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**Existence theorems for elliptic and evolutionary variational and quasi-variational inequalities.**
*(English)*
Zbl 1337.49012

Summary: This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge’s fixed-point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

### MSC:

49J40 | Variational inequalities |

49J53 | Set-valued and variational analysis |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

47H05 | Monotone operators and generalizations |

47H04 | Set-valued operators |

### Keywords:

evolutionary variational inequalities; quasi-variational inequalities; hemivariational inequalities; monotonicity; pseudo-monotonicity; set-valued maps; coercivity; asymptotic recessivity
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\textit{A. A. Khan} and \textit{D. Motreanu}, J. Optim. Theory Appl. 167, No. 3, 1136--1161 (2015; Zbl 1337.49012)

Full Text:
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### References:

[1] | Alber, Y.I., Butnariu, D., Ryazantseva, I.: Regularization of monotone variational inequalities with Mosco approximations of the constraint sets. Set Valued Anal. 13, 265-290 (2005) · Zbl 1129.49003 |

[2] | Alber, Y.I., Notik, A.I.: Perturbed unstable variational inequalities with unbounded operators on approximately given sets. Set Valued Anal. 1, 393-402 (1993) · Zbl 0815.49010 |

[3] | Azevedo, A., Miranda, F., Santos, L.: Variational and quasivariational inequalities with first order constraints. J. Math. Anal. Appl. 397, 738-756 (2013) · Zbl 1258.49004 |

[4] | Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984) · Zbl 0551.49007 |

[5] | Barrett, J.W., Prigozhin, L.: Lakes and rivers in the landscape: a quasi-variational inequality approach. Interfaces Free Bound. 16(2), 269-296 (2014) · Zbl 1297.35122 |

[6] | Bensoussan, A., Lions, J.L.: Nouvelles méthodes en contrôle impulsionnel, Appl. Math. Optim., 1, 289-312 (1974/75) · Zbl 0339.93033 |

[7] | Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications. Springer, New York (2007) · Zbl 1109.35004 |

[8] | Giannessi, F.: Separation of sets and gap functions for quasi-variational inequalities. Variational Inequalities and Network Equilibrium Problems (Erice, 1994), pp. 101-121. Plenum, New York (1995) · Zbl 0847.49008 |

[9] | Giannessi, F., Khan, A.A.: Regularization of non-coercive quasi-variational inequalities. Control Cybern. 29, 91-110 (2000) · Zbl 1006.49004 |

[10] | Giannessi, F., Khan, A.A.: On the envelope of a variational inequality. Nonlinear Anal. Var. Probl. 35, 285-293 (2010) · Zbl 1184.49010 |

[11] | Goeleven, D., Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities: Theory, Methods and Applications, vol. I. Unilateral Analysis and Unilateral Mechanics, Nonconvex Optimization and its Applications, 69, Kluwer Academic Publishers, Boston (2003) · Zbl 1259.49002 |

[12] | Goeleven, D., Motreanu, D.: Variational and Hemivariational Inequalities: Theory, Methods and Applications, vol. II. Unilateral Problems. Nonconvex Optimization and its Applications, 70, Kluwer Academic Publishers, Boston (2003) · Zbl 1259.49001 |

[13] | Jadamba, B., Khan, A.A., Raciti, F., Rouhani, D.B.: Generalized solutions of multi-valued monotone quasi-variational inequalities. Optim. optim. control 39, 227-240 (2010) · Zbl 1229.90233 |

[14] | Jadamba, B., Khan, A.A., Sama, M.: Generalized solutions of quasi-variational inequalities. Optim. Lett. 6(7), 1221-1231 (2012) · Zbl 1259.90143 |

[15] | Kano, R., Kenmochi, N., Murase, Y.: Existence theorems for elliptic quasi-variational inequalities in Banach spaces. In Recent Advances in Nonlinear Analysis, pp. 149-169. World Scientific Publishing, New Jersey (2008) · Zbl 1351.49010 |

[16] | Kenmochi, N.: Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations. Hiroshima Math. J. 4, 229-263 (1974) · Zbl 0284.47030 |

[17] | Khan, A.A., Sama, M.: Optimal control of multivalued quasi-variational inequalities. Nonlinear Anal. 75(3), 1419-1428 (2012) · Zbl 1236.49019 |

[18] | Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization. An Introduction with Applications. Springer, Berlin (2015) · Zbl 1308.49004 |

[19] | Khan, A.A., Tammer, C., Zalinescu, C.: Regularization of quasi-variational inequalities. Optimization 64, 1703-1724 (2015) · Zbl 1337.49005 |

[20] | Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (1987) · Zbl 0457.35001 |

[21] | Kluge, R.: On some parameter determination problems and quasi-variational inequalities, Theory of nonlinear operators. In: Proc. Fifth Internat. Summer School, Central Inst. Math. Mech. Acad. Sci. GDR, Berlin, 1977, pp. 129-139. Akademie-Verlag, Berlin (1978) · Zbl 1047.49009 |

[22] | Kravchuk, A.S., Neittaanmäki, P.J.: Variational and Quasi-Variational Inequalities in Mechanics. Springer, Dordrecht (2007) · Zbl 1131.49001 |

[23] | Lenzen, F., Lellmann, J., Becker, F., Schnorr, C.: Solving quasi-variational inequalities for image restoration with adaptive constraint sets. SIAM J. Imaging Sci. 7(4), 2139-2174 (2014) · Zbl 1308.49008 |

[24] | Liu, Z.: Generalized quasi-variational hemi-variational inequalities. Appl. Math. Lett. 17, 741-745 (2004) · Zbl 1058.49006 |

[25] | Liu, Z.: Existence results for evolution noncoercive hemivariational inequalities. J. Optim. Theory Appl. 120, 417-427 (2004) · Zbl 1047.49009 |

[26] | Lunsford, M.L.: Generalized variational and quasi-variational inequalities with discontinuous operators. J. Math. Anal. Appl. 214, 245-263 (1997) · Zbl 0945.49003 |

[27] | Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 512-585 (1969) · Zbl 0192.49101 |

[28] | Mosco, U.: Implicit variational problems and quasi-variational inequalities. In: Nonlinear Operators and the Calculus of Variations, Lecture Notes in Mathematics, vol. 543, pp. 83-156. Springer, Berlin (1976) · Zbl 1258.49004 |

[29] | Motreanu, V.V.: Existence results for constrained quasivariational inequalities. In Abstract and Applied Analysis, Art. ID 427908 (2013) · Zbl 1292.49012 |

[30] | Zeidler, E.: Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators. Springer, New York (1990) · Zbl 0684.47029 |

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