Geodesic
A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2016), pp. 41-52

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We study the boundary-value problem for hyperbolic equation with nonlocal with respect to time-variable condition in integral form. We obtain sufficient conditions for the unique solvability of the nonlocal problem. The proof is based on possibility to reduce a nonlocal condition of the first kind to the second kind one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the posed problem.
Keywords: hyperbolic equation, nonlocal problem, integral conditions, generalized solution. 
@article{IVM_2016_10_a5,
     author = {L. S. Pul'kina and A. E. Savenkova},
     title = {A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {41--52},
     publisher = {mathdoc},
     number = {10},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_10_a5/}
}
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L. S. Pul'kina; A. E. Savenkova. A~problem with a~nonlocal with respect to time condition for multidimensional hyperbolic equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2016), pp. 41-52. http://geodesic.mathdoc.fr/item/IVM_2016_10_a5/