A Nash type solution for hemivariational inequality systems

Abstract

In this paper we prove an existence result for a general class of hemivariational inequalities systems using the Ky Fan version of KKM theorem (1984) or the Tarafdar fixed point theorem (1987). As application we give an infinite dimensional version for existence result of Nash generalized derivative points introduced recently by Krist\'{a}ly (2010) and also we give an application to a general hemivariational inequalities system

A system of evolutionary problems driven by a system of hemivariational inequalities

“…One is closely related to KKM theorems and fixed point theorems, which are used by Panagiotopulos [24], Repovs and Varga [26], Costea and Radulescu [4], and Zhang and He [35] to study stationary hemivariational inequalities and systems of stationary hemivariational inequalities. The other is closely related to surjectivity theorems concerning pseudomonotone and coercive operators, which are captured by Xiao and Huang [29], and Liu [15] for various types of stationary hemivariational inequalities and evolutionary hemivariational inequalities.…”

Well-posedness for systems of time-dependent hemivariational inequalities in Banach spaces

“…One is closely related to Knaster-Kuratowski-Mazurkiewicz theorems and fixed point theorems, which are used by Panagiotopulos et al [29], Repovs and Varga [30], Costea and Radulescu [10], and Zhang and He [41] to study stationary hemivariational inequalities and systems of stationary hemivariational inequalities. The other is closely related to surjectivity theorems concerning pseudomonotone and coercive operators, which are captured by Migorski et al [24], Carl et al [1], Naniewicz and Panagiotopulos [26], Xiao and Huang [33], and Liu [22] for various types of stationary hemivariational inequalities and evolutionary hemivariational inequalities.…”

A System of Generalized Variational-Hemivariational Inequalities with Set-Valued Mappings

“…They proved the existence and uniqueness of the weak solution for the problem by using a surjectivity result for operators of pseudomonotone type. In 2011, Repovš and Varga [6] studied the Nash equilibrium point by using the Ky Fan version of the KKM theorem and the Tarafdar fixed point theorem for a class of hemivariational inequality system. It is obvious that some problems studied in literatures are special cases of our system of generalized variational-hemivariational inequalities under some special conditions, such as = 1, are single-valued, or are indicators of some convex subsets for = 1, 2, .…”

“…, . Although it seems that our problem (P) cannot include the problem studied in [6] as a special case, we remark here that, in essence, the problem (P) is a generalization of the problem in [6] since, when ( = 1, 2, . .…”

“…. , ) are single-valued and are the indicators of the convex subsets , the problem (P) reduces to the problem studied by Repovš and Varga [6] with = and ∘ ( ; V − ) being incorporated into ∘ ( ; V − ) under the regularity condition. For more information on the research of hemivariational inequalities and systems of hemivariational inequalities, we can refer to [7][8][9][10][11][12][13][14][15][16] and references therein.…”