On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay

The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.


1.
Introduction. The variation-of-constants formula is one of the principal tools of the theory of differential equations (ordinary, functional, partial, etc.) for deriving statements about the qualitative properties of the considered equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications, see e.g. [2], [4], [5], [6], [16], [19], [24].
A very instructive paper on this subject is [16]. The authors studied the inhomogeneous linear system under suitable conditions (here A (t) and B (t, s) are n × n matrices), and they derived the following variation-of-constants formula: the unique solution x of the initial value problem (1) can be obtained by 2010 Mathematics Subject Classification. 34K06, 45D05. Key words and phrases. Volterra type operator, linear functional differential and integral systems, variation-of-constants formula.
The research of the authors has been supported by Hungarian National Foundations for Scientific Research Grant No. K120186. * Corresponding author.

ISTVÁN GYŐRI AND LÁSZLÓ HORVÁTH
where R (t, s) is the so called resolvent. They formally defined the resolvent by where I is the identity matrix and We can see the essence of the variation-of-constants formula from (2): the function t → R (t, 0) x 0 , t ≥ 0, is the solution of the homogeneous linear system taking the value x 0 at 0, and we are able to generate a particular solution of the inhomogeneous linear system (1) in the form There is another approach to the problem of giving a variation-of-constants formula for (1) in [4] based on the fundamental matrix solution of the homogeneous linear system (3). This is another way to look at the resolvent R (t, s) and that is from the standpoint of linear systems of ordinary differential equations. The fundamental matrix solution of (3) is the n × n matrix function Z (t, s) whose columns are linearly independent solutions of (3) and whose value at t = s is the identity matrix I. In this case the author obtains (2) in the same form It is proved in [4] that the fundamental matrix solution and the resolvent are one and the same.
We mention that the conditions for the functions f , A and B are different in [16] and [4].
A variation-of-constants formula for linear functional differential equations with delay can be found e.g. in [24] by using the second dual of the phase space, and in [19] by using the theory of semigroups.
Our main purpose here is to give a variation-of-constants formula for inhomogeneous linear functional differential systems with delay determined by general Volterra-type operators. Our treatment of the delay in the considered systems is completely different from the usual methods. The proofs are essentially based on measure-and integral-theoretic considerations.
In Section 2 the terminology and notation is introduced, and the generality of the considered systems is illustrated by comparing them with usually studied systems. We deal with the representation of the studied Volterra-type operators and give some special realizations. By using a noncanonical approach to functional differential equations with delay, we are able to establish a variation-of-constant formula under very general assumptions. In Section 3 some preliminary results are found, some of them are interesting in their own right. The main part of Section 4 contains existence and uniqueness theorems for inhomogeneous linear functional integral systems with delay determined by the introduced Volterra-type operators. As a consequence, we have the basic existence and uniqueness theorems for the considered differential systems.
The literature on this topic for linear integral and differential equations with abstract Volterra operators offers a lot of books, papers and results. For equations without delay, see e.g. the books [7] and [15], and the papers [10], [13], [14] and [25], and the references therein. Several concrete causal differential and integral equations with delay are available in the literature, see e.g. the books [9] and [8], and the papers [12] and [27], and the references therein, but our approach to generate a causal operator with delay may be new. The usual idea of the proofs of the existence results is to apply fixed-point theorems. In this paper, the successive approximations method is applied. Moreover, some differential and integral inequalities are also studied. In Section 5 we obtain a variation-of-constants formula for the investigated systems. Our result extends and generalizes the similar results in [16] and [4].
Finally, some applications can be found in Section 6. We give some conditions for some special homogeneous linear functional differential systems with delay under which all solutions tend to zero at infinity. We also study the boundedness of the solution of the corresponding inhomogeneous systems. As a consequence we have a result about the persistence and permanence of the systems. These results are closely related to some results in [17] and [18], but in these papers systems are not studied. 2. Terminology and notation. The set of nonnegative integers will be denoted by N. R d stands for the set of all d-dimensional column vectors with real components. The norm of a vector x = (x 1 , . . . , The vector x is nonnegative if x i ≥ 0 (i = 1, . . . , d). In this case we write x ≥ 0. A partial ordering is defined on R d by letting x ≤ y if and only if Let t 0 ∈ R and τ 0 ≥ 0 be fixed. We introduce two function spaces which are needed in the sequel. The general theoretical background of topological vector spaces can be found, for example, in the book [20].
