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

Abstract

In this paper, several metric characterizations of well-posedness for systems of generalized mixed quasivariational inclusion problems and for optimization problems with systems of generalized mixed quasivariational inclusion problems as constraints are given. The equivalence between the well-posedness of systems of generalized mixed quasivariational inclusion problems and the existence of solutions of systems of generalized mixed quasivariational inclusion problems is given under suitable conditions.

Existence of solutions for a class of variational-hemivariational-like inequalities in Banach spaces

“…Since J(ϕ) = Ω j(x, ϕ(x))dx and j satisfies the conditions (4) and (5) or (4) and (6)-(7), by Lemma 2.2, we have…”

“…As pointed out in Liu and Motreanu [19], problem (P) at resonance as well as nonresonance has a striking theoretic interest and a strong motivation due to its applications in mechanics and engineering. Recently, problem (P) has been studied by many authors, see e.g., [2,5,15,21,23,26] and the references therein. Now let us recall some important techniques for mixed variational(-like) inequality problems.…”

On the Hadamard Well-Posedness of Generalized Mixed Variational Inequalities in Banach Spaces

“…ey also established some characterizations of Hadamard well-posedness for a general mixed variational inequality. Very recently, some scholars still focused on the study of the well-posedness of various classes of variational inequalities, see e.g., generalized variational-hemivariational inequalities with perturbations in [13], completely generalized mixed variational inequalities in [14], noncompact generalized mixed variational inequalities in [15], generalized variational inequality with generalized mixed variational inequality constraint in [16], systems of generalized mixed quasivariational inclusion problems in [17], systems of time-dependent hemivariational inequalities in [18], and generalized hemivariational inequalities in [19]. Motivated and inspired by the research work going on this field, we introduce a new concept of Hadamard wellposedness for a generalized mixed variational inequality in a Banach space.…”