Sharshanov A. A.
Ukr. Mat. Zh. - 1968. - 20, № 3. - pp. 427–430
Ukr. Mat. Zh. - 1962. - 14, № 1. - pp. 68-86
The following problem is considered. Given functions $u, v$ are depending on the arguments $x, y, z$ partially differentiate with respect to z and periodic with respect to this argument with a period equal to unity. In addition the functions $u, v$ are analytical in respect to the two other arguments, the Jacobian of these variables is positive with any $z$ for all values of $x, у$ of the region under consideration. Then, we may construct functions $u, v$ satisfying certain conditions depending on $x, y, z, t$, analytical in $x, у$ and differentiable a sufficient number of times with respect to $t$, the Jacobian of these functions in $x, у$ differing from zero for all real values of $z, t$ and the considered $x, y$.