Geodesic
Initial problem for a nonlinear integro-differential equation with a higher-order hyperbolic operator and with reflection of the argument
Daghestan Electronic Mathematical Reports, Tome 13 (2020), pp. 31-56

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In this paper it is studied the questions of one value solvability of initial value problem for nonlinear integro-differential equation with hyperbolic operator of the higher order, with degenerate kernel and reflective argument for regular values of spectral parameter. It is expressed the partial differential operator on the left-hand side of equation of higher order by the superposition of first-order partial differential operators. This is allowed us to present the considering integro-differential equation as an integral equation, describing the change of the unknown function along the characteristic. Further is applied the method of degenerate kernel. In proof of the theorem on one-value solvability of initial value problem is applied the method of successive approximations. Also is proved the stability of this solution with respect to the initial functions.
Keywords: initial value problem, degenerate kernel, superposition of differential operators, hyperbolic operator of higher order, reflecting argument, one value solvability. 
@article{DEMR_2020_13_a2,
     author = {T. K. Yuldashev (Iuldashev) and Zh. A. Artykova},
     title = {Initial problem for a nonlinear integro-differential equation with a higher-order hyperbolic operator and with reflection of the argument},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {31--56},
     publisher = {mathdoc},
     volume = {13},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2020_13_a2/}
}
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T. K. Yuldashev (Iuldashev); Zh. A. Artykova. Initial problem for a nonlinear integro-differential equation with a higher-order hyperbolic operator and with reflection of the argument. Daghestan Electronic Mathematical Reports, Tome 13 (2020), pp. 31-56. http://geodesic.mathdoc.fr/item/DEMR_2020_13_a2/