GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 5, 1999, 429-440

ON BOUNDED SOLUTIONS OF SYSTEMS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

R. HAKL

Abstract. Sufficient conditions of the existence and uniqueness of bounded on real axis solutions of systems of linear functional differential equations are established.

1. Formulations of the Main Results

Let R be the set of real numbers, C_{loc}(R, R) be the space of continuous functions u : R → R with the topology of uniform convergence on every compact interval and L_{loc}(R, R) be the space of locally summable functions u : R → R with the topology of convergence in the mean on every compact interval. Consider the system of functional differential equations

x′_{i}(t) = ∑_{k=1}^{n} l_{ik}(x_{k})(t) + q_{i}(t) (i = 1, …, n),  (1.1)

where l_{ik} : C_{loc}(R, R) → L_{loc}(R, R) (i, k = 1, …, n) are linear continuous operators and q_{i} ∈ L_{loc}(R, R) (i = 1, …, n). Moreover, there exist linear positive operators \overline{l}_{ik} : C_{loc}(R, R) → L_{loc}(R, R) (i, k = 1, …, n) such that for any u ∈ C_{loc}(R, R) the inequalities

|l_{ik}(u)(t)| ≤ \overline{l}_{ik}(|u|)(t) (i, k = 1, …, n)  (1.2)

are fulfilled almost everywhere on R.

A simple but important particular case of (1.1) is the linear differential system with deviated arguments

x′_{i}(t) = ∑_{k=1}^{n} p_{ik}(t)x_{k}(τ_{ik}(t)) + q_{i}(t) (i = 1, …, n),  (1.3)

429 where p_{ik} ∈ L_{loc}(R, R), q_{i} ∈ L_{loc}(R, R) (i, k = 1, …, n) and τ_{ik} : R → R (i, k = 1, …, n) are locally measurable functions.

1991 Mathematics Subject Classification. 34K10.

Key words and phrases. System of linear functional differential equations, linear differential system with deviated arguments, bounded solution.

1072-947X/99/0900-0429$16.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

A locally absolutely continuous vector function (x_{i})_{i=1}^{n} : R → R is called a bounded solution of system (1.1) if it satisfies this system almost everywhere on R and

sup { ∑_{i=1}^{n} |x_{i}(t)| : t ∈ R} < +∞.

I. Kiguradze [1, 2] has established optimal in some sense sufficient conditions of the existence and uniqueness of a bounded solution of the differential system

\frac{dx_{i}(t)}{dt} = ∑_{k=1}^{n} p_{ik}(t)x_{k}(t) + q_{i}(t) (i = 1, …, n).

In the present paper these results are generalized for systems (1.1) and (1.3). Before formulating the main results we want to introduce some notation. δ_{ik} is Kronecker’s symbol, i.e., δ_{ii} = 1 and δ_{ik} = 0 for i ≠ k. A = (a_{ik})_{i,k=1}^{n} is a n × n matrix with components a_{ik} (i, k = 1, …, n). r(A) is the spectral radius of the matrix A. If t_{i} ∈ R ∪ {−∞, +∞} (i = 1, …, n), then

N_{0}(t_{1}, …, t_{n}) = {i : t_{i} ∈ R}.

If u ∈ L_{loc}(R, R), then

η(u)(s, t) = ∫_{t}^{s} u(ξ) dξ for t and s ∈ R.  (1.4)

Theorem 1.1. Let there exist t_{i} ∈ R ∪ {−∞, +∞} (i = 1, …, n), a nonnegative constant matrix A = (a_{ik})_{i,k=1}^{n} and a nonnegative number a such that

r(A) < 1,  (1.5)

| ∫_{t_{i}}^{t} exp ( ∫_{s}^{t} l_{ii}(1)(ξ)dξ)[ \overline{l}_{ii}(|η(\overline{l}_{ik}(1))(·, s)|)(s) + (1 − δ_{ik})|\overline{l}_{ik}(1)(s)|]ds| ≤

≤ a_{ik} for t ∈ R (i, k = 1, …, n),  (1.6)

