Calmness of Set-Valued Mappings Between Asplund Spaces and Application to Equilibrium Problems

Necessary optimality conditions for countably infinite Lipschitz problems with equality constraint mappings

“…2. f is w * -strictly Lipschitzian atx if there is a neighborhood V of the origin in X such that for any v ∈ X and any sequences x k −→x, t k ↓ 0, and y * k w * −→ 0 one has y * k , y k −→ 0 as k −→ ∞, where y k are defined in (12). Obviously every strictly Lipschitzian mapping atx is w * -strictly Lipschitzian at this point and the opposite holds if B Y * is weak * sequentially compact.…”

“…Let us explain the motivation of the above form. As mentioned before, the original pattern of our approach is based on two theorems: [8,Theorem 5.48] and [12,Theorem 5]. With a little attention, it can be seen that the main focus in both of the above results is on the multifunction M in (3).…”

“…The author in [12], established a new sufficient condition for the calmness of multifunction M in (3), which was derived in terms of the coderivative and w * boundary of the normal cone to constraint set. In order to achieve this goal, a new concept so-called "sequential normal smoothness" for the sets in Asplund spaces was introduced in [12]. This property, provided an efficient tool for passing to the w * boundary of the Mordukhovich normal cone.…”

“…In [12], it was shown that the closed convex subsets of Asplund spaces are SNS at each of their points and this class is stable under Cartesian products. The following theorem from [12], plays a key role in the proof of our main result.…”