From now on the phrase "a function satisfies a property locally" means that the function satisfies this property on every compact subset of its domain. In the sequel, if not otherwise explicitly stated, by "measurable" and by "integrable" we will always mean "Lebesgue measurable" and "Lebesgue measurable and Lebesgue integrable", respectively.
A function x : [t 0 − τ 0 , ∞[ → R d is said to belong to B t0−τ0 if x is Borel measurable and locally bounded. We define a family of seminorms on B t0−τ0 in the following way These seminorms generate a locally convex and metrizable topology T p on B t0−τ0 (the topology of uniform convergence on compact sets). The elements of L t0 are those functions x : [t 0 , ∞[ → R d for which x is locally integrable. We obtain a family of seminorms on L t0 by setting Equipped with this family of seminorms, L t0 becomes a locally convex space. The topology is denoted by T q (the topology of pointwise convergence).
Let V : B t0−τ0 → L t0 be a linear operator which is of Volterra-type. The late property means the next: if t ≥ t 0 and x 1 , x 2 ∈ B t0−τ0 such that the restriction of Since the family {p t | t ≥ t 0 } of seminorms on B t0−τ0 is increasing, and V is of Volterra-type, the operator V is T p −T q continuous if and only if for every seminorm q t (t ≥ t 0 ) there exists a positive number M t such that Based on this we say that V is g-continuous if there exists a nonnegative and locally integrable function g : Remark 1. It is not hard to think that if V is nonnegative (this means that V (x) is nonnegative for all nonnegative x ∈ B t0−τ0 ) and T p − T q continuous, then it is g-continuous with g := V (1) , where 1 ∈ B t0−τ0 is the constant function with value 1.
In this paper we study inhomogeneous linear functional differential systems of the form under the following conditions (A 1 ) The operator V : B t0−τ0 → L t0 is linear, of Volterra-type, and g-continuous.
The function ρ belongs to L t0 . We say that a function x ∈ B t0−τ0 is a solution of (5) if x is locally absolutely continuous on [t 0 , ∞[ and it satisfies (5) almost everywhere on [t 0 , ∞[.
is a linear operator such that (A 2 ) holds, then V is obviously T p − T q continuous.
(b) Let C t0−τ0 denote the subspace of B t0−τ0 consisting of all functions which are Borel measurable and bounded on [t 0 − τ 0 , t 0 ] and locally absolutely continuous on [t 0 , ∞[. Assume (A 1 ) and (A 3 ). We shall show (see Theorem 4.2) that for each ρ ∈ L t0 there exists a solution x ∈ C t0−τ0 of (5), and this is often expressed as follows: by considering equation (5), the pair (L t0 , C t0−τ0 ) is admissible for the operator V (see [23]). Some mention should be made here of the generality of the considered systems. This is best understood when viewed as a generalization of the following frequently studied inhomogeneous linear functional differential system where L : Banach space of continuous functions endowed with the uniform norm topology ( · ∞ ), and x t is defined by It is easy to think that It can be seen that the system (5) contains the system (6) as a simple case, and the properties (i) and (ii) are reformulated in the assumptions (A 1 ) and (A 2 ) in a very general manner.
The next result shows that the considered operators can be built up from more elementary operators with the same properties. In the left part of this section the dimension of the used space will be indicated explicitly, so we shall write B d t0−τ0 , L d t0 and p d t .
where the operators V ij : Then V is an operator from B d t0−τ0 to L d t0 , and it is obvious that V is linear and of Volterra-type.
From the foregoing it is clear that V satisfies (A 1 ). It is not hard to see that V satisfies (A 2 ) too. The proof is complete.
