# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021027

## On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces

 Department of Applied Mathematics, Faculty of Applied Sciences, University Politehnica of Bucharest, Bucharest 060042, Romania

* Corresponding author: Savin Treanţă

Received  November 2020 Revised  March 2021 Published  May 2021

A class of differential quasi-variational-hemivariational inequalities (DQVHI, for short) is studied in this paper. First, based on the Browder's result, KKM theorem and monotonicity arguments, we prove the superpositionally measurability, convexity and strongly-weakly upper semicontinuity for the solution set of a general quasi-variational-hemivariational inequality. Further, by using optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of (DQVHI) is nonempty and compact. This kind of evolutionary problems incorporates various classes of problems and models.

Citation: Savin Treanţă. On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces. Evolution Equations & Control Theory, doi: 10.3934/eect.2021027
##### References:
 [1] F. E. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 71 (1965), 780-785.  doi: 10.1090/S0002-9904-1965-11391-X.  Google Scholar [2] J. Cen, C. Min, V. T. Nguyen and G.-J. Tang, On the well-posedness of differential quasi-variational-hemivariational inequalities, Open Mathematics, 18 (2020), 540-551.  doi: 10.1515/math-2020-0028.  Google Scholar [3] X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 146 (2014), 379-408.  doi: 10.1007/s10107-013-0689-1.  Google Scholar [4] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.  Google Scholar [5] J. Gwinner, On the $p$-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction, Appl. Numer. Math., 59 (2009), 2774-2784.  doi: 10.1016/j.apnum.2008.12.027.  Google Scholar [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.  Google Scholar [7] L. Han and J.-S. Pang, Non-Zenoness of a class of differential quasi-variational inequalities, Math. Program., 121 (2010), 171-199.  doi: 10.1007/s10107-008-0230-0.  Google Scholar [8] M. I. Kamemskii, V. V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Berlin: De Gruyter Series in Nonlinear Analysis and Applications, 7, 2001. doi: 10.1515/9783110870893.  Google Scholar [9] 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.  Google Scholar [10] X. J. Li and J. M. Yong, Optimal Control Theory for infinite Dimensional Systems, Boston: Birkhaüser, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar [11] X.-S. Li, N.-J. Huang and D. O'Regan, Differential mixed variational inequalities in finite dimensional spaces, Nonlinear Anal., 72 (2010), 3875-3886.  doi: 10.1016/j.na.2010.01.025.  Google Scholar [12] Z. Liu, S. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differ. Equ., 260 (2016), 6787-6799.  doi: 10.1016/j.jde.2016.01.012.  Google Scholar [13] Z. Liu and S. Zeng, Differential variational inequalities in infinite Banach spaces, Acta Mathematica Scientia, 37 (2017), 26-32.  doi: 10.1016/S0252-9602(16)30112-6.  Google Scholar [14] Z. Liu, D. Motreanu and S. Zeng, On the well-posedness of differential mixed quasi-variational inequalities, Topol. Method Nonl. Anal., 51 (2018), 135-150.  doi: 10.12775/tmna.2017.041.  Google Scholar [15] Z. Liu, N. V. Loi and V. Obukhovskii, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Int. J. Bifurcation Chaos, 23 (2013), ID: 1350125. doi: 10.1142/S0218127413501253.  Google Scholar [16] Z. Liu and B. Zeng, Existence results for a class of hemivariational inequalities involving the stable $(G, F, \alpha)$-quasimonotonicity, Topol. Methods Nonlinear Anal., 47 (2016), 195-217.   Google Scholar [17] 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: 10.1016/j.na.2015.03.019.  Google Scholar [18] S. Migórski and Y. Bai, Well-posedness of history-dependent evolution inclusions with applications, Z. Angew. Math. Phys., 70 (2019), 114. doi: 10.1007/s00033-019-1158-3.  Google Scholar [19] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar [20] J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program., 113 (2008), 345-424.  doi: 10.1007/s10107-006-0052-x.  Google Scholar [21] J.-S. Pang and D. E. Stewart, Solution dependence on initial conditions in differential variational variational inequalities, Math. Program., 116 (2009), 429-460.  doi: 10.1007/s10107-007-0117-5.  Google Scholar [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [23] K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces, Nonlinear Anal., 75 (2012), 2701-2712.  doi: 10.1016/j.na.2011.10.049.  Google Scholar [24] S. Treanţă, On a class of differential variational inequalities in infinite-dimensional spaces, Mathematics, 9 (2021), 266. doi: 10.3390/math9030266.  Google Scholar [25] X. Wang and N.-J. Huang, A class of differential vector variational inequalities in finite dimensional spaces, J. Optim. Theory Appl., 162 (2014), 633-648.  doi: 10.1007/s10957-013-0311-y.  Google Scholar [26] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, NewYork, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

