Denote the eigenvalues of the matrix H$_{j}$ by λ ( j ) 2 , . . . , λ ( j ) n . By means of the Lemma 4.1 and 4.2, there exits a number α > 0 , such that Re ( λ ( j ) k ) < − α, k = 2 , 3 , . . . , n, where Re ( z ) means the real part of the number, z. Taking into account that the matrices P$_{j}$ and $_{P}$− 1 j are regular and periodic, the following estimates can be calculated where K ( j ) is a positive real constant. Denote the first column of the fundamental matrix Y by χ $^{1}$. By the equation (4.34), χ 1 is equal to the first column of P$_{j}$, this means that it is a ω -periodic solution of (3.13). By assumptions of the theorem the variational system (4.35) satisfies the conditions of Lemma 4.2, and one can verify that the following estimate is true [17] where K ( j ) 1 is a positive constant. Let us setup the following integral equation where a = [0 , a$_{2}$, . . . , a$_{n}$ ] , a$_{i}$ ∈ $^{R}$, i = 2 , 3 , . . . , n, are orthogonal to Ψ ′ (0) , i.e. with the zero first coordinate. Let z ( j ) 0 ( t, a ) ≡ 0 , and consider the following successive approximations for k = 1 , 2 , . . . . By using the approximation (4.42) and estimation (4.40), one can verify that We will show that the bounded solution of (4.41) exists and satisfies (4.35). For arbitrary positive small number L , there exists a number δ = δ ( L ) such that for | z$_{1}$ | < δ, | z$_{2}$ | < δ 24 | G ( j ) 1 ( t, s ) | ≤ K ( j $^{)}$exp( − α ( t − s )) , (4.38) | G ( j ) 2 ( t, s ) | ≤ K ( j $^{)}$, (4.39) | Y$_{j}$ ( t ) | ≤ K ( j ) 1 exp( − αt ) for t ≥ 0 , (4.40) z ( j $^{)}$( t, a ) = Y$_{j}$ ( t ) a + t ∫ 0 G ( j ) 1 ( t, s ) r ( s, z ( s )) ds − ∞ ∫ t G ( j ) 2 ( t, s ) r ( s, z ( s )) ds + ∑ 0 <θ$_{k}$<t G ( j ) 1 ( t, θ$_{k}$ +) q$_{k}$ ( z ( θ$_{k}$ )) − ∑ t<θ$_{k}$$^{<}$∞$^{G}$$^{(}$$^{j}$$^{)}$ 2 ( t, θ$_{k}$ +) q$_{k}$ ( z ( θ$_{k}$ )) , (4.41) z ( j ) k ( t, a ) = Y$_{j}$ ( t ) a + ∞ ∫ 0 G ( j $^{)}$( t, s ) r ( $^{s, z$_{k}$}$− $_{1}$( s )) ds + ∞ ∑ k =1 G ( j $^{)}$( t, θ$_{k}$ +) q$_{k}$ ( $^{z$_{k}$}$− $_{1}$( θ$_{k}$ )) , (4.42) | z ( j ) 1 ( t, a ) | ≤ K ( j ) 1 | a | exp( − αt/ 2) . (4.43) | r ( t, z$_{1}$ ) − r ( t, z$_{2}$ ) | ≤ L | z$_{1}$ − z$_{2}$ | (4.44)