∑_{i=1}^{n} | ∫_{t_{i}}^{t} exp ( ∫_{s}^{t} l_{ii}(1)(ξ)dξ)[ \overline{l}_{ii}(|η(|q_{i}|)(·, s)|)(s) + |q_{i}(s)|]ds| ≤

≤ a for t ∈ R  (1.7)

and

sup { ∫_{t_{i}}^{t} l_{ii}(1)(s)ds : t ∈ R} < +∞ for i ∈ N_{0}(t_{1}, …, t_{n}). (1.8)

Then for any c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) system (1.1) has at least one bounded solution satisfying the conditions

x_{i}(t_{i}) = c_{i} for i ∈ N_{0}(t_{1}, …, t_{n}).  (1.9)

Theorem 1.2. Let all the conditions of Theorem 1.1 be fulfilled and

lim inf_{t→t_{i}} ∫_{t}^{0} l_{ii}(1)(s)ds = −∞ for i ∈ {1, …, n} \ N_{0}(t_{1}, …, t_{n}). (1.10)

Then for any c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) system (1.1) has one and only one bounded solution satisfying conditions (1.9).

If t_{i} ∈ {−∞, +∞} (i = 1, …, n), then N_{0}(t_{1}, …, t_{n}) = ∅. In that case in Theorems 1.1 and 1.2 conditions (1.8) and (1.9) become unnecessary so that these theorems are formulated as follows.

Theorem 1.1′. Let there exist t_{i} ∈ {−∞, +∞} (i = 1, …, n), a nonnegative constant matrix A = (a_{ik})_{i,k=1}^{n} and a nonnegative number a such that conditions (1.5)–(1.7) are fulfilled. Then system (1.1) has at least one bounded solution.

Theorem 1.2′. Let all the conditions of Theorem 1.1′ be fulfilled and

lim inf_{t→t_{i}} ∫_{t}^{0} l_{ii}(1)(s) ds = −∞ (i = 1, …, n).

Then system (1.1) has one and only one bounded solution.

The above theorems yield the following statements for system (1.3).

Corollary 1.1. Let there exist t_{i} ∈ R ∪ {−∞, +∞} (i = 1, …, n), a nonnegative constant matrix A = (a_{ik})_{i,k=1}^{n} and a nonnegative number a such that r(A) < 1,

| ∫_{t_{i}}^{t} exp ( ∫_{s}^{t} p_{ii}(ξ)dξ)[|p_{ii}(s) ∫_{s}^{τ_{ii}(s)}|p_{ik}(ξ)|dξ| + (1 − δ_{ik})|p_{ik}(s)|]ds| ≤

≤ a_{ik} for t ∈ R (i, k = 1, …, n),  (1.11)

∑_{i=1}^{n} | ∫_{t_{i}}^{t} exp ( ∫_{s}^{t} p_{ii}(ξ)dξ)[|p_{ii}(s) ∫_{s}^{τ_{ii}(s)}|q_{i}(ξ)|dξ| + |q_{i}(s)|]ds ≤

≤ a for t ∈ R  (1.12)

and

sup { ∫_{t_{i}}^{t} p_{ii}(s)ds : t ∈ R} < +∞ for i ∈ N_{0}(t_{1}, …, t_{n}). (1.13)

Then for any c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) system (1.3) has at least one bounded solution satisfying conditions (1.9).

Corollary 1.2. Let all the conditions of Corollary 1.1 be fulfilled and

lim inf_{t→t_{i}} ∫_{t}^{0} p_{ii}(s)ds = −∞ for i ∈ {1, …, n} \ N_{0}(t_{1}, …, t_{n}). (1.14)

Then for any c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) system (1.3) has one and only one bounded solution satisfying conditions (1.9).