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##### References:
 [1] F. E. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc., 71 (1965), 780-785.  doi: 10.1090/S0002-9904-1965-11391-X.  Google Scholar [2] J. Cen, C. Min, V. T. Nguyen and G.-J. Tang, On the well-posedness of differential quasi-variational-hemivariational inequalities, Open Mathematics, 18 (2020), 540-551.  doi: 10.1515/math-2020-0028.  Google Scholar [3] X. Chen and Z. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 146 (2014), 379-408.  doi: 10.1007/s10107-013-0689-1.  Google Scholar [4] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.  Google Scholar [5] J. Gwinner, On the $p$-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction, Appl. Numer. Math., 59 (2009), 2774-2784.  doi: 10.1016/j.apnum.2008.12.027.  Google Scholar [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.  Google Scholar [7] L. Han and J.-S. Pang, Non-Zenoness of a class of differential quasi-variational inequalities, Math. Program., 121 (2010), 171-199.  doi: 10.1007/s10107-008-0230-0.  Google Scholar [8] M. I. Kamemskii, V. V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Berlin: De Gruyter Series in Nonlinear Analysis and Applications, 7, 2001. doi: 10.1515/9783110870893.  Google Scholar [9] 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.  Google Scholar [10] X. J. Li and J. M. Yong, Optimal Control Theory for infinite Dimensional Systems, Boston: Birkhaüser, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar [11] X.-S. Li, N.-J. Huang and D. O'Regan, Differential mixed variational inequalities in finite dimensional spaces, Nonlinear Anal., 72 (2010), 3875-3886.  doi: 10.1016/j.na.2010.01.025.  Google Scholar [12] Z. Liu, S. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differ. Equ., 260 (2016), 6787-6799.  doi: 10.1016/j.jde.2016.01.012.  Google Scholar [13] Z. Liu and S. Zeng, Differential variational inequalities in infinite Banach spaces, Acta Mathematica Scientia, 37 (2017), 26-32.  doi: 10.1016/S0252-9602(16)30112-6.  Google Scholar [14] Z. Liu, D. Motreanu and S. Zeng, On the well-posedness of differential mixed quasi-variational inequalities, Topol. Method Nonl. Anal., 51 (2018), 135-150.  doi: 10.12775/tmna.2017.041.  Google Scholar [15] Z. Liu, N. V. Loi and V. Obukhovskii, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Int. J. Bifurcation Chaos, 23 (2013), ID: 1350125. doi: 10.1142/S0218127413501253.  Google Scholar [16] Z. Liu and B. Zeng, Existence results for a class of hemivariational inequalities involving the stable $(G, F, \alpha)$-quasimonotonicity, Topol. Methods Nonlinear Anal., 47 (2016), 195-217.   Google Scholar [17] 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: 10.1016/j.na.2015.03.019.  Google Scholar [18] S. Migórski and Y. Bai, Well-posedness of history-dependent evolution inclusions with applications, Z. Angew. Math. Phys., 70 (2019), 114. doi: 10.1007/s00033-019-1158-3.  Google Scholar [19] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar [20] J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program., 113 (2008), 345-424.  doi: 10.1007/s10107-006-0052-x.  Google Scholar [21] J.-S. Pang and D. E. Stewart, Solution dependence on initial conditions in differential variational variational inequalities, Math. Program., 116 (2009), 429-460.  doi: 10.1007/s10107-007-0117-5.  Google Scholar [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [23] K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces, Nonlinear Anal., 75 (2012), 2701-2712.  doi: 10.1016/j.na.2011.10.049.  Google Scholar [24] S. Treanţă, On a class of differential variational inequalities in infinite-dimensional spaces, Mathematics, 9 (2021), 266. doi: 10.3390/math9030266.  Google Scholar [25] X. Wang and N.-J. Huang, A class of differential vector variational inequalities in finite dimensional spaces, J. Optim. Theory Appl., 162 (2014), 633-648.  doi: 10.1007/s10957-013-0311-y.  Google Scholar [26] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators, Springer-Verlag, NewYork, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar
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