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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.16141 |
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Table of Contents:
- In this paper, we study the existence of solutions in non-monotone variational inequalities (VIs) through the normal mapping properties. In particular, we show that when the normal mapping $F_K^{\rm nor}(\cdot)$ is norm coercive over a set $K$, and the generalized Jacobian of the normal mapping has a full rank at points $x$ where $F_K^{\rm nor}(x)\ne0$, then the VI$(K,F)$ has a solution. We then investigate conditions on the mapping $F(\cdot)$ and its Jacobian that imply the full rank condition for the generalized Jacobian, such as the uniform P-function and the uniform P-matrix condition. Subsequently, we focus on VIs arising from games and interpret our main result in a game setting. Based on the P$_Υ$-matrix condition, we provide a sufficient condition for a game to have a Nash equilibrium. Additionally, through examples we show that our sufficient conditions can be used to assert the existence of a solution to a VI, or a quasi-Nash in a game, while the existing results relying on the uniform P-function property or the P$_Υ$-matrix condition cannot be employed.