Periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence

Abstract

We establish conditions of the existence of solutions periodic in t with period T for a weakly nonlinear system of partial differential equations with pulse influence.Systems of ordinary differential equations and partial differential equations with pulse influence are encountered in studying various control systems.The existence of periodic solutions of systems of ordinary differential equations with pulse influence was studied in many works (see, e.g., [1][2][3][4]).In the present paper, we investigate the existence of periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence of the form O2uAOu = +f (t,x,u(t,x), Ux(t,x)), t:/:t i, Ot ~x ~x

Задача З Косою Похідною Для Параболічних Рівнянь З Імпульсними Умовами І Виродженням

“…Питання iснування перiодичних розв'язкiв рiвняння iз частинними похiдними гiпер-болiчного типу з iмпульсною дiєю вивчалися у працях [7][8][9]. Побудовi теорiї коректностi задачi Кошi для параболiчних систем з iмпульсною дiєю у максимально широких про-сторах Дiнi присвячено другий роздiл монографiї [10].…”

Periodic solutions of a system of nonlinear integro-differential equations with pulse action

“…We rewrite Eq. (6) in the form x = Rx, where R is the integral operator in (6). We estimate the norm…”

“…The numerical-analytic method for the investigation of periodic solutions [4] was used for the construction of periodic solutions of a system of integro-differential equations in [5]. Periodic solutions of systems of integro-differential equations with pulse action were found in [6].…”

Numerical-Analytic Method for the Investigation of Periodic Solutions of Partial Integro-Differential Equations with Pulse Action

“…In [4][5][6], Nurzhanov, Turbaev, Perestyuk, Sarafova, and Khekimova studied conditions for the existence of periodic solutions of systems of integro-differential equations with pulse action.…”