Abstract
The main purpose of this paper is threefold. One is to study the existence and convergence problem of solutions for a class of generalized mixed quasi-variational hemivariational inequalities. The second one is to study the existence of optimal control for such kind of generalized mixed quasi-variational hemivariational inequalities under given control u∈U. The third one is to study the relationship between the optimal control and the data for the underlying generalized mixed quasi-variational inequality problems and a class of minimization problem. As an application, we utilize our results to study the elastic frictional problem in a class of Hilbert spaces. The results presented in the paper extend and improve upon some recent results.
Highlights
Variational inequality theory is a very effective and powerful tool for studying a wide range of problems that arise in differential equations, mechanics, contact problems in elasticity, the optimization and control problem, as well as unilateral, obstacle and moving problems
Hemivariational inequalities, which were first initiated by Panagiotopoulos [9], deal with certain mechanical problems involving nonconvex and nonsmooth energy functions
If the energy function is convex, the hemivariational inequalities reduce to the variational inequalities that have been previously considered by many authors
Summary
Variational inequality theory is a very effective and powerful tool for studying a wide range of problems that arise in differential equations, mechanics, contact problems in elasticity, the optimization and control problem, as well as unilateral, obstacle and moving problems (see, for example, [1,2,3,4,5,6,7,8]). G◦(x; y) is the Clarkes generalized directional derivative of the locally Lipschitz mapping G : Lp( ; R ) → R at the point x ∈ Lp( ; R ) with respect to direction y ∈ Lp( ; R ). (iii) Pseudomonotone, if (a) For each x ∈ Ω, the set F (x) is nonempty, bounded, closed and convex; (b) The mapping F is u.s.c. from each finite-dimensional subspace of E to E endowed with the weak topology; (c) If {xn} ⊂ E with xn x ∈ E, and xn ∈ F (xn) such that lim sup xn, xn − x ≤ 0, n→∞. A mapping : Ω → 2Ω is called M-continuous, if the following conditions hold:. (M2) For yn ∈ (xn) with xn x and yn y, we have y ∈ (x)
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