On continuous and Lipschitz selections of multivalued mappings given by systems of inequalities

We consider a multivalued mapping of the following form
$$ a(x)=\{ y \in Y \,|\,\, f_i(x,y) \leq 0, \ i\in I\}, \ \ x \in X, $$
where $X \subset \mathbb{R}^m$ is compact; $Y \subset \mathbb{R}^n$ is convex compact; the gradients $f'_{iy}(x,y),$ $i \in I,$ of the functions $f_i(x,y)$ along $y$ satisfy the Lipschitz condition on $Y$; $I$ is a finite set of indices. Using the linearization method, existence theorems for continuous and Lipschitz selectors passing through any point of the graph of the multivalued mapping $a$ are proved. Both local and global theorems are obtained. Examples are given that confirm the significance of the assumptions made, as well as examples illustrating the application of the obtained statements to optimization problems.