Periodic Solutions of Operator-Differential Equations

Also let f t,x,y) satisfy the Lipschitz condition with respect to x and y, i.e., 

Assume that each coordinate Aj (f = 1. s) of the operator A = (Aj, A2. Aj. Af) is defined on the class of functions continuous on (aj, bj) and maps this class onto the class of functions continuous on Cj, dp. Further, we assume that the operator A transforms every continuous co-periodic function x (r) = (Xi(t). x (r)) into a continuous oa-periodic function y(t) = (yi(t). y/t)) with a finite or infinite number of coordinates. In addition, let Ax(r) Dj for every continuous (o-periodic function x(f) with values in the region and let 

Suppose also that the vector M and matrices K, Q and R satisfy the following conditions  

we prove that the recursion relation (13.8) defines the functions Xnfi for all m 1 and that each of these functions takes values in the region D. hi fact, the operator A is defined for Xq e ) - Af (o/2, and hence, the function /(f, Xq, Axq) is also defined moreover, l/(r, xq, Axq) I M. According to Lemma 1.1, the difference Xi(r,Xo)-Xo can be estimated as follows 

This implies that lxi(r, Xq) -Xq I M co/2, and therefore, the function Xj(r, xq) takes values in the region D and is periodic in t with period (O. One can easily prove by indue- 

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