Vol 72, No 5 (2017)

Lower and upper bounds are proved for the norms of functions being sums of double series in cosines.

Fixed point and coincidence theorems for mappings of ordered sets and their metric counterparts generalizing the well-known Caristi fixed point theorem are presented.

A topological space is said to be paranormal if every countable discrete collection of closed sets {D_n: n < ω} can be expanded to a locally finite collection of open sets {U_n: n < ω}, i.e., D_n ⊂ U_n and D_m ∩ U_n ≠ 0 if and only if D_m = D_n. It is proved that if F: Comp → Comp is a normal functor of degree ≥ 3 and the compact space F(X) is hereditarily paranormal, then the compact space X is metrizable.

An optimal control problem of minimization of the sum of a linear-quadratic integral functional and a terminal of arbitrary form is considered. It is proved that in this case a control can be also constructed in an explicit analytic form.

Immersions of graphs into the projective plane are studied. A classification of immersions up to a regular homotopy is obtained. A complete invariant of immersions up to a regular homotopy is constructed. The case of immersions of graphs into any compact surface differing from the projective plane was known previously.