REPRESENTATION AND STABILITY OF SOLUTIONS OF SYSTEMS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

This paper is devoted to the study of systems of nonlinear functional differential equations with time-dependent coefficients and multiple variable increasing delays represented by functions gi(t) < t. The solution is found in terms of a piecewise-defined matrix function. Using our representation of the solution and Gronwall’s, Bihari’s and Pinto’s integral inequalities, asymptotic stability results are proved for some classes of nonlinear functional differential equations with multiple variable delays and linear parts given by pairwise permutable constant matrices. The derived theory is illustrated on nontrivial examples.


Introduction
The classical method of steps [8] where the initial value problem ẋ(t) = Bx(t − τ ), t ≥ 0 (1.1) is solved by subsequent integrating of equation (1.1) on intervals [0, τ ), [τ, 2τ ), [2τ, 3τ ),. . .was renown in 2003 by Khusainov and Shuklin [13].Applying this method, they constructed so-called delayed matrix exponential e Bt τ defined as where Θ, E are the zero and the identity N × N matrix, respectively.Let us recall their result.Using the variation of constants formula for retarded differential equations with constant delay [8] and e Bt τ , they stated the solution of nonhomogeneous equation ẋ(t) = Ax(t) + Bx(t − τ ) + f (t) with continuous function f : [0, ∞) → R N , satisfying initial condition (1.2).Later, their result was used to establish sufficient conditions for the exponential stability of the trivial solution of the nonlinear equation with various functions f in [16].The results from [13] were generalized to delay differential equations with multiple fixed delays and pairwise permutable matrices in [14] and analogical theory was developed for retarded oscillating systems and difference equations with one or more fixed delays (cf.[7,11,12,15]).Recently, the matrix representation of solutions of systems of differential equations with a single fixed delay was applied to boundary-value problems in [4,5,6].
In this paper, we consider the functional differential equation (FDE) with one or multiple time-dependent delays.More precisely, we deal with equations of the form ẋ(t) = B 1 (t)x(g 1 (t)) + • • • + B n (t)x(g n (t)), t ≥ 0 where B i ∈ C([0, ∞), L(R N , R N )), g i ∈ G 0 for i = 1, . . ., n and In Section 2, we derive the solution of a corresponding nonhomogeneous equation.Later, in Section 3 we use the property of commutativity of matrices to transform the nonlinear FDE with multiple delays and linear term Ax(t) to a nonlinear FDE with multiple delays but without a delayindependent linear term.After this transformation, we can apply the theory of Section 2, and so establish sufficient conditions for the exponential stability of the trivial solution of nonlinear FDE with multiple delays and linear term Ax(t) added on the right-hand side, supposing that the linear parts are given by pairwise permutable constant matrices.So, in Section 3, we study the exponential stability of the trivial solution of systems of FDEs with linear parts given by pairwise permutable matrices (for stability criteria for scalar equations with variable coefficients see e.g.[1,2,10,19]).
Definition 1.2.Given continuous function F , under the solution of a general FDE ) (at 0 we take the righthand derivative) which solves equation (1.3) and satisfies (1.2).

