Eckland and Bishop-Phelps variational principles in partially ordered spaces

In this paper, an assertion about the minimum of the graph of a mapping acting in partially ordered spaces is obtained. The proof of this statement uses the theorem on the minimum of a mapping in a partially ordered space from [A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy. Caristi-like condition and the existence of minima of mappings in partially ordered spaces // Journal of Optimization Theory and Applications. 2018. V. 180. Iss. 1, 48–61]. It is also shown that this statement is an analogue of the Eckland and Bishop–Phelps variational principles which are effective tools for studying extremal problems for functionals defined on metric spaces. Namely, the statement obtained in this paper and applied to a partially ordered space created from a metric space by introducing analogs of the Bishop–Phelps order relation, is equivalent to the classical Eckland and Bishop–Phelps variational principles.