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A generalized mixed vector variational-like inequality problem. (English) Zbl 1172.49011

Summary: We introduce relaxed \(\eta\)-\(\alpha\)-\(P\)-monotone mapping, and by utilizing KKM technique and Nadler’s Lemma we establish some existence results for the generalized mixed vector variational-like inequality problem. Further, we give the concepts of \(\eta \)-complete semicontinuity and \(\eta \)-strong semicontinuity and prove the solvability for generalized mixed vector variational-like inequality problem without monotonicity assumption by applying Brouwer’s fixed point theorem. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.

MSC:

49J40 Variational inequalities
47H10 Fixed-point theorems
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J45 Methods involving semicontinuity and convergence; relaxation
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