Important special cases of elementary Volterra-type operators (from B 1 t0−τ0 to L 1 t0 ) satisfying conditions (A 1 ) and (A 2 ) are described in the next examples. By a real measure we shall mean a finite signed measure. If µ is a real measure, then µ means the total variation of µ.
Assume a real measure µ t on B t is assigned to every t ≥ t 0 such that for all belongs to L 1 t0 . Clearly the operator V is linear and of Volterra-type. The definition of V , and Lebesgue's dominated convergence theorem show that (A 2 ) holds for V too, and hence V is T p − T q continuous. Since it follows that if the function Conversely, it follows from Corollary 4 (b) and (c) in the sequel that if V : is a linear and Volterra-type operator which satisfies (A 2 ), then there exist real measures µ t (t ≥ t 0 ) on B t such that (10) holds. If µ t is a finite measure for all t ≥ t 0 , then by Remark 1, V is g-continuous with g := |V (1)|.
where ε t is the unit mass at t on B t . Then does not belong to L 1 t0 . (c) By Example 1, and by (b), it is an interesting problem to characterize those functions t → µ t , µ t is a real measure on B t , t ≥ t 0 for which the function defined in (10) belongs to L 1 t0 for all x ∈ B 1 t0−τ0 . We now construct some special operators from B 1 t0−τ0 to L 1 t0 satisfying (A 1 ) and (A 2 ) in the following three examples. These operators contain frequently used operators. Each of these operators can serve as V ij in (8).
where (B 1 ) The delay τ : [t 0 , ∞[ → R + is nonnegative, measurable and it obeys the inequality It is easy to check the fulfilment of properties (A 1 ) and (A 2 ).
Example 3. Let λ denote the Lebesgue measure on the Borel measurable subsets where (B 3 ) The delay η : [t 0 , ∞[ → R + is nonnegative, Borel measurable and it obeys the inequality ν is a σ-finite measure on the Borel measurable subsets of [t 0 , ∞[. (B 5 ) Let H := (t, s) ∈ R 2 | t ≥ s ≥ t 0 , . The function b : H → R is Borel measurable, locally λ × ν-integrable, and for each t ≥ t 0 the sections First we show that I b,η,ν is well defined. Let x ∈ B 1 t0−τ0 , and t ≥ t 0 be fixed. Since the function exists and finite. The assumption on b implies that the function It is obvious that I b,η,ν is linear and of Volterra-type. Another application of Fubini's theorem shows that the function is locally integrable, and therefore I b,η,ν is g-continuous.
Let (x n ) be a sequence of functions from B 1 t0−τ0 such that (x n ) is uniformly bounded on every compact subinterval of [t 0 − τ 0 , ∞[, and it has a pointwise limit x. Lebesgue's dominated convergence theorem can now be applied to the integral in (11), and it implies that (A 2 ) also holds.
where (B 4 ) and (B 5 ) are satisfied. Similarly to Example 3, we can prove that (A 1 ) and (A 2 ) hold for C b,ν .

Remark 4. An operator of the form
can serve as V ij in (8). Here n 1,i,j , n 2,i,j and n 3,i,j belong to N, and the value of any empty sum of operators is taken to be zero. This demonstrates the generality of the studied system (5).
be the vector space of all functions z defined on [t 0 − τ 0 , t] and taking their values in R d , such that they are Borel measurable and bounded.
, · ∞ ) is described in the following basic result.
Theorem 3.1. (see [22] and [28]) If Λ is a continuous linear functional from The next three results are natural extensions of the previous theorem. Being not so easy to find similar results in the literature, and for the sake of completeness, we present the proofs. Corollary 1. If Λ is a continuous and σ-continuous linear functional from the Proof. By Theorem 3.1, there exists a unique finitely additive real measure µ : It is enough to show that µ is σ-additive. This is satisfied if and only if for every Denote by 1 Hn the characteristic function of H n . By using (12) and the σcontinuity of Λ, we obtain The series (13) is absolutely convergent, since µ has finite total variation. The proof is complete.
