On the existence of infinite-dimensional invariant tori of nonlinear countable systems of difference-differential equations

Abstract

UDC 517.9In the space of bounded number sequences, we establish sufficient conditions for the existence of invariant tori for nonlinear countable systems of difference-differential equations defined on infinite-dimensional tori and containing an infinite set of constant deviations of a scalar argument.In the present paper, in the space of bounded number sequences we establish sufficient conditions for the existence of Lipschitz invariant tori for nonlinear countable systems of difference-differential equations of the general form defined on infinite-dimensional tori and containing an infinite set of constant deviations of a scalar argument of different signs. The theorems proved in this paper agree well with the results of [1, 2] concerning the existence of invariant tori of countable systems of differential and difference equations defined on tori. This paper continues the investigations carried out in [3][4][5], where analogous problems were solved for linear and quasilinear systems of the indicated type. Statement of the ProblemConsider the system of equations(1) where D . 1 ; 2 ; 3 ; : : :/ and x D .x 1 ; x 2 ; x 3 ; : : :/ belong to the Banach space M of bounded number sequences with the standard norm kxk D sup i fjx i jg; the mapping a. ; x/ D fa 1 . ; x/; a 2 . ; x/; a 3 . ; x/; : : :g is defined by functions a i . ; x/W M D ! R 1 ; D D fx 2 M j kxk Ä d D const > 0g; 2 -periodic with respect to the coordinates j ; j D 1; 2; 3; : : : ; i 2 N; N is the set of natural numbers, .t/ D t . / D