Abstract
In this work, we give necessary and sufficient conditions for the existence of solutions to the variational inequality problem: find \(x_0 \in K\) such that \(\langle F(x_0),y-x_0 \rangle \ge 0\), for every \(y \in K\), where K is a nonempty closed convex subset of a real Hilbert space H and \(F:K \rightarrow H\) is a monotone and continuous operator. These characterizations are given in terms of approximate fixed points sequences, as well as by Leray–Schauder condition. We apply our obtained results in a constrained convex minimization problem.
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Acknowledgements
The research of the authors has been partially supported by SEP-CONACYT Grant A1-S-53349. The authors would like to thanks the referees for their valuable comments and recommendations.
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Hernández-Linares, C.A., Martínez-Anteo, E. & Muñiz-Pérez, O. Characterizations of the existence of solutions for variational inequality problems in Hilbert spaces. J. Fixed Point Theory Appl. 25, 66 (2023). https://doi.org/10.1007/s11784-023-01067-7
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DOI: https://doi.org/10.1007/s11784-023-01067-7
Keywords
- Variational inequality
- fixed point
- monotone mapping
- metric projection
- closed convex cones
- convex minimization problem
- Hilbert spaces