Abstract
We introduce the concepts of ordered variational inequalities and ordered complementarity problems with both domain and range in Banach lattices. Then we apply the Fan-KKM theorem and KKM mappings to study the solvability of these problems.
Highlights
Let X be a real Banach space with its norm dual X
We introduce the concepts of ordered variational inequalities and ordered complementarity problems with both domain and range in Banach lattices
The variational inequality problem associated with C and f, denoted as VI(C, f), is to find an x∗ ∈ C such that
Summary
Let X be a real Banach space with its norm dual X. Let C be a nonempty convex subset of X and f : C → X a singlevalued mapping. Since most classical Banach spaces are Banach lattices equipped with some lattice orders on which the positive operators appear naturally, the domain of an ordinal variational inequality defined in (1) and the complementarity problem defined in (2) may be in a Banach lattice (in particular, a Hilbert lattice). In this case, to investigate the properties of the solution set of (1) related to the partial order may be an important topic in economics theory and other appliedmathematics fields.
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