Then Λ i is a σ-continuous linear functional (i = 1, . . . , d), and therefore Corollary 1 guarantees the existence of a real measure µ i on B t such that for all z ∈ Since the result follows from (14). The proof is complete.
If T is a continuous and σ-continuous linear oper- Proof. Apply Corollary 2 to the components of T .
With the aid of Corollary 3 we get an essential representation result for the studied operators: be a linear and Volterra-type operator which satisfies (A 2 ). Introduce the operator Then (a) The operator T t is linear and σ-continuous.
By using the previous corollary, we obtain the following result which will play an important role in the proof of our variation-of-constants formula.
Then the function Proof. By using some results from the theory of product measure, it is easy to check Let t > t 0 be fixed. Let and let µ t ij (i, j = 1, . . . , d, t ≥ t 0 ) be the real measures on B t corresponding to V and defined in Corollary 4 (b).
Then H t is Borel measurable and v j is λ t × µ t+ ij and λ t × µ t− ij integrable on H t , where λ t is the Lebesgue measure on B t and µ t+ ij , µ t− ij are the positive and negative variations of µ t ij , respectively (i, j = 1, . . . , d).
Since v is Borel measurable and locally bounded, we can apply Fubini's theorem which gives according to (15) In the last step Corollary (4) was used.
The last equality follows from another application of Corollary (4). The proof is complete.
We end this section by mentioning two results which will be used later.
It is known (see [1]) that the so-called Charatheodory functions (functions which are continuous in one variable and measurable in another) are jointly measurable in many important cases. The following result is a variant of this under weaker conditions.
Proof. For every positive integer n, define the function f n : and the result follows.
The proof is complete.
The final result deals with differentiability of functions defined by integrals. Then the function is locally absolutely continuous and Proof. This is an immediate consequence of Theorem 2.7 (b) and (c) in [26].

Existence and uniqueness results. A function
if c is Borel measurable and locally bounded. Equip the vector space B t0 with the topology of uniform convergence on compact sets which will be denoted by To obtain existence and uniqueness results for the considered differential system (5), we first study more general integral systems of the form where c ∈ B t0 and (A 1 ) holds. A solution of (16) is a function x ∈ B t0−τ0 that satisfies the equation for all As usual, if a solution of (16) is sought in F ϕ , then we can say that the initial condition is associated with (16).
Then (a) The integral equation (16) has at least one solution x Then (x n ) ∞ n=0 is a sequence in B ϕ t0 which converges in the topology T p (and thus pointwise) to a solution x ϕ ∈ B ϕ t0 of the integral equation (16).
(c) Let F be a subset of B t0 which is closed in the topology T p . Assume further that for all x ∈ F ϕ the functions also belong to F . Then belongs to SB t0−τ0 . If x ∈ SB t0−τ0 such that Proof. (a) To prove the uniqueness of the solutions, suppose that x 1 , x 2 ∈ B t0−τ0 are solutions of (16) satisfying By the g-continuity of V , Since the integral in (22) is increasing, and therefore an application of the Gronwall's inequality implies the result.
(b) First we show that the sequence (x n ) ∞ n=0 is well defined, and the functions x n (n ∈ N) belong to B ϕ t0 . By definition, x 0 ∈ B ϕ t0 . Proceeding inductively, suppose that x n ∈ B ϕ t0 for some n ∈ N. Since the function is locally absolutely continuous, we have that x n+1 ∈ B ϕ t0 . Let t 1 > t 0 be fixed.
The function x 0 is bounded on [t 0 − τ 0 , t 1 ], the function c − h is bounded on [t 0 , t 1 ], and therefore By (4), This and It follows from the g-continuity of V and (23) that Since By using an induction argument, we can prove similarly that for every n ≥ 1 It follows from this (see for example [21]) that for every n ≥ 1 We obtain from (23) and (24) that Since and since t 1 is arbitrarily chosen from [t 0, ∞[, we have from (25) and (18) that there exists a function x ϕ : and Since the functions x n (n ∈ N) belong to B ϕ t0 , we conclude from (26), (27) and (28) that x ϕ ∈ B ϕ t0 . For every n ∈ N and for every integer m ≥ n we have according to (24) that and hence This shows (t 1 can be chosen arbitrarily) that which implies that (x n ) converges to x ϕ in the topology T p . Since gives It now follows from (18) that x ϕ is a solution of (16).
