Karisti inequality and $\alpha$-contractive mappings

The article considers a new Caristi-like inequality and proves some development of the Caristi theorem on fixed points of mappings of complete metric spaces (both in the single-valued and multi-valued case). Based on the obtained theorem, we study mappings of complete metric spaces that are contractive with respect to a certain $\alpha$ function of 2 vector arguments $\alpha$-contractive mappings). This function may not be a metric or even a continuous function. Proved theorems are generalizations of the Banach principle of contraction maps of and the Nadler theorem.