Existence results for a class of quasi-variational inequalities and applications to a noncooperative model

Abstract

The aim of the paper is to investigate the existence of solutions for the class of strongly pseudomonotone quasi-variational inequalities. The weak Mosco continuity of multifunctions has a key role to reach this aim. As an application, the existence of equilibrium distributions for a dynamic oligopolistic market equilibrium problem with adaptive set of feasible solutions is obtained.

Keywords:

• Quasi-variational inequalities
• weak mosco continuity
• existence results
• oligopolistic market equilibrium problem

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A continuously differentiable function $v_i\left(x\right)$ is called pseudoconcave with respect to $x_i,$ $i=1,\dots ,m$, iff $\sum _{j=1}^n\frac{\mathrm{\partial }{v}_i\left(x\right)}{\mathrm{\partial }{x}_{\mathrm{ij}}}(x_{\mathrm{ij}}-y_{\mathrm{ij}})\ge 0\Rightarrow v_i(x_1,\dots ,x_i,\dots ,x_m)\ge v_i(x_1,\dots ,y_i,\dots ,x_m).$

2 In the Hilbert space $L^2\left(\right[0,T],{\mathbb{R}}^{\mathrm{mn}}),$ we define the canonical bilinear form on $L^2\left(\right[0,T],{\mathbb{R}}^{\mathrm{mn}}{)}^{\ast }\times L^2([0,T],{\mathbb{R}}^{\mathrm{mn}}),$ by $\ll \varphi ,\mathrm{w}\gg ={\int }_0^T〈\varphi \left(t\right),\mathrm{w}\left(t\right)〉\mathrm{d}t,$where $\varphi \in \left(L^2\right([0,T],{\mathbb{R}}^{\mathrm{mn}}){)}^{\ast }=L^2([0,T],{\mathbb{R}}^{\mathrm{mn}}),$ $\mathrm{w}\in {\mathrm{L}}^2\left(\right[0,\mathrm{T}],{\mathbb{R}}^{\mathrm{mn}})$ and $〈\varphi \left(t\right),\mathrm{w}\left(t\right)〉=\sum _{l=1}^k{\varphi }_l\left(t\right){\mathrm{w}}_l\left(t\right).$

Additional information

Funding

The research of the author has been partially supported by the Research Project PRIN 2022 Nonlinear differential problems with applications to real phenomena (Grant number 2022ZXZTN2) funded by the Italian Ministry of University and Research (MUR). The author is member of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of INdAM.