Corollary 1.3. Let there exist t_{i} ∈ R ∪ {−∞, +∞}, b_{i} ∈ [0, +∞[, b_{ik} ∈ [0, +∞[ (i, k = 1, …, n) such that the real part of every eigenvalue of the matrix (−δ_{ik}b_{i} + b_{ik})_{i,k=1}^{n} is negative and the inequalities

σ(t, t_{i})p_{ii}(t) ≤ −b_{i}, |p_{ii}(t) ∫_{t}^{τ_{ii}(t)}|p_{ik}(s)|ds| + (1 − δ_{ik})|p_{ik}(t)| ≤

≤ b_{ik} (i, k = 1, …, n)

hold almost everywhere on R, where σ(t, t_{i}) ≡ sgn(t−t_{i}) if t_{i} ∈ R, σ(t, t_{i}) ≡ 1 if t_{i} = −∞ and σ(t, t_{i}) ≡ −1 if t_{i} = +∞. Moreover, let

sup { ∫_{t}^{t+1}[|p_{ii}(s) ∫_{s}^{τ_{ii}(s)}|q_{i}(ξ)|dξ| + |q_{i}(s)|]ds : t ∈ R} <

< +∞ (i = 1, …, n).  (1.15)

Then for any c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) system (1.3) has one and only one bounded solution satisfying conditions (1.9).

Corollary 1.1′. Let there exist t_{i} ∈ {−∞, +∞} (i = 1, …, n), a nonnegative constant matrix A = (a_{ik})_{i,k=1}^{n} and a nonnegative number a such that r(A) < 1 and conditions (1.11) and (1.12) are fulfilled. Then system (1.3) has at least one bounded solution.

Corollary 1.2′. Let all the conditions of Corollary 1.1′ be fulfilled and

lim inf_{t→t_{i}} ∫_{t}^{0} p_{ii}(s) ds = −∞ (i = 1, …, n).

Then system (1.3) has one and only one bounded solution.

Corollary 1.3′. Let there exist σ_{i} ∈ {−1, 1}, b_{i} ∈ [0, +∞[, b_{ik} ∈ [0, +∞[ (i, k = 1, …, n) such that the real part of every eigenvalue of the matrix (−δ_{ik}b_{i} + b_{ik})_{i,k=1}^{n} is negative and the inequalities

σ_{i}p_{ii}(t) ≤ −b_{i}, |p_{ii}(t) ∫_{t}^{τ_{ii}(t)}|p_{ik}(s)|ds| + (1 − δ_{ik})|p_{ik}(t)| ≤

≤ b_{ik} (i, k = 1, …, n)

hold almost everywhere on R. Moreover, if conditions (1.15) are fulfilled, then system (1.3) has one and only one bounded solution.

2. Lemma of the Existence of a Bounded Solution of System (1.1)

Let t_{i} ∈ R ∪ {−∞, +∞} (i = 1, …, n) and (t_{0m})_{m=1}^{+∞} and (t_{m}^{0})_{m=1}^{+∞} be arbitrary sequences of real numbers such that

\begin{matrix}t_{0m} < t_{m}^{0}, t_{0m} ≤ t_{i} ≤ t_{m}^{0} ( i ∈ N_{0}(t_{1}, …, t_{n}); m = 1, 2, …), \\ lim_{m→+∞} t_{0m} = −∞, lim_{m→+∞} t_{m}^{0} = +∞. \end{matrix} (2.1)

For any natural number m and arbitrary functions u ∈ C_{loc}(R, R) and h ∈ L_{loc}(R, R) set

t_{im} = \begin{cases}t_{i} & for t_{i} ∈ R \\ t_{0m} & for t_{i} = −∞ \\ t_{m}^{0} & for t_{i} = +∞ \end{cases} ,  (2.2)

e_{m}(u)(t) = \begin{cases}u(t) & for t_{0m} ≤ t ≤ t_{m}^{0} \\ u(t_{0m}) & for t < t_{0m} \\ u(t_{m}^{0}) & for t > t_{m}^{0} \end{cases} ,  (2.3)

l_{ikm}(u)(t) = l_{ik}(e_{m}(u))(t) (i, k = 1, …, n)  (2.4)

and

ν_{im}(h) = max \{| ∫_{t_{im}}^{t} exp ( ∫_{s}^{t} l_{ii}(1)(ξ) dξ)[ \overline{l}_{ii}(|η(|h|)(·, s)|)(s) +

+|h(s)|] ds| : t_{0m} ≤ t ≤ t_{m}^{0}}.  (2.5)

