Vector variational inequality and its duality.

*(English)*Zbl 0809.49009Some 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.

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.

Reviewer: Z.Wyderka (Katowice)

##### MSC:

49J40 | Variational inequalities |

90C29 | Multi-objective and goal programming |

46A40 | Ordered topological linear spaces, vector lattices |

##### Keywords:

dual variational inequality; hemi-continuity; monotonicity; coercivity; vector variational inequalities; weak order relation; multicriteria optimization problems
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\textit{X. Q. Yang}, Nonlinear Anal., Theory Methods Appl. 21, No. 11, 869--877 (1993; Zbl 0809.49009)

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