1. Statement of Problem and Formulation of Main

Unimprovable effective efficient conditions are estab- lished for the unique solvability of the periodic problem u '(t) = i+1 X j=2 li,j(uj)(t) + qi(t) for 1 ≤ i ≤ n − 1, u ' (t) = n X j=1 ln,j(uj)(t) + qn(t),


Statement of Problem and Formulation of Main Results
Consider on [0, ω] the system ℓ n,j (u j )(t) + q n (t), (1.1) with the periodic boundary conditions where n ≥ 2, ω > 0, ℓ i,j : C([0, ω]) → L([0, ω]) are linear bounded operators and q i ∈ L([0, ω]).By a solution of the problem (1.1), (1.2) we understand a vector function u = (u i ) n i=1 with u i ∈ C([0, ω]) (i = 1, n) which satisfies system (1.1) almost everywhere on [0, ω] and satisfies conditions (1.2).EJQTDE, 2009 No. 59, p. 1 Much work had been carried out on the investigation of the existence and uniqueness of the solution for a periodic boundary value problem for systems of ordinary differential equations and many interesting results have been obtained (see, for instance, [1][2][3][7][8][9]11,12,17] and the references therein).However, an analogous problem for functional differential equations, remains investigated in less detail even for linear equations.In the present paper, we study problem (1.1) (1.2) under the assumption that ℓ n,1 , ℓ i,i+1 (i = 1, n − 1) are monotone linear operators.We establish new unimprovable integral conditions sufficient for unique solvability of the problem (1.1),(1.2) which generalize the wellknown results of A. Lasota and Z. Opial (see Remark 1.1) obtained for ordinary differential equations in [13], and on the other hand, extend results obtained for linear functional differential equations in [5,[14][15][16].These results are new not only for the systems of functional differential equations (for reference see [2,4,6,10] ), but also for the system of ordinary differential equations of the form where q i , p i,j ∈ L([0, ω]) (see, for instance, [2,[7][8][9] and the references therein).The method used for the investigation of the problem considered is based on that developed in our previous papers [14][15][16] for functional differential equations.
The following notation is used throughout the paper: N(R) is the set of all the natural (real) numbers; R n is the space of n-dimensional column vectors x = (x i ) n i=1 with elements Definition 1.1.We will say that a linear operator ℓ : i,1 = 0 for 2 i,j = 0 for i + 2 ≤ j ≤ n, a i,j = ||ℓ i,j || for 3 ≤ j + 1 ≤ i ≤ n. and the matrices A k = (a given by the recurrence relations (1.10) Then problem (1.1), (1.2) has a unique solution.
For the cyclic feedback system , q 1 ≡ 0 and q 2 ≡ q.
From the propositions i. and ii. by the the method of mathematical induction we obtain that the inequalities (2.9 1 ), (2.9 k ) and (2.10 k ) (k = k 0 , n) hold.
On the other hand from (1.4)-(1.6)and Lemma 2.1 it is clear that a k,k+1 = a Thus from the conditions (1.17) and (1.18) it follows that (1.9) and (1.10) hold.Consequently all the conditions of Theorem 1.1 are fulfilled for the system (1.16).
ω is satisfied.We will say that an operator ℓ is monotone if it is either nonnegative or nonpositive.