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

Abstract

In this paper, we generalize the concept of α-well-posedness to a system of time-dependent hemivariational inequalities without Volterra integral terms in Banach spaces. We establish some metric characterizations of α-well-posedness and prove some equivalence results of strong α-well-posedness (resp., in the generalized sense) between a system of time-dependent hemivariational inequalities and its derived system of inclusion problems.

Optimal control of feedback control systems governed by systems of evolution hemivariational inequalities

“…is a reflexive Banach space ( [6]). For a C 0 -semigroup {T i (t)} t≥0 , there exist constants ω i and ρ i > 0 such that T i (t) ≤ ρ i e ω i t for 0 ≤ t < ∞ and we set sup t∈I T i (t) ≤ sup t∈I ρ i e ω i t ≤ M i with M i > 0 ( [26]).…”

“…Let φ : (0, 1) × (0, π) × R → R be a function satisfying the following assumptions: Assume that for i = 1, 2, U i is a reflexive Banach space, u i : (0, 1) → U i a control function and B i : U i → R a bounded linear operator. Thus, combining (8)-(9), system (7) turns to be system (6).…”

“…In this section we study the existence of feasible pairs for system (6). At the first, we study the existence of solutions of system (6). We will make the following conditions.…”

Well-posedness for systems of generalized mixed quasivariational inclusion problems and optimization problems with constraints

“…The concept of well-posedness has been generalized to some other problems: variational inequality problems [5,6,9,10,13,16,19,29], saddle point problems [4], Nash equilibrium problems [20,21], inclusion problems [9,15], and fixed point problems [9,15,22].…”

“…For more details, we refer the reader to [4,11,12,24,27,28,30,31] and the references therein. In 2009, Long and Huang [18] generalized the concept of α-well-posedness to symmetric quasiequilibrium problems in Banach spaces, which include equilibrium problems, Nash equilibrium problems, quasivariational inequalities, variational inequalities, and fixed-point problems as special cases.…”