Existence of solutions for lower semicontinuous quasi-equilibrium problems

Gap functions for quasi-equilibria

“…If C is given by (16) and 0 ∈ int K, then x ∈ X if and only if s(x) ≥ γ K (x), where γ K (x) := inf{λ > 0 : x ∈ λK} denotes the gauge of K.…”

“…As a consequence, y − y (16), then s(x)y * /s(x * ) ∈ C(x) and s(x * )y/s(x) ∈ C(x * ) so that summing the optimality conditions for y and y * with these choices of z after being multiplied respectively by s(x * ) and s(x) gives ∇p(y) − ∇p(y * ), s(x)y * − s(x * )y ≥ 0.…”

“…Clearly, the error bound requires the existence of a solution of (QEP) (see, for instance, [11,16] for existence results). Also, beyond suitable monotonicity and continuity conditions on the data, the result relies on the possibility to control the optimal solution…”

“…Unlikely QVI and GNEP, the QEP format did not receive much attention: existence results have been given in [11,16] and an extragradient type algorithm has been developed in [58]. To the best of our knowledge, no other algorithm has been devised up to now.…”

Random Generalized Bi-linear Mixed Variational-like Inequality for Random Fuzzy Mappings

“…In recent years, classical variational inequality theories have been generalized and applied in various directions, the readers refer to [1]- [3] and the references therein. A useful and important generalization of variational inequalities is the mixed variational-like inequalities, which have potential and significant applications in optimization theory [4,5], structural analysis [6], and economics [7,8]. It is noted that there are many effective numerical methods for finding approximate solutions of various variational inequalities.…”