The existence of solutions for the system of vector quasi-equilibrium problems in topological order spaces

On system of vector quasi-equilibrium problems for multivalued mappings

“…In this case the SVQEP reduces to the vector quasi-equilibrium problem studied (for instance) in [5,9,20,22]. Moreover, the (SVQEP) will collapse to the problems given in [1,4,14,23] when the algebraic interior of C i is replaced by the interior of C i . It is easy to present some examples in order to show that the converse of Proposition 1.1 may drop.…”

“…Hence the C-quasiconcave is a special case of Definition 1.3 by taking F (x) = {f (x)}. Moreover, it is not difficult to verify that Definition 2.2 of [23] implies Definition 1.3 when the multivalued mapping F reduces to a single valued mapping. Definition 1.4.…”

“…The following proposition is the multivalued version of Lemma 2.1 of [23] by relaxing the locally convexity of the space and replacing the topological interior of the cone C by the algebraic interior of C. Proposition 1.1. Let X be a sup-semilattice, Y a vector space and C a subset of Y .…”

“…The system of vector quasi-equilibrium problems was introduced by Ansari et al [1] with applications in Debreu-type equilibrium problem for vector-valued functions. The system of vector quasi-equilibrium problems (SVQEP, in short) is a unified way to research some nonlinear problems such as vector equilibrium problems (VEP), vector variational inequality [23], and vector complementarity problems [24] and so on. In all the above problems, the authors obtained some existence results in the setting of topological linear structure.…”

“…For a nonempty set X, 2 X denotes the class of all nonempty subsets of X. A partially ordered set (X, ≤) is called a sup-semilattice if any two elements x and y of X have a least upper bound, denoted by x ∨ y = sup{x, y} (see, for instance, [18,23]). If x and x are elements in a partially ordered set (X, ≤) with x ≤ x , then the set…”