Solutions of systems of FDEs
In this section we derive a representation of a solution of FDE with single variable delay using a piecewise-defined matrix function, which is analogical to delayed matrix exponential e Bt τ for equations with constant delay.Later, we find a solution of FDE with multiple delays as it was done in [14].Throughout this part, we widely use the method of steps and variation of constants formula for FDEs (cf.[8,9]).We note that the existence and uniqueness of solutions of problems of this section are obvious.First, we find the fundamental solution of linear FDE with one delay satisfying the below-stated initial condition (2.2).
Proof.From definition of X B g (t, s), the initial condition is immediately verified.If s ≤ t < g −1 (s) then g(s) ≤ g(t) < s and B(q 2 ) . . .
Hence equation (2.1) is verified and the proof is finished.
Now, when we have the fundamental solution, we can derive the solution of corresponding nonhomogeneous equation (see [9]).Without any loss of generality we assume the initial function to be given on [g(0), 0].
Then the solution of the initial value problem has the form (2.6) what after differentiating with respect to t yields this, we differentiate the solution (2.7) for t ∈ [0, g −1 (0)) and In fact, we get the equality since ψ(0) = 0. Now if g −k (0) ≤ t < g −(k+1) (0) for some k ∈ N, then by differentiating formula (2.6) we obtain ẋ(t) = ẊB g (t, 0)ϕ(0) + B(t)ψ(g(t)) where we used the properties of X B g (t, s) from Theorem 2.1.Next, we apply the identity For t ≥ g −1 (0) it holds ψ(g(t)) = 0 and equation (2.4) B(q l )dq l . . .dq 1 for l = 0, . . ., k, i.e. the lower index l denotes the number of integrals in the sum, then x(t) can be written as Here we used the form of X B g (t, s) for fixed t and variable s (in (2.3) it was given for fixed s and variable t): (2.8) Now we provide an application of Theorem 2.2 on a problem with a bounded delay.
Corollary 2.5.The solution of Example 2.4 has the form In the next step, we shall use the solution of the nonhomogeneous initial value problem to construct the fundamental solution of FDE with two delays.Let us consider a matrix equation and s instead of 0 yields Hence for such t we get Analogically we proceed on other intervals [g −k 2 (s), g (s)) with k = 2, 3, . . . .By this process we obtain (2.10) In conclusion, we have proved that X(t) solves equation (2.9) for any t ≥ s.
Remark 2.7.Sometimes, it may be easier to use the "fixed t" form of X B 1 ,B 2 g 1 ,g 2 (t, s) analogical to (2.8) instead of "fixed s" given by (2.10).
Matrix function X B 1 ,B 2 g 1 ,g 2 (t, s) has some important properties which are concluded in the next lemma.
Then the following statements hold true for any t ∈ R: (1 All statements of the lemma follow from the uniqueness of a solution of a corresponding initial value problem.For instance in 1., both sides of the identity solve equation together with initial condition (2.2).
As before, we obtain a result on the solution of nonhomogeneous equation, this time with two delays.
Then the solution of the initial value problem has the form (2.13) Proof.Clearly, the initial condition is satisfied.In verification of equation (2.11) we consider four cases with respect to t. EJQTDE, 2012 No. 54, p. 9 After differentiating so one can see that x(t) really solves (2.11).The case g −1 2 (0) ≤ t < g −1 1 (0) can be proved analogically to the previous one using the change )) = 0 and direct differentiation of (2.13) gives the desired result.Now, we apply formula (2.13) on a problem with concrete unbounded delays.
Example 2.10.Let us consider the following initial value problem ) 2 and we can assume s ≥ 0. Hence by (2.3), where For the convenience, these sets are sketched in Figure 1.
Corollary 2.11.The solution of Example 2.10 has the form (2.16) For the graph of the solution with concrete functions One can proceed inductively from X B 1 ,B 2 g 1 ,g 2 (t, s) and, with the aid of the latter theorem, construct the fundamental solution of FDE with any finite number n ≥ 3 of variable delays.So one obtains (2.17) where Y (t, s) = X B 1 ,...,B n−1 g 1 ,...,g n−1 (t, s).
..,Bn g 1 ,...,gn (t, s) is the matrix solution of equation Proof.The case n = 2 was proved in Theorem 2.6.So here we suppose that the statement is true for n − 1 and we show that it holds also for n.
(s) for some k ∈ N. Then from (2.17) using the inductive hypothesis, we get for the derivative Y (g n (t), q 2 )B n (q 2 ) . . .
By collecting terms beginning with B i (t) we obtain for each i = 1, . . ., n − 1 exactly B i (t)X(g i (t)) since Y (g i (t), q 1 ) = Θ for g i (t) < q 1 (hence the upper boundary of integrals is changed from t to g i (t)).Next, g , thus collecting terms beginning with B n (t) yields B n (t)X(g n (t)) (in comparison to X(t), the number of integrals in X(g n (t)) is decreased by one).In conclusion, the last identity is precisely the equation which X(t) has to satisfy.
Matrix function X B 1 ,...,Bn g 1 ,...,gn (t, s) has properties that are analogical to those of X B 1 ,B 2 g 1 ,g 2 (t, s) provided in Lemma 2.8.We conclude them into a lemma without a proof.
In Section 3 we shall seek conditions for the exponential stability of the trivial solution of FDE with constant coefficients at linear terms.Here we find the solution of such an equation.
Then the solution of the equation satisfying initial condition (2.12) has the form (2.22) where f (t) = e −At f (t), ϕ(t) = e −At ϕ(t).Applying Theorem 2.14 to this problem yields Bn g 1 ,...,gn (t, s) f (s)ds, 0 ≤ t where When one returns to x(t), the formula (2.22) is obtained.