On the interval [t_{0m}, t_{m}^{0}] consider the boundary value problem

y′_{i}(t) = ∑_{k=1}^{n} l_{ikm}(y_{k})(t) + h_{i}(t) (i = 1, …, n),  (2.6_{m})

\begin{matrix}y_{i}(t_{im}) = c_{i} for i ∈ N_{0}(t_{1}, …, t_{n}), \\ y_{i}(t_{im}) = 0 for i ∈ \{1, …, n\} \ N_{0}(t_{1}, …, t_{n}). \end{matrix}  (2.7_{m})

Lemma 2.1. Let there exist a positive number ρ such that for any h_{i} ∈ L_{loc}(R, R) (i = 1, …, n), c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) and natural m every solution (y_{i})_{i=1}^{n} of problem (2.6_{m}), (2.7_{m}) admits the estimate

∑_{i=1}^{n} |y_{i}(t)| ≤ ρ ∑_{i=1}^{n} (|c_{i}| + ν_{im}(h_{i})) for t_{0m} ≤ t ≤ t_{m}^{0},  (2.8)

where c_{i} = 0 as i ∈ {1, …, n} \ N_{0}(t_{1}, …, t_{n}). Moreover, let conditions (1.7) hold. Then for any c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) system (1.1) has at least one bounded solution satisfying conditions (1.9).

Proof. If c_{i} = 0 and h_{i}(t) ≡ 0 (i = 1, …, n), then (2.8) implies that y_{i}(t) ≡ 0 (i = 1, …, n), i.e., the homogeneous problem

y′_{i}(t) = ∑_{k=1}^{n} l_{ikm}(y_{k})(t) (i = 1, …, n),

y_{i}(t_{im}) = 0 (i = 1, …, n)

has only the trivial solution. On the other hand, by (1.2), (2.3) and (2.4) for any u ∈ C([t_{0m}, t_{m}^{0}], R) the inequalities

|l_{ikm}(u)(t)| ≤ \overline{l}_{ik}(1)(t)∥u∥ (i, k = 1, …, n)  (2.9)

hold almost everywhere on [t_{0m}, t_{m}^{0}], where ∥u∥ = max{|u(t)| : t_{0m} ≤ t ≤ t_{m}^{0}}. These facts imply that for any h_{i} ∈ L_{loc}(R, R), c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) and natural m the boundary value problem has one and only one solution (see [3], Theorem 1.1).

For arbitrarily fixed c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) and natural m denote by (x_{im})_{i=1}^{m} the solution of the problem

x′_{im}(t) = ∑_{k=1}^{n} l_{ikm}(x_{km})(t) + q_{i}(t) (i = 1, …, n),  (2.10)
x_{im}(t_{im}) = c_{i} (i = 1, …, n),  (2.11)

where

c_{i} = 0 as i ∈ {1, …, n} \ N_{0}(t_{1}, …, t_{n}),

and extend x_{im} (i = 1, …, n) on R by the equalities

x_{im}(t) = e_{m}(x_{im})(t) for t ∈ R (i = 1, …, n).  (2.12)

Then according to (1.7), (2.5) and (2.8) we have

∑_{i=1}^{n} |x_{im}(t)| ≤ ρ ∑_{i=1}^{n} (|c_{i}| + ν_{im}(q_{i})) ≤ ρ^{∗} for t ∈ R (m = 1, 2, …), (2.13)

where ρ^{∗} = ρ(∑_{i=1}^{n} |c_{i}| + a) is a nonnegative number independent of m.

