# Existence and Uniqueness of Equilibrium points for Concave N-Person Games

I am reading a paper. I have a problems with understanding their lemma

Lemma: The nonzero elements of every vector $$u \in U(x)$$ are given by a vector $$\bar{u} \in E^k, \bar{k} \leqslant k$$, where $$\bar{u}=-\left(\bar{H}^{\prime} \bar{H}\right)^{-1} \bar{H}^{\prime} g(x, \bar{r}) \geqslant 0$$. The $$m \times \bar{k}$$ matrix $$\bar{H}=\bar{H}(x)$$ consists of $$\bar{k}$$ linearly independent columns of $$H(x)$$ selected from $$\nabla h_j(x)$$ for $$j \in J$$.

Let me recall some useful notations

We define an $$m \times k$$ matrix $$H(x)$$ whose $$jth$$ column is $$\nabla h_j(x)$$ $$H(x)=\left[\begin{array}{lll} \nabla h_1(x) & \nabla h_2(x) \ldots \nabla h_k(x) \end{array}\right]$$

$$g(x,r)$$ is a mapping of $$E^m$$ into itself $$g(x, r)=\left[\begin{array}{c} r_1 \nabla_1 \varphi_1(x) \\ r_2 \nabla_2 \varphi_2(x) \\ \vdots \\ r_n \nabla_n \varphi_n(x) \end{array}\right]$$

We define the mapping $$f(x,u,r)$$ of $$E^{m+k} \to E^m$$ for each fixed $$\bar{r}>0$$ as follows $$f(x, u, \bar{r})=g(x, \bar{r})+H(x) u$$

where $$u \in U(x) \subset E^k$$ such that $$U(x)=\left\{u \mid\|f(x, u, \bar{r})\|=\min _{\substack{v_j \geqslant 0, j \in J \\ v_j=0, j \neq J}}\|f(x, v, \bar{r})\|\right\}$$

$$J=J(x)=\left\{j \mid h_j(x) \leqslant 0\right\}$$

Thank you very much for your help. I appreciate it a lot