Some problems concerning the vector variational inequalities in Banach spaces are studied. If \(X\) is a Banach space, \((Y,P)\) is an ordered Banach space such that the cone \(P\) is a convex, closed set with non- empty interior, \(C\subset X\) is a closed, convex set, \(T: X\to L(X,Y)\) and \(f: X\to Y\), then the variational inequality under consideration is: Find \(x_ 0\in C\) such that \[ \langle T(x_ 0), x- x_ 0\rangle\not< f(x_ 0)- f(x)\quad\text{for all } x\in C.\tag{1} \] Here \(\langle\cdot,\cdot\rangle\) denotes the duality mapping and the weak order relation “\(\not<\)” is defined by \[ y\not< x\Leftrightarrow x- y\not\in\text{int }P.\tag{2} \] Next, some properties of \(T\), such as: \(v\)- hemi-continuity, \(v\)-monotonicity, \(v\)-coercivity, and condition (L) are defined.
If in (1) the right-hand-side is replaced by \(0\) and \(T\) is continuous, \(v\)-coercive, then the solution of (1) exists. Some special cases are also discussed.
At the end, the dual inequality to (1) is introduced and some interconnections between the solutions of the inequality (1) and its dual are proved. Some applications to multicriteria optimization problems are presented, too.