J Optim Theory Appl DOI 10.1007/s10957-013-0270-3 Quasimonotone Quasivariational Inequalities: Existence Results and Applications D. Aussel · J. Cotrina Received: 7 July 2011 / Accepted: 5 January 2013 © Springer Science+Business Media New York 2013 Abstract A quasivariational inequality is a variational inequality in which the con- straint set depends on the variable. Based on fixed point techniques, we prove var- ious existence results under weak assumptions on the set-valued operator defining the quasivariational inequality, namely quasimonotonicity and lower or upper sign- continuity. Applications to quasi-optimization and traffic network are also consid- ered. Keywords Quasivariational inequality · Quasimonotone operator · Quasi-optimization · Traffic network 1 Introduction Quasivariational inequalities correspond to variational inequalities in which the con- straint set is also depending on the variable. This dependence allows one to model complex phenomena as generalized Nash equilibrium problems in economy or con- tact problems with deformation in mechanics. Contrary to the particular case of the variational inequalities, for which there exists an important literature dealing with existence of solutions (see, e.g., [1] or Facchinei– Pang [2] and references therein), there is only few contributions proving existence of solutions for quasivariational inequalities (see [3–6]). Communicated by Igor Konnov. D. Aussel ( ) Lab. PROMES, UPR CNRS 8521, Université de Perpignan, Perpignan, France e-mail: aussel@univ-perp.fr J. Cotrina IMCA, Instituto de Matemática y Ciencias Afines, Universidad Nacional de Ingeniería, Calle Los Biólogos 245 Urb. San Cesar, La Molina, Lima 12, Peru e-mail: jecotrina@imca.edu.pe