The Existence of Solutions to an Inhomogeneous Higher Order Differential Equation in the Schwartz Space

Valerii Samoilenko

Taras Shevchenko National University of Kyiv, 60 Volodymyrska St., Kyiv, 01601,Ukraine
E-mail: valsamyul@gmail.com

Yuliia Samoilenko

Taras Shevchenko National University of Kyiv, 60 Volodymyrska St., Kyiv, 01601,Ukraine
E-mail: yusam@univ.kiev.ua

Received September 15, 2019, revised January 14, 2020.

Abstract

he paper deals with the problem of the existence of solutions to an inhomogeneous linear differential equation of higher even order. The problemarises while studying soliton and soliton-like solutions to partial differential equations of integrable type. The theorem on necessary and sufficient conditions of the existence of solutions to the differential equation in the Schwartz space of rapidly decreasing functions is proved by means of theoryof pseudodifferential operators.

Mathematics Subject Classification 2010: 35A01, 35J30, 47G30 Key words: existence of solutions, higher order differential equations, the Schwartz space of rapidly decreasing functions, pseudodifferential operators