By virtue of (2.9) and (2.13) from (2.10) we obtain

∑_{i=1}^{n} |x′_{im}(t)| ≤ q(t) for almost all t ∈ R (m = 1, 2, …),

where

q(t) = ∑_{i=1}^{n} [ρ^{∗} ∑_{k=1}^{n} \overline{l}_{ik}(1)(t) + |q_{i}(t)|]

and q ∈ L_{loc}(R, R). Consequently, the sequences (x_{im})_{m=1}^{+∞} (i=1, …, n) are uniformly bounded and equicontinuous on every compact interval. Without loss of generality, by Arzela–Ascoli’s lemma we can assume that (x_{im})_{m=1}^{+∞} (i = 1, …, n) are uniformly convergent on every compact interval. Put

lim_{m→+∞} x_{im}(t) = x_{i}(t) for t ∈ R (i = 1, …, n).  (2.14)

Then by (2.1), (2.3), and (2.12)

lim_{m→+∞} e_{m}(x_{im})(t) = x_{i}(t)

uniformly on every compact interval (i = 1, …, n).  (2.15)

Let m_{0} be a natural number such that

t_{0m} < 0 < t_{m}^{0} (m = m_{0}, m_{0} + 1, …).

Then by (2.10) we have

x_{im}(t) = x_{im}(0) + ∫_{0}^{t} [ ∑_{k=1}^{n} l_{ikm}(x_{km})(s) + q_{i}(s)] ds

for t_{0m} ≤ t ≤ t_{m}^{0} (i = 1, …, n).

According to conditions (2.1), (2.4), (2.15) and the continuity of the operators l_{ik} : C_{loc}(R, R) → L_{loc}(R, R) (i, k = 1, …, n) these equalities imply that

x_{i}(t) = x_{i}(0) + ∫_{0}^{t} [ ∑_{k=1}^{n} l_{ik}(x_{k})(s) + q_{i}(s)] ds for t ∈ R (i = 1, …, n),

i.e., (x_{i})_{i=1}^{n} is a solution of system (1.1). On the other hand, by virtue of (2.1), (2.2), and (2.14) from (2.11) and (2.13) we conclude that the vector function (x_{i})_{i=1}^{n} is bounded and satisfies conditions (1.9).

3. Proof of the main results

Along with the notation introduced in Section 1, we shall also use some additional notation.

Z^{−1} is the matrix, inverse to the nonsingular n × n matrix Z.

E is the n × n unit matrix.

If Z = (z_{ik})_{i,k=1}^{n}, then ∥Z∥ = ∑_{i,k=1}^{n} |z_{ik}|.

The inequalities between the real column vectors z = (z_{i})_{i=1}^{n} and \overline{z} = (\overline{z}_{i})_{i=1}^{n} are understood componentwise, i.e.,

z ≤ \overline{z} ⇔ z_{i} ≤ \overline{z}_{i} (i = 1, …, n).

Proof of Theorem 1.1. By (1.8) there exists a constant ρ_{0} > 1 such that

exp ( ∫_{t_{i}}^{t} p_{ii}(s)ds) < ρ_{0} for t ∈ R (i ∈ N_{0}(t_{i}, …, t_{n})). (3.1)

On the other hand, by (1.5) the matrix E − A is nonsingular and its inverse matrix (E − A)^{−1} is nonnegative. Put

ρ = ρ_{0}∥(E − A)^{−1}∥.  (3.2)

Let (t_{0m})_{m=1}^{∞} and (t_{m}^{0})_{m=1}^{∞} be arbitrary sequences of real numbers satisfying conditions (2.1) and t_{im}, e_{m}, l_{ikm} and ν_{im} (i, k = 1, 2; m = 1, 2, …) are the numbers and operators given by equalities (2.2)–(2.5). By Lemma 2.1, to prove Theorem 1.1 it is sufficient to show that for any h_{i} ∈ L_{loc}(R, R)

(i = 1, …, n), c_{i} ∈ R (i ∈ N_{0}(t_{1}, …, t_{n})) and natural m an arbitrary solution (y_{i})_{i=1}^{n} of problem (2.6_{m}), (2.7_{m}) admits estimate (2.8), where c_{i} = 0 as i ∈ {1, …, n} \ N_{0}(t_{1}, …, t_{n}).