(c) Assume h ∈ F . By using (19), it is easy to show that the successive approximations (x n ) ∞ n=1 defined by (18) belong to F ϕ . By (b) (see (30)), the sequence (x n ) converges to x ϕ in the topology T p . Since F is closed in T p , x ϕ ∈ F ϕ .
(d) Since SB t0−τ0 is a vector space and V is nonnegative on SB t0−τ0 , V is monotonic on SB t0−τ0 .
It follows from (20) that the function belongs to SB t0−τ0 . Consider the successive approximations (18) determined by this function. According to (20) x n ∈ SB t0−τ0 for all n ∈ N.
Next we show that x ≤ x n , n ∈ N.
By (21), this is true if n = 0. Also, if n is a nonnegative integer for which (31) holds, then the monotonicity of V implies that so that (31) holds for n + 1, and therefore for all n ∈ N.
The proof is complete.
Now we give some consequences of Theorem 4.1. To achieve this introduce the following subsets of B t0 : (a) A function x ∈ B t0 is said to belong to C t0 if x is continuous. The subspace C t0 of B t0 is closed in T p .
(b) Let s ≥ t 0 be fixed, and define the subspace B s of B t0 by This subspace is closed in T p . (c) A function x ∈ B t0 is said to belong to N t0 if x is nonnegative. The set N t0 is closed in T p and it is closed under addition. ( (a) Let F denote either B t0 or C t0 , and let c ∈ F . Then for every ϕ ∈ B ([t 0 − τ 0 , t 0 [) the integral equation (16) has a unique solution x ϕ ∈ F ϕ satisfying (b) If c ∈ B s , then the integral equation (16) has a unique solution x ∈ B t0−τ0 satisfying , the function g is integrable and ∞ t0 g (s) ds ≤ 1, and there exists K > 0 such that then x ϕ is also bounded.
(a) It follows that if x ∈ F ϕ , then c +x belongs to F too. (b) The operator V is of Volterra-type and hence for every x ∈ B t0−τ0 satisfying (33) we have c+x ∈ B s .
(c) Since the operator V is nonnegative, c ∈ N t0 and ϕ is also nonnegative, and we can follow as in (d).
Now the results follows from Theorem 4.1 (b) and (c). The proof is complete.
In the next result we consider the inhomogeneous linear functional differential equation (5). (a 1 ) The differential equation (5) has exactly one solution x ϕ satisfying then x ϕ is also bounded.

ISTVÁN GYŐRI AND LÁSZLÓ HORVÁTH
(a 4 ) If the operator V is nonnegative, x ∈ B t0−τ0 such that x is locally absolutely continuous on [t 0 , ∞[, and (b) Let s ≥ t 0 be fixed. If ρ = 0 on [t 0 , s[ and c ∈ R d , then the inhomogeneous linear functional differential equation has exactly one solution x s satisfying Proof. (a) It is easy to check that x : [t 0 − τ 0 , ∞[ → R d is a solution of the differential equation (5) satisfying (35) if and only if x is a solution of the integral equation for which To prove (a 1 ) and (a 2 ), Corollary 5 (a) and (c) can be applied to the integral equation (39) by considering (a 1 ) The functionρ always belongs to B t0 (moreover,ρ ∈ C t0 ). (a 4 ) According to (36) and thus Theorem 4.1 (d) can be applied.
It is also not hard to see that x is a solution of the differential equation (37) satisfying (38) if and only if x is a solution of the integral equation for which x (t) = 0 for all t ∈ [t 0 − τ 0 , t 0 [, or equivalently x is a solution of the integral equation for which We can apply Corollary 5 (b), since the functionρ belongs to B s . The proof is complete.

5.
A variation-of-constants formula. Assume (A 1 ), and consider the homogeneous linear functional differential system It follows from Theorem 4.2 (b) that for every s ≥ t 0 and i = 1, . . . , d the system where e 1 , . . . , e d is the standard basis in R d . This means that for every s ∈ [t 0 , ∞[ and (43) holds (i = 1, . . . , d).
