Volume 8 (2001), Number 4, 645–664

ON THE *ξ-EXPONENTIALLY ASYMPTOTIC STABILITY OF*
