Levitin-Polyak Well-Posedness for Equilibrium Problems with Functional Constraints

Abstract

We generalize the notions of Levitin-Polyak well-posedness to an equilibrium problem with both abstract and functional constraints. We introduce several types of generalized Levitin-Polyak well-posedness. Some metric characterizations and sufficient conditions for these types of wellposedness are obtained. Some relations among these types of well-posedness are also established under some suitable conditions.

Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints

“…4, by virtue of a nonlinear scalarization function and a gap function for generalized vector quasi-equilibrium problems, we show equivalent relations between the Levitin-Polyak well-posedness of the optimization problem and the Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems. The results in this paper unify, generalize and extend some known results in [6][7][8][9][32][33][34].…”

“…type II, generalized type I, generalized type II) LP well-posedness of (GQVEP) defined in Definition 2.2 reduces to the type I (resp. type II, generalized type I, generalized type II) LP well-posedness of the scalar equilibrium problem with abstract set constraints and functional constraints introduced by Long et al [32]. (vi) Let Z = L(X, W ) be the space of all the linear continuous operators from X to W, S(x) = X 0 for all x ∈ X 1 and C(x) = C for all x ∈ X , and let z, x denote the function value z(x), where z ∈ L(X, W ), x ∈ X 1 .…”

“…A great deal of results of various kinds of vector equilibrium problems have been obtained, such as existence or stability of solutions (see, for example, and the references therein). Long et al [32] introduced and studied four types of Levitin-Polyak wellposedness of equilibrium problems with abstract set constraints and functional constraints. Li and Li [33] introduced and researched two types of Levitin-Polyak well-posedness of vector equilibrium problems with abstract set constraints.…”

Well-posedness for parametric strong vector quasi-equilibrium problems with applications

“…Subsequently, the well-posedness has been extensively studied by many authors for various equilibrium problems, such as scalar equilibrium problems [28], vector equilibrium problems [29][30][31], parametric vector equilibrium problems [32], vector quasi-equilibrium problems [33], generalized vector quasi-equilibrium problems [34], and symmetric quasi-equilibrium problems [35]. It is worth mentioning that the most of results of well-posedness are for scalar equilibrium problem or for weak vector equilibrium problem which depends on intC ≠ ∅.…”

The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems

“…It is worth noting that the recent study for various types of well-posedness have been generalized to variational inequality problems [14][15][16][17][18], generalized variational inequality problems [19,20], quasi-variational inequalities [21], generalized quasi-variational inequalities [22], generalized vector variational inequality problems [23], vector quasi-variational inequality problems [24], equilibrium problems [25], vector equilibrium problems [26], vector quasiequilibrium problems [27], generalized vector quasi-equilibrium problems [28] and many other problems.…”