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June  2022, 11(3): 827-836. 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  June 2022 Early access  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 and Control Theory, 2022, 11 (3) : 827-836. 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.

[2]

J. CenC. MinV. 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.

[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.

[4]

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.

[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.

[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]

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.

[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.

[9]

T. D. KeN. 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.

[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.

[11]

X.-S. LiN.-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.

[12]

Z. LiuS. 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.

[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.

[14]

Z. LiuD. 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.

[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.

[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. 

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[24]

S. Treanţă, On a class of differential variational inequalities in infinite-dimensional spaces, Mathematics, 9 (2021), 266. doi: 10.3390/math9030266.

[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.

[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.

show all references

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.

[2]

J. CenC. MinV. 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.

[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.

[4]

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.

[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.

[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]

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.

[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.

[9]

T. D. KeN. 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.

[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.

[11]

X.-S. LiN.-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.

[12]

Z. LiuS. 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.

[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.

[14]

Z. LiuD. 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.

[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.

[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. 

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[24]

S. Treanţă, On a class of differential variational inequalities in infinite-dimensional spaces, Mathematics, 9 (2021), 266. doi: 10.3390/math9030266.

[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.

[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.

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