On generalized equilibrium points (Q1877810)

Let \(X=X_{1}\times X_{2}\times\cdots\times X_{n}\) be a product of Euclidean spaces and \(\varphi:X\times X\rightarrow\mathbb{R}\) be a function. This paper introduces the notion of \(\varepsilon\)-equilibrium point of \(f\), where \(\varepsilon=(\varepsilon_{1},\varepsilon_{2},\ldots\varepsilon_{n} )\in\mathbb{R}^{n}\), as follows: \(x=(x_{1},x_{2},\ldots x_{n})\in X\) is an \(\varepsilon\)-equilibrium point if for all \(y=(y_{1},y_{2},\ldots y_{n})\in X\), \(\varphi(x,y)\geq\sum_{i=1}^{n}\varepsilon_{i}\left\| x_{i} -y_{i}\right\| ^{2}\). The main result of the paper is an existence theorem for \(\varepsilon\)-equilibrium points. As an application, the particular cases of \(\varepsilon\)-saddle points and \(\varepsilon\)-Nash equilibria are examined and corresponding existence theorems are established.