The function v is called the fundamental kernel of (41).
The essential properties of the fundamental kernel are described in the next theorem. (c) The functions (i = 1, . . . , d) are locally integrable.
Proof. We can obviously suppose that i = 1.
The sequence (w n ) ∞ n=0 is well defined (see (40)). First we show that the functions w n (t, ·) (n ∈ N) are continuous on It follows that The function defined by the right hand side of (47) is obviously locally integrable. Now we prove that the function w n+1 (t, ·) is continuous on [t 0 , t] for every t ∈ ]t 0 , ∞[.

It comes from (4) that
Lebesgue's dominated convergence theorem can now be applied to the integral in (45), and it yields that which gives that w n+1 t , · is continuous atŝ.
By considering (40), it follows from the proof of Theorem 4.1 (b) that for every Since for every t ≥ t 0 we obtain from (29) (K 2 = 0) that and hence for every fixed t ≥ t 0 the sequence (w n (t, ·)) converges uniformly to v 1 (t, ·) in s on [t 0 , t].
Since the set is Borel measurable, v 1 is also Borel measurable.
By (47) and (48), v 1 is locally bounded. (c) Letŝ ∈ ]t 0 , ∞[, and let (s k ) ∞ k=0 be a sequence from [t 0 ,ŝ] with limitŝ. By using (a) and the definition of v 1 , we have that which gives that the function s → V (v 1 (·, s) By applying Theorem 3.4 to the function defined in (44) in case of i = 1, we obtain the (joint) Lebesgue measurability of this function.
Since V is g-continuous, and this yields the integrability condition by using the local boundedness of v 1 . The proof is complete.
Now we arrive the main result of this section.
for which the restriction of is the unique solution of the inhomogeneous linear functional differential equation (5) for which the restriction of x to [t 0 − τ 0 , t 0 ] is ϕ, then x can be obtained by Proof. By Theorem 5.2 (a), the integral in (50) exists. By Theorem 5.2 (c), we can apply Theorem 3.5 which shows that Theorem 3.3 now yields that for every j = 1, . . . , d t t0 V (v j (·, s) ρ j (s)) (t) ds = V (w j ) (t) , t ≥ t 0 .
By using this in (51), we obtain The proof is complete.

6.
Applications. In this section we study the inhomogeneous linear functional differential system and the corresponding homogeneous linear functional differential system where (C 1 ) The functions a i : [t 0 , ∞[ → R (i = 1, . . . , d) are locally integrable, and Λ (t) is a diagonal d × d matrix whose diagonal entries starting in the upper left corner are a 1 (t) , . . . , a d (t) for all t ≥ t 0 .
It is obvious that V is linear, of Volterra-type and satisfies (A 2 ). By Remark 1, N is N (1) -continuous. This implies that V is g-continuous with . . . , d).
The fundamental kernel of (53) is denoted by v : In the first main result of this section we study the behavior of the solutions of the homogeneous system (53). Theorem 6.1. Assume (C 1 ), (C 2 ) and (a) Every nonnegative solution y = (y 1 , . . . , y d ) T of the homogeneous system (53) is bounded. Moreover, y j (s) , t ≥ t 0 , i = 1, . . . , d.
(b) Assume further that there exists a nonnegative delay τ : [t 0 , ∞[ → R + such that τ is measurable, it obeys the inequality and for all x ∈ B t0−τ0 and t ≥ t 0 the vector N (x) (t) depends only on the restriction of x to [t − τ (t) , t].
(b 1 ) If there exists a nonnegative and locally integrable function δ : then then every solution of the homogeneous system (53) tends to zero at infinity.
This at once leads to the question whether the conditions in the previous theorem guarantee the existence of a function δ satisfying (55) and (56). To answer this question, we can use the results in [17].
