ON PERIODIC-TYPE SOLUTIONS OF SYSTEMS OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS

When Λ1 and Λ2 are unit matrices, this problem becomes the well-known problem on a periodic solution which has been the subject of numerous studies (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and the references therein). In this paper, sufficient conditions for the unique solvability of problem (1.1), (1.2) are established, which are nonimprovable in a certain sense and in particular provide new results on the existence of a unique ω-periodic solution of system (1.1). The following notation is used in the paper: (1) R is the set of real numbers; (2) Rn is the n-dimensional real Euclidean space; (3) x = (ξi)i=1 ∈Rn is the column vector with components ξ1, . . . ,ξn,


Formulation of the problem and statement of the main results
Let n 1 and n 2 be natural numbers, ω > 0, Λ i ∈ R ni×ni (i = 1,2) nonsingular matrices, and ᏼ ik : R → R ni×nk (i,k = 1,2) and q i : R → R ni (i = 1,2) matrix and vector functions whose components are Lebesgue integrable on each compact interval.We consider the problem on the existence and uniqueness of a solution of the linear differential system dx i dt = ᏼ i1 (t)x 1 + ᏼ i2 (t)x 2 + q i (t) (i = 1,2), (1.1) satisfying the conditions When Λ 1 and Λ 2 are unit matrices, this problem becomes the well-known problem on a periodic solution which has been the subject of numerous studies (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and the references therein).
In this paper, sufficient conditions for the unique solvability of problem (1.1), (1.2) are established, which are nonimprovable in a certain sense and in particular provide new results on the existence of a unique ω-periodic solution of system (1.1).
The following notation is used in the paper: (1) R is the set of real numbers; (2) R n is the n-dimensional real Euclidean space; (3) x = (ξ i ) n i=1 ∈ R n is the column vector with components ξ 1 ,...,ξ n , (1.4) (6) X * is the transposed matrix of the matrix X; (7) E n is the unit n × n matrix; (8) det(X) is the determinant of the matrix X; (9) r(X) is the spectral radius of the matrix Inequalities between the matrices and the vectors are understood componentwise.Throughout the paper, it will be assumed that For each i ∈ {1, 2}, consider the differential system and denote by X i its fundamental matrix satisfying the initial condition If, however, the matrix Λ i − X i (ω) is nonsingular, then it is assumed that For each i ∈ {1, 2}, we define a matrix function Λ i0 : [0,3ω] → R ni×ni in the following manner: , and ᏼ i2 (t) = −p i (t), where p i : R →]0, +∞[ (i = 1,2) are the integrable on [0, ω] ω-periodic functions.Then conditions (1.5), (1.11), and (1.12), where A = 1, are fulfilled.On the other hand, in the considered case, system (1.1) has the form and therefore problem (1.1), (1.2) has an infinite set of solutions This example shows that the condition r(A) < 1 in Theorem 1.1 is nonimprovable and it cannot be replaced by the condition r(A) ≤ 1.
Example 1.4.Consider the problem where ε is a positive constant, If the latter inequality is fulfilled, then, by Theorem 1.3, there exists ε 0 > 0 such that, for arbitrary for arbitrary ε, problem (1.20) has an infinite set of solutions where x 10 ∈ R n1 is the eigenvector of the matrix B 1 B −1 B 2 corresponding to the zero eigenvalue and x 20 = −εB −1 B 2 x 10 .
Example 1.4 shows that condition (1.16) is essential and cannot be omitted.

Auxiliary propositions
In this section, we consider the problem dx dt = ᏼ(t)x + q(t), (2.1) assuming that Λ ∈ R n×n is a nonsingular matrix, and ᏼ : R → R n×n and q : R → R n are matrix and vector functions with components Lebesgue integrable on [0, ω] and satisfying the conditions (2.4) We denote by X the fundamental matrix of the homogeneous differential system satisfying the initial condition 3) immediately implies the following lemma.
Lemma 2.1.The matrix function X satisfies the identity Proof.Let x be an arbitrary solution of system (2.5).Then where c ∈ R n .Hence, by Lemma 2.1, it follows that x is a solution of problem (2.5), (2.2) if and only if (2.10) However, for the latter identity to be fulfilled, it is necessary and sufficient that c be a solution of the system of algebraic equations where (2.14) If, along with these identities, we also take into consideration condition (2.4), then, from (2.12), we obtain (2.15) Thus x is a solution of problem (2.1), (2.2).

Proofs of the main results
Proof of Theorem 1.1.By Lemma 2.3, it is sufficient to show that the homogeneous problem ) has only a trivial solution.
Let (x 1 ,x 2 ) be an arbitrary solution of this problem.By virtue of Lemma 2.3, condition (1.11) and the equalities guarantee the validity of the representations Let (3.6) Then by (1.9), (1.10) for i = 1, we have If, along with this, we also take into consideration inequality (1.12), then, from representation (3.5), we obtain Hence ρ ≤ Aρ and, therefore, According to the condition r(A) < 1 and the nonnegativeness of the matrix A, the matrix E n1 − A is nonsingular and (E n1 − A) −1 is nonnegative.Hence the multiplication of the 402 On periodic-type solutions latter vector inequality by (E n1 − A) −1 gives ρ ≤ 0. Therefore, ρ = 0, that is, By virtue of this equality, from (3.4), it follows that x i (t) = 0 for t ∈ R (i = 1,2).
Proof of Theorem 1.3.Let (x 1 ,x 2 ) be an arbitrary solution of problem (3.1), (3.2).Then by the Cauchy formula, we have where c ∈ R n1 .On the other hand, by Lemma 2.3, the nonsingularity of the matrix Λ 2 − X 2 (ω) and the equality guarantee the validity of the representation Hence, by virtue of equalities (1.17) and (3.11), it follows that where By Lemma 2.1 and the equality X 1 (ω) = Λ 1 , we have Therefore, from (3.11), we find Hence, by (3.2), it follows that If, along with this identity, we also take into account identities (1.5) and (3.16), then we obtain Therefore, from (1.17) and (3.21), we have (3.24) By virtue of this fact and condition (1.16), from (3.11), (3.14), and (3.20), we get Hence it is clear that ρ ≤ Aρ and, therefore, By virtue of the condition r(A) < 1 and the nonnegativeness of the matrix A, the latter inequality implies ρ = 0. Therefore,  On the other hand, by virtue of (1.24) and (3.2), we have t,s)ᏼ 21 (s)X 1 (s)ds.