By (1.4) and (2.4), equation (2.6_{m}) implies

l_{iim}(y_{i})(t) = l_{iim}(1)(t)y_{i}(t) + l_{iim}(y_{i}(·) − y_{i}(t))(t) =
= l_{ii}(1)(t)y_{i}(t) + l_{iim}(η(y′_{i})(·, t))(t) =
= l_{ii}(1)(t)y_{i}(t) + ∑_{k=1}^{m} l_{iim}(η(l_{ikm}(y_{k}))(·, t))(t) +
+ l_{iim}(η(h_{i})(·, t))(t),
y′_{i}(t) = l_{ii}(1)(t)y_{i}(t) + h̃_{i}(t) (i = 1, …, n),

and

y_{i}(t) =c_{i} exp ( ∫_{t_{im}}^{t} l_{ii}(1)(ξ)dξ) +

+ ∫_{t_{im}}^{t} exp ( ∫_{s}^{t} l_{ii}(1)(ξ)dξ)h̃_{i}(s)ds (i = 1, …, n),  (3.3)

where

h̃_{i}(t) = ∑_{k=1}^{m} [l_{iim}(η(l_{ikm}(y_{k}))(·, t))(t) + (1 − δ_{ik})l_{ikm}(y_{k})(t)] +
+ l_{iim}(η(h_{i})(·, t))(t) + h_{i}(t) (i = 1, …, n).

Set

γ_{i} = max {|y_{i}(t)| : t_{0m} ≤ t ≤ t_{m}^{0}}, γ = (γ_{i})_{i=1}^{n}.  (3.4)

Then according to (1.2), (1.4), (2.3), and (2.4) we obtain

|q̃_{i}(t)| ≤ ∑_{k=1}^{m} [\overline{l}_{ii}(|η(\overline{l}_{ik}(1))(·, t)|)(t) + (1 − δ_{ik})\overline{l}_{ik}(1)(t)]γ_{k} +
+ \overline{l}_{ii}(|η(|h_{i}|)(·, t)|)(t) + |h_{i}(t)| (i = 1, …, n).

If along with these inequalities we take into account conditions (1.6) and (3.1) and notation (2.5), then from (3.3) we find

γ_{i} ≤ ∑_{k=1}^{n} a_{ik}γ_{k} + ρ_{0}|c_{i}| + ν_{im}(h_{i}) ≤ ∑_{k=1}^{n} a_{ik}γ_{k} + ρ_{0}(|c_{i}| + ν_{im}(h_{i})),

i.e.,

(E − A)γ ≤ ρ_{0}(|c_{i}| + ν_{im}(h_{i}))_{i=1}^{n}.

But, as mentioned above, the matrix E − A is nonsingular and (E − A)^{−1} is nonnegative. Therefore the last inequality implies that

γ ≤ ρ_{0}(E − A)^{−1}(|c_{i}| + ν_{im}(h_{i}))_{i=1}^{n}.

Hence by (3.2) and (3.4) we obtain estimate (2.8).

Proof of Theorem 1.2. By Theorem 1.1, system (1.1) has at least one bounded solution satisfying conditions (1.9). Consequently, to prove Theorem 1.2 it is sufficient to show that the homogeneous problem

x′_{i}(t) = ∑_{k=1}^{n} l_{ik}(x_{k}) (i = 1, …, n),  (3.5)
x_{i}(t_{i}) = 0 for i ∈ N_{0}(t_{1}, …, t_{n})  (3.6)

has no nontrivial bounded solution.

Let (x_{i})_{i=1}^{n} be a bounded solution of problem (3.5), (3.6) and

γ_{i} = sup {|x_{i}(t)| : t ∈ R}, γ = (γ_{i})_{i=1}^{n}.

Then

l_{ii}(x_{i})(t) = l_{ii}(1)(t)x_{i}(t) + l_{ii}(x_{i}(·) − x_{i}(t))(t) =

= l_{ii}(1)(t)x_{i}(t) + l_{ii}(η(x′_{i})(·, t))(t) =

= l_{ii}(1)(t)x_{i}(t) + ∑_{k=1}^{n} l_{ii}(η(l_{ik}(x_{k}))(·, t))(t)

and

x′_{i}(t) = l_{ii}(1)(t)x_{i}(t) + Δ_{i}(t) (i = 1, …, n),  (3.7)

where

Δ_{i}(t) = ∑_{k=1}^{n} [l_{ii}(η(l_{ik}(x_{k}))(·, t))(t) + (1 − δ_{ik})l_{ik}(x_{k})(t)] (i = 1, …, n)

and

|Δ_{i}(t)| ≤ ∑_{k=1}^{n} [\overline{l}_{ii}(|η(l_{ik}(1))(·, t)|)(t) + (1 − δ_{ik})\overline{l}_{ik}(1)(t)]γ_{k}