Assume further that the function τ : [t 0 , ∞[ → R + is measurable and it obeys the inequality (a) There exists a nonnegative and locally integrable functionδ : [t 0 − τ 0 , ∞[ → R + which is unique and satisfies the integral equation and hence there exists a nonnegative and locally integrable function δ : (57) and there is q ∈ ]0, 1[ such that In the second main result of this section the boundedness of the solutions of the system (52) are studied. It is an essential generalization of Theorem 2.1 in [18]. Theorem 6.3. Assume (C 1 -C 3 ). Assume further that there exists t 1 ≥ t 0 for which and every solution of the homogeneous system (53) tends to zero at infinity. Then for every solution x : [t 0 − τ 0 , ∞[ → R d of the inhomogeneous system (52), we obtain , i = 1, . . . , d.
As an immediate consequence of the previous theorem we have a result about persistence and permanence of the system (52). First we give the definition of these concepts (see e.g. [3] and [11]).

ISTVÁN GYŐRI AND LÁSZLÓ HORVÁTH
The system (52) is said to be permanent if there are constants K > k > 0 such that, given any solution x, there is t (x) ≥ t 0 such that Corollary 6. Assume (C 1 -C 3 ). Assume further that there exists t 1 ≥ t 0 for which and every solution of the homogeneous system (53) tends to zero at infinity.
(a) If then the system (52) is uniformly persistent.
then the system (52) is permanent.
To prove these results, we need three lemmas.
Lemma 6.5. Assume (C 1 -C 3 ), and assume further that ρ is nonnegative. Let x be the solution of the inhomogeneous system (52) satisfying Since N is of Volterra-type and nonnegative, and hence which is a contradiction.
Since N is of Volterra-type and nonnegative, and therefore which is a contradiction. Let (ε n ) Evidently, By using these establishments, we can apply Lebesgue dominated convergence theorem and the monotone convergence theorem in x εn (t) = ϕ εn (t) , t 0 − r ≤ t < t 0 , n ≥ 0, and obtain that The proof is complete.
We now establish some useful properties of the homogeneous system (53).
Lemma 6.6. Assume (C 1 ) and (C 2 ). (a)Then every solution of (53) tends to zero at infinity if and only if every nonnegative solution of (53) tends to zero at infinity.
(b) The fundamental kernel v of (53) is nonnegative in the sense that Proof. (a) The "only if" part is trivial. Conversely, let y : [t 0 − τ 0 , ∞[ → R d be a solution of (53), and let z : [t 0 − τ 0 , ∞[ → R d be the solution of (53) satisfying Since (53) is homogeneous and an application of Lemma 6.5 implies that z − y and z + y are nonnegative solutions of (53), and hence they tend to zero at infinity. By using y = 1 2 ((z + y) − (z − y)) , the result follows. (b) By using the definition of v, it follows immediately from Lemma 6.5. The proof is complete.
Lemma 6.7. Assume (C 1 -C 3 ). Assume further that every solution of the homogeneous system (53) tends to zero at infinity. Fix T ≥ t 0 . Then (a) Every solution of the homogeneous system tends to zero at infinity too. Proof. (a) By Lemma 6.6, it is enough to consider nonnegative solutions of (60). Let y T be a nonnegative solution of (60), and let y be the solution of (53) satisfying Since y is continuous on [t 0 , ∞[, (61) and Lemma 6.5 (a) imply that there exists c ∈ R d , c > 0 such that It follows from this and from the fact that y T is nonnegative and bounded on [t 0 − τ 0 , T ] that there is a positive number k for which The systems (53) and (60) are homogeneous, and therefore ky is a solution of both systems.
Since ky −y T is also a solution of (60), by (62), and by (ky − y T ) (T ) > 0, Lemma 6.5 (a) implies that Now the result follows, because y T is nonnegative and lim t→∞ ky (t) = 0.
and let z : [t 0 − τ 0 , ∞[ → R d be given by By Theorem 5.3, z is the solution of the equation According to the definition ofθ, it can be seen that z is a solution of (60), and therefore (a) implies the result.
