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Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems. (English) Zbl 1433.35462

The paper presents existence results for a nonlinear quasi-hemivariational inequality and for the associated inverse problem. These results improve previous works regarding the nonlinear dependence on the sought parameter and relaxation of convexity requirements. An application to an identification inverse problem is also given.

MSC:

35R30 Inverse problems for PDEs
49J40 Variational inequalities
49J53 Set-valued and variational analysis

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[1] Acar R and Vogel C R 1994 Analysis of bounded variation penalty methods for ill-posed problems Inverse problems10 1217-29 · Zbl 0809.35151 · doi:10.1088/0266-5611/10/6/003
[2] Alleche B and Rădulescu V D 2015 The Ekeland variational principle for equilibrium problems revisited and applications Nonlinear Anal.: RWA23 17-25 · Zbl 1318.49011 · doi:10.1016/j.nonrwa.2014.11.006
[3] Alleche B and Rădulescu V D 2016 Set-valued equilibrium problems with applications to Browder variational inclusions and to fixed point theory Nonlinear Anal.: RWA28 251-68 · Zbl 1327.49029 · doi:10.1016/j.nonrwa.2015.10.002
[4] Alleche B and Rădulescu V D 2017 Further on set-valued equilibrium problems and applications to Browder variational inclusions J. Optim. Theory Appl.175 39-58 · Zbl 1377.26002 · doi:10.1007/s10957-017-1169-1
[5] Attouch H, Buttazzo G and Michaille G 2006 Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization(MPS/SIAM Series on Optimization vol 6) (Philadelphia, PA: SIAM) · Zbl 1095.49001 · doi:10.1137/1.9780898718782
[6] Aussel D, Gupta R and Mehra A 2013 Gap functions and error bounds for inverse quasi-variational inequality problems J. Math. Anal. Appl.407 270-80 · Zbl 1311.49016 · doi:10.1016/j.jmaa.2013.03.049
[7] Barbu V 1982 Boundary control problems with nonlinear state equation SIAM J. Control Optim.20 125-43 · Zbl 0493.49024 · doi:10.1137/0320010
[8] Clason C, Khan A A, Sama M and Tammer C 2018 Contingent derivatives and regularization for noncoercive inverse problems Optimization68 1337-64 · Zbl 1422.90060 · doi:10.1080/02331934.2018.1442448
[9] Clason C and Valkonen T 2017 Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization SIAM J. Optim.27 1314-39 · Zbl 1369.49040 · doi:10.1137/16M1080859
[10] Clason C, Tameling C and Wirth B 2018 Vector-valued multibang control of differential equations SIAM J. Optim.56 2295-326 · Zbl 1393.49007 · doi:10.1137/16M1104998
[11] Dudek S and Migórski S 2018 Evolutionary Oseen model for generalized Newtonian fluid with multivalued nonmonotone friction law J. Math. Fluid Mech.20 1317-33 · Zbl 06955609 · doi:10.1007/s00021-018-0367-4
[12] Duvaut G and Lions J-L 1976 Inequalities in Mechanics and Physics (Berlin: Springer) · Zbl 0331.35002 · doi:10.1007/978-3-642-66165-5
[13] Gasiński L and Ochal A 2015 Dynamic thermoviscoelastic problem with friction and damage Nonlinear Anal.: RWA21 63-75 · Zbl 1334.74062 · doi:10.1016/j.nonrwa.2014.06.004
[14] Gasiński L and Papageorgiou N S 2013 A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities J. Global Optim.56 1347-60 · Zbl 1277.35182 · doi:10.1007/s10898-011-9841-8
[15] González Granada J R, Gwinner J and Kovtunenko V A 2018 On the shape differentiability of objectives: a Lagrangian approach and the Brinkman problem Axioms7 76 · Zbl 1432.49060 · doi:10.3390/axioms7040076
[16] Gwinner J 2018 An optimization approach to parameter identification in variational inequalities of second kind Optim. Lett.12 1141-54 · Zbl 1421.90146 · doi:10.1007/s11590-017-1173-2
[17] Gwinner J 2018 On two-coefficient identification in elliptic variational inequalities Optimization67 1017-30 · Zbl 1397.49014 · doi:10.1080/02331934.2018.1446955
[18] Gwinner J, Jadamba B, Khan A A and Sama M 2018 Identification in variational and quasi-variational inequalities J. Convex Anal.25 545-69 · Zbl 1391.49012
[19] Han W, Cong W and Wang G 2006 Mathematical theory and numerical analysis of bioluminescence tomography Inverse Problems22 1659-75 · Zbl 1106.35124 · doi:10.1088/0266-5611/22/5/008
[20] Han W, Sofonea M and Barboteu M 2017 Numerical analysis of elliptic hemivariational inequalities SIAM J. Numer. Anal.55 640-63 · Zbl 1362.74033 · doi:10.1137/16M1072085
[21] Huang N J and Fang Y P 2005 On vector variational inequalities in reflexive Banach spaces J. Global Optim.32 495-505 · Zbl 1097.49009 · doi:10.1007/s10898-003-2686-z
[22] Kassay G and Rădulescu V 2018 Equilibrium Problems and Applications (Elsevier: Academic)
[23] Khan A A, Migórski S and Sama M 2019 Inverse problems for multi-valued quasi-variational inequalities and noncoercive variational inequalities with noisy data Optimization68 1897-931 · Zbl 1426.49040 · doi:10.1080/02331934.2019.1604706
