# On solvability of boundary value problems with integral-type boundary condition for odd-order equations with changing time direction

### Abstract

Local boundary value problems for equations with changing time direction have been studied in many papers. The solvability of nonlocal in time boundary value problems was studied for second order parabolic equations. For a third-order non-classical equation, the regular solvability of boundary value problems with integral boundary conditions over time was considered. In this paper, in a cylindrical domain of $R^{n+1}$, we study a boundary value problem with an integral boundary condition in time for an equation of odd order with changing time direction. Under certain conditions on the coefficients of the equation and the data of the boundary value problem, the regular solvability of the nonlocal boundary value problem under consideration is proved. Proving the regular solvability of the given non-local boundary value problem, we introduce an auxiliary local boundary value problem for an equation of odd order with changing time direction. Also, for the auxiliary boundary value problem, a convergence estimate is obtained, which is used to establish an estimate of the convergence of approximate solutions to the exact solution of the nonlocal boundary value problem under consideration.

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*Mathematical notes of NEFU*, 26(1), pp. 6-13. doi: https://doi.org/10.25587/SVFU.2019.101.27242.

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