(c) It is obvious that the constant function t → 1 d (t ≥ t 0 − τ 0 ) is the solution of the system Since every solution of (53) tends to zero at infinity, Theorem 5.3 gives that

ISTVÁN GYŐRI AND LÁSZLÓ HORVÁTH
By using the result follows from (63) and (b). The proof is complete.
Finally, we give the proofs of the main results of this section.
Proof of Theorem 6.1. First we prove a useful property of N . Let x = (x 1 , . . . , x d ) where K > 0 and t K > t 0 . Define the function z = (z 1 , . . . , z d ) Because x ≤ z and N is linear, nonnegative and of Volterra-type, we have (a) Let y = (y 1 , . . . , y d ) T be a fixed nonnegative solution of (53), and let y j (s) .
The proof splits into two cases.
(i) Special case: strict inequality holds in (54) that is Assume the result is not true. Then there is a number K 2 > K 1 and an index i ∈ {1, . . . , d} such that y i (t) = K 2 for some t > t 0 .
The continuity of y on [t 0 , ∞[ and the definition of K 1 show that for each K ∈ ]K 1 , K 2 ] there is at least one indices i K ∈ {1, . . . , d} and a number t K > t 0 such that It is well known that if f : [a, b] → R is an absolutely continuous function, then for every set H ⊆ [a, b] with Lebesgue measure 0, f (A) also has Lebesgue measure 0. Since y is absolutely continuous on [t 0 , t K2 ], it now follows that there exists i ∈ {1, . . . , d} such that the outer measure of y i (H) is positive. This implies that there is K ∈ ]K 1 , K 2 ] such that y i differentiable at t K and By using (66), (64) with x = y, and (65), we have which is a contradiction.
(ii) The general case: Let ε > 0, and let Λ ε (t) be a diagonal d × d matrix whose diagonal entries starting in the upper left corner are a 1 (t) + ε, . . . , a d (t) + ε for all t ≥ t 0 . Consider the homogeneous linear functional differential system and denote by y ε that solution of (67) for which By Lemma 6.5, y ε is also nonnegative. Since a i (t) + ε − N i (1) (t) > 0, t ≥ t 0 , i = 1, . . . , d, the first part of the proof and (68) show that According to Lemma 6.5, the fundamental kernel v of (53) is nonnegative. We have from Theorem 5.3 that y ε (t) = y (t) − ε t t0 v (t, s) y ε (s) ds, t ≥ t 0 .
Since v is nonnegative, it now follows from (69) that Then it is easy to check that z is a nonnegative solution of the homogeneous system z i (t) = − (a i (t) − δ (t)) z i (t) + exp We know from (a) that y is bounded, and hence by (71) and (56), it is enough to prove that z is also bounded.
The functions a i : [t 0 , ∞[ → R, a i (t) = a i (t) − δ (t) , i = 1, . . . , d are obviously locally integrable. It is not hard to show that the operator N : B t0−τ0 → L t0 defined by is linear, of Volterra-type, N (1) -continuous, nonnegative, and satisfies (A 2 ). The previous two establishments insure that the system (72) has the same structure as the system (53), that is (C 1 ) and (C 2 ) hold for it with a i , i = 1, . . . , d and N .
It now follows from (a) (by applying it to the system (72)) that z is bounded if the properties a i (t) − N i (1) (t) ≥ 0, t ≥ t 0 , i = 1, . . . , d are valid. Let u ≥ t 0 be fixed, and let x u , z u ∈ B t0−τ0 be defined by   .
Since for all x ∈ B t0−τ0 and t ≥ t 0 the vector N (x) (t) depends only on the restriction of x to [t − τ (t) , t], N (1) (t) = N (z u ) (t) , t ≥ u.
(b 2 ) It is an immediate consequence of (b 1 ). The proof is complete.
By applying Theorem 5.3, we obtain and thus where v i (t, s) denotes the ith row of v (t, s).
We now use the assumption that lim Note that for all t ≥ T ε . . .
By Lemma 6.6 (b), v i (t, s) is nonnegative, and thus we have    Now the result follows from (78), (79) and from Lemma 6.7 (c). We can prove similarly if (76) is satisfied. The proof is complete.