(i = 1, …, n).  (3.8)

By (1.10) there exist t_{im} ∈ R (i ∈ {1, …, n} \ N_{0}(t_{1}, …, t_{n}); m = 1, 2, …) such that

lim_{m→+∞} t_{im} = t_{i}, lim_{m→+∞} ∫_{t_{im}}^{0} l_{ii}(1)(s)ds = −∞

for i ∈ {1, …, n} \ N_{0}(t_{1}, …, t_{n}).  (3.9)

Set

t_{im} = t_{i} (i ∈ N_{0}(t_{1}, …, t_{n}); m = 1, 2, …).  (3.10)

From (3.7) we have

x_{i}(t) = x_{i}(t_{im}) exp ( ∫_{t_{im}}^{t} l_{ii}(1)(ξ)dξ) +

+ ∫_{t_{im}}^{t} exp ( ∫_{s}^{t} l_{ii}(1)(ξ)dξ)Δ_{i}(s)ds (i = 1, …, n).

Hence by virtue of conditions (3.6) and (3.8)–(3.10) we find

x_{i}(t) = ∫_{t_{i}}^{t} exp ( ∫_{s}^{t} l_{ii}(1)(ξ)dξ)Δ_{i}(s)ds (i = 1, …, n).

These equalities and conditions (1.6), (3.6) and (3.8)–(3.10) yield

γ_{i} ≤ ∑_{k=1}^{n} a_{ik}γ_{k} (i = 1, …, n),

i.e.,

(E − A)γ ≤ 0.

Hence the nonnegativity of the matrix (E − A)^{−1} and vector γ implies that γ = 0, i.e., x_{i}(t) ≡ 0 (i = 1, …, n).

If

l_{ik}(u)(t) ≡ p_{ik}(t)u(τ_{ik}(t)) (i, k = 1, …, n),

then system (1.1) admits form (1.3). In that case

\overline{l}_{ik}(u)(t) ≡ |p_{ik}(t)|u(τ_{ik}(t)) (i, k = 1, …, n),

\overline{l}_{ii}(|η(\overline{l}_{ik}(1))(·, t)|)(t) ≡ |p_{ii}(t) ∫_{t}^{τ_{ii}(t)}|p_{ik}(ξ)|dξ| (i, k = 1, …, n)

and

\overline{l}_{ii}(|η(|q_{i}|)(·, t)|)(t) ≡ |p_{ii}(t) ∫_{t}^{τ_{ii}(t)}|q_{i}(ξ)|dξ| (i = 1, …, n)

and conditions (1.6)–(1.8) and (1.10) take the form of (1.11)–(1.13) and (1.14). Theorems 1.1 and 1.2 (Theorems 1.1′ and 1.2′) give rise to Corollaries 1.1 and 1.2 (Corollaries 1.1′ and 1.2′).

Finally, note that if the conditions of Corollary 1.3 (Corollary 1.3′) hold, then the conditions of Corollary 1.2 (Corollary 1.2′) hold too.^{∗}

References

1. I. Kiguradze, Boundary value problems for systems of ordinary differential equations. J. Soviet Math. 43(1988), No. 2, 2259–2339.

2. I. Kiguradze, Initial and boundary value problems for systems of ordinary differential equations, I. (Russian) Metsniereba, Tbilisi, 1997.

3. I. Kiguradze and B. Půža, On boundary value problems for systems of linear functional differential equations. Czechoslovak Math. J. 47(1997), No. 2, 341–373.

(Received 1.05.1998)

Author’s address:

Department of Mathematical Analysis

Masaryk University

Janáčkovo nám. 2a, 66295 Brno

Gzech Republic

∗See [2], the proof of Corollary 6.11.