Vector Variational Inequalities and Vector Optimization by Qamrul Hasan Ansari Elisabeth Köbis & Jen-Chih Yao

Author:Qamrul Hasan Ansari, Elisabeth Köbis & Jen-Chih Yao
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof

The result follows from Proposition 5.7 and Lemma 5.2 (b). □

Analogously, by using Proposition 5.8 and Lemma 5.2 (a), we derive the following existence result for a solution of VVIP.

Theorem 5.13

Let X, Y, K and C be the same as in Proposition 5.8 . Suppose that T: K → ℒ(X, Y ) is v-hemicontinuous such that both L-condition and v-coercive condition C 2 hold on K and the set

is weakly closed. Then VVIP has a solution.

In Theorem 5.10, if we replace weakly v-coercive condition C 1 by weakly v-coercive condition C 2, then the existence theorem for a solution of WVVIP can be obtained as follows.

Theorem 5.14

Let X, Y, K, C, W, and be the same as in Proposition 5.7 . For each x ∈ K, suppose that T: K → ℒ(X, Y ) is weakly C x -pseudomonotone and v-hemicontinuous on K, and assume in addition that T satisfies weakly v-coercive condition C 2 on K. Then WVVIP has a solution.

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