.K. Golmankhaneh, On electromagnetic field in fractional space, Nonlinear Anal. 11 (2010), 288–292.
J. Banas and K. Goebal, Measure of Noncompactness in Banach Spaces, Marcel Dekker Inc., New York, 1980.
I. Benedetti, N.V. Loi and L. Malaguti, Nonlocal problems for differential inclusions in Hilbert spaces, Set-Valued Var. Anal. 22 (2014), 639–656.
X.J. Chen and Z.Y. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program. 146 (2014), 379–408.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
L.C. Evans, Partial Differential Equations, American Mathematical Society, 1998.
S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis (Theory), Kluwer Academic Publishers, Dordrecht Boston, London, 1997.
T.D. Ke, N.V. Loi and V. Obukhovskiı̆, Decay solutions for a class of fractional differential variational inequalities, Fract. Calc. Appl. Anal. 18 (2015), 531–553.
M. Kamenskiı̆, V. Obukhovskiı̆ and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Water de Gruyter, Berlin, 2001.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elservier Science B.V., Amsterdam, 2006.
X.W. Li, Y.X. Li, Z.H. Liu and J. Li, Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions, Fract. Calc. Appl. Anal. 21 (2018), no. 6, 1439–1470.
X.W. Li and Z.H. Liu, Sensitivity analysis of optimal control problems described by differential hemivariational inequalities, SIAM J. Control Optim. 56 (2018), no. 5, 3569–3597.
X.W. Li, Z.H. Liu and M. Sofonea, Unique solvability and exponential stability of differential hemivariational inequalities, Appl. Anal. 99 (2020), no. 14, 2489–2506.
Y.J. Liu, Z.H. Liu, C.F. Wen, J.C. Yao and S.D. Zeng, Existence of solutions for a class of noncoercive variational–hemivariational inequalities arising in contact problems, Appl. Math. Optim. 84 (2021), 2037–2059.
Z.H. Liu, N.V. Loi and V. Obukhovskiı̆, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23 (2013), no. 7, 1350125, 1–10.
Z.H. Liu, S. Migóski and S.D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations 263 (2017), 3989–4006.
Z.H. Liu, D. Motreanu and S.D. Zeng, Generalized penalty and regularization method for differential variational- hemivariational inequalities, SIAM J. Optim. 31 (2021), no. 2, 1158–1183.
Z.H. Liu, B. Zeng, Existence and controllability for fractional evolution inclusions of Clarke’s subdifferential type, Appl. Math. Comput. 257 (2015), 178–189.
Z.H. Liu, S.D. Zeng and Y. Bai, Maximum principles for multi-term space-time variableorder fractional diffusion equations and their applications, Fract. Calc. Appl. Anal. 19 (2016), 188–211.
Z.H. Liu, S.D. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations 260 (2016), 6787–6799.
Z.H. Liu, S.D. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal. 7 (2018), no. 4, 571–586.
N.V. Loi, On two parameter global bifurcation of periodic solutions to a class of differential variational inequalities, Nonlinear Anal. 122 (2015), 83–99.
N.V. Loi, T.D. Ke, V. Obukhovskı̆ and P. Zecca, Topological methods for some classes of differential variational inequalities, J. Nonlinear Convex Anal. 17 (2016), 403–419.
S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal. 41 (2009), 1415–1435.
S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26, Springer, New York, 2013.
S. Migórski, A. Ochal and M. Sofonea, A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elast. 127 (2017), 151–178.
Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, vol. 188, CRC Press, 1994.
T.V.A. Nguyen and D.K. Tran, On the differential variational inequalities of parabolic elliptic type, Math. Meth. Appl. Sci. 40 (2017), 4683–4695.
J.S. Pang and D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2008), 345–424.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
R. Sakthivel, Y. Ren and N.I. Mahmudov, On the approximate controllability of semilinear fractional differential systems, Comput. Math. Appl. 62 (2011), 1451–1459.
A. Strauss and J.A. Yorke, Perturbing uniform asymptotically stable nonlinear systems, J. Differential Equations 6 (1969), 452–483.
S.D. Zeng, S. Migórski and Z.H. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariation al inequalities, SIAM J. Optim. 31 (2021), no. 4, 2829–2862.
H.C. Zhou and B.Z. Guo, Boundary feedback stabilization for an unstable time fractional reaction diffusion equation, SIAM J. Control Optim. 56 (2018), no. 1, 75–101.
Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), 1063–1077.
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