Exponential stability of nonlinear FDEs
In this section, we apply the theory derived in the preceding section to establish criteria for the exponential stability of the trivial solution of nonlinear FDE with multiple variable delays where the linear parts are given by pairwise permutable constant matrices.First, we estimate the fundamental solutions X B g (t, s) and X B 1 ,...,Bn g 1 ,...,gn (t, s) with the aid of the next lemma.
Proof.We prove the lemma via induction with respect to k. Denote EJQTDE, 2012 No. 54, p. 16 for any t ≥ s.
Proof.It is sufficient to prove the statement for g : [s, ∞) → [g(s), ∞) surjective.By this, the other case is also covered.Let t ≥ g −1 (s) be arbitrary and fixed, k ∈ N be such that g −k (s) ≤ t < g −(k+1) (s).Then from (2.3) we know that B(q j )dq j . . .dq 1 for j = 1, . . ., k.Since g is increasing and according to Lemma 3.1 we derive As a consequence, Obviously, the last estimate holds for each k ∈ N and hence for any dq so it remains true for such t.Proof.As before, it is enough to prove the lemma for surjective for each i = 1, . . ., n.
We show that if the statement holds for n − 1 delays, then it is true for n.Let k ∈ N be such that g −k n (s) ≤ t < g −(k+1) n (s) for arbitrary and fixed t ≥ g −1 n (s).Then from (2.17) we know that where Y (g n (q j−1 ), q j )B n (q j )Y (g n (q j ), s)dq j . . .dq 1 ..,g n−1 (t, s).For X j (t, s) we get (3.1) Y (g n (q j−1 ), q j ) B n (q j ) Y (g n (q j ), s) dq j . . .dq 1 .
Thus the statement is proved for t ≥ g −1 n (s).Analogically, one can prove it for t ≥ g −1 i (s) for any i = 1, . . ., n − 1 by the change of order described in Lemma 2.13.If s ≤ t < min{g −1 1 (s), . . ., g −1 n (s)}, X B 1 ,...,Bn g 1 ,...,gn (t, s) = E, hence the statement holds.Now, we define what exactly we shall understand under the notion of exponential stability.Then we use the estimations of fundamental solutions to derive the sufficient conditions for the exponential stability of FDEs with different types of nonlinearities (see [14,15,16,17] for analogical criteria for delay differential and difference equations with constant delays).Definition 3.4.Let n ∈ N, A, B 1 , . . ., B n be pairwise permutable N × N constant matrices, i.e.AB i = B i A, B i B j = B j B i for each i, j ∈ {1, . . ., n}, g i ∈ G 0 for i = 1, . . ., n, γ := min{g 1 (0), . . ., g n (0)}, ϕ ∈ C([γ, 0], R N ) and g i ∈ G 0 for i = 1, . . ., n and there be k 1 < k such that for all t ≥ 0, where B i (t) = e −A(t−g i (t)) B i for i = 1, . . ., n.Then if f (x) = o( x ), the trivial solution of equation  (2.14).From the property of eigenvalues of A it follows that there are positive constants k, K such that e At ≤ Ke −kt for all t ≥ 0. Next, since f (x) = o( x ), for any P > 0 there is δ > 0 such that if x < δ, then f (x) < P x .Applying these two estimations, Lemma 3.3 and assuming that x(s) is sufficiently small for s ∈ [0, t], t ≥ 0 we derive EJQTDE, 2012 No. 54, p. 20 Now, the property of k 1 implies that for each i ∈ {1, . . ., n} function g i : [0, ∞) → [g i (0), ∞) is surjective and, especially, g −1 i (0) < ∞.Indeed, suppose in contrary that there exists Q ∈ R such that g i (t) < Q for all t ≥ 0 and some i ∈ {1, . . ., n}.The property of eigenvalues of A yields the existence of a positive constant L i such that L i e kt ≤ e −At B i for all t ≥ 0 (assuming B i = Θ).Consequently, for all t ≥ 0, a contradiction results.So using the definition of ψ(t) we can estimate for all t ≥ 0 and i = 1, . . ., n, where the right-hand side is constant.Next, from (3.4) we get u(t) ≤ M + KP t 0 u(s)ds where (3.5) Finally, applying Gronwall's inequality, u(t) ≤ Me KP t which for x(t) means then for max{ ϕ(0) , M} < δ it holds x(t) ≤ Me −ηt for all t ≥ 0 with η = k − k 1 − KP > 0, i.e. the trivial solution of (3.3) is exponentially stable.Theorem 3.6.Let the assumptions of Theorem 3.5 be fulfilled.Moreover, let S i.e. for any P > 0 there exists δ > 0 such that then the trivial solution of equation (3.2) is exponentially stable.

Example 2 . 4 .
Let us consider the following initial value problem ẋ