Notations and Basic Ideas. Auxiliary Lemmas

The integral time average of a vector function periodic in t with period cf  

Assume that f t, x, y) is a continuous function with respect to its variables t, x, and y such that 

Further, we shall consider only those systems (1.1) for which the region D, vector M, matrices K, K2, i A andpeiod (O satisfy the following conditions  

As follows from Perron s theorem (Bellman, Kalaba, 1968), the largest eigenvalue of the matrix Q is real and nonnegative (since its elements are nonnegative). It can be estimated from above by 

Note also that, for solutions of (1.1) periodic in t with period (O, we can assume without loss of generality that 0 A w. In fact, A can otherwise be represented as A = m(0 + Aj, where m is an integer, m , and Aj satisfies the inequality 0 Ax to. Since the solution of (1.1) is periodic in t with p od to, the to-periodic solutions of the equation 

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