[24] Khan A A and Motreanu D 2015 Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities J. Optim. Theory Appl.167 1136-61 · Zbl 1337.49012 · doi:10.1007/s10957-015-0825-6
[25] Khan A A and Motreanu D 2018 Inverse problems for quasi-variational inequalities J. Global Optim.70 401-11 · Zbl 1387.35620 · doi:10.1007/s10898-017-0597-7
[26] Khan A A and Sama M 2012 Optimal control of multivalued quasi variational inequalities Nonlinear Anal.: TMA75 1419-28 · Zbl 1236.49019 · doi:10.1016/j.na.2011.08.005
[27] Khan A A, Tammer C and Zalinescu C 2015 Regularization of quasi-variational inequalities Optimization64 1703-24 · Zbl 1337.49005 · doi:10.1080/02331934.2015.1028935
[28] Kinderlehrer D and Stampacchia G 2000 An Introduction to Variational Inequalities and their Applications(Classics in Applied Mathematics vol 31) (Philadelphia, PA: SIAM) · Zbl 0988.49003 · doi:10.1137/1.9780898719451
[29] Kluge R (ed) 1978 On some parameter determination problems and quasi-variational inequalities Theory of Nonlinear Operators vol 6 (Berlin: Akademie) pp 129-39 · Zbl 0431.49006
[30] Liu Z H 2004 Generalized quasi-variational hemi-variational inequalities Appl. Math. Lett.17 741-5 · Zbl 1058.49006 · doi:10.1016/S0893-9659(04)90115-2
[31] Liu Z H, Zeng S D and Motreanu D 2018 Partial differential hemivariational inequalities Adv. Nonlinear Anal.7 571-86 · Zbl 1404.49004 · doi:10.1515/anona-2016-0102
[32] Liu Z H, Migórski S and Zeng S D 2017 Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces J. Differ. Equ.263 3989-4006 · Zbl 1372.35008 · doi:10.1016/j.jde.2017.05.010
[33] Liu Z H, Motreanu D and Zeng S D 2018 Nonlinear evolutionary systems driven by quasi-hemivariational inequalities Math. Methods Appli. Sci.41 1214-29 · Zbl 1390.35441 · doi:10.1002/mma.4660
[34] Liu Z H and Zeng B 2015 Optimal control of generalized quasi-variational hemivariational inequalities and its applications Appl. Math. Optim.72 305-23 · Zbl 1326.49017 · doi:10.1007/s00245-014-9281-1
[35] Liu Z H, Zeng S D and Motreanu D 2016 Evolutionary problems driven by variational inequalities J. Differ. Equ.260 6787-99 · Zbl 1341.47088 · doi:10.1016/j.jde.2016.01.012
[36] Migórski S, Khan A A and Zeng S D 2019 Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems of p -Laplacian type Inverse Problems35 14 · Zbl 1516.35530
[37] Migórski S and Zeng S D 2018 Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model Nonlinear Anal. RWA43 121-43 · Zbl 1394.35290 · doi:10.1016/j.nonrwa.2018.02.008
[38] Migórski S and Zeng S D 2018 A class of differential hemivariational inequalities in Banach spaces J. Glob. Optim.72 761-79 · Zbl 1475.49015 · doi:10.1007/s10898-018-0667-5
[39] Migórski S and Ochal A 2004 Boundary hemivariational inequality of parabolic type Nonlinear Anal.: TMA57 579-96 · Zbl 1050.35043 · doi:10.1016/j.na.2004.03.004
[40] Migórski S and Ochal A 2010 An inverse coefficient problem for a parabolic hemivariational inequality Appl. Anal.89 243-56 · Zbl 1185.35330 · doi:10.1080/00036810902889559
[41] Migórski S, Ochal A and Sofonea M 2013 Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems(Advances in Mechanics and Mathematics vol 26) (New York: Springer) · Zbl 1262.49001 · doi:10.1007/978-1-4614-4232-5
[42] Migórski S and Paczka D 2018 On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces Nonlinear Anal.: RWA39 337-61 · Zbl 1448.76010 · doi:10.1016/j.nonrwa.2017.07.003
[43] Naniewicz Z and Panagiotopoulos P D 1995 Mathematical Theory of Hemivariational Inequalities and Applications (New York: Marcel Dekker, Inc.) · Zbl 0968.49008
[44] Nashed M Z and Scherzer O 1998 Least squares and bounded variation regularization with nondifferentiable functionals Numer. Funct. Anal. Optim.19 873-901 · Zbl 0914.65067 · doi:10.1080/01630569808816863
[45] Panagiotopoulos P D 1983 Nonconvex energy functions, hemivariational inequalities and substationary principles Acta Mech.42 160-83
[46] Panagiotopoulos P D 1985 Nonconvex problems of semipermeable media and related topics Z. Angew. Math. Mech.65 29-36 · Zbl 0574.73015 · doi:10.1002/zamm.19850650116
[47] Panagiotopoulos P D 1985 Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions (Basel: Birkhäuser) · Zbl 0579.73014 · doi:10.1007/978-1-4612-5152-1
[48] Panagiotopoulos P D 1993 Hemivariational Inequalities, Applications in Mechanics and Engineering (Berlin: Springer) · Zbl 0826.73002 · doi:10.1007/978-3-642-51677-1
[49] Zeng B and Migórski S 2018 Variational-hemivariational inverse problems for unilateral frictional contact Appl. Anal. 1-20 (accepted) (https://doi.org/10.1080/00036811.2018.1491037) · Zbl 1436.49050
[50] Zeng S D, Liu Z H and Migórski S 2018 A class of fractional differential hemivariational inequalities with application to contact problem Z. Angew. Math. Phys.69 23 · Zbl 1392.35238 · doi:10.1007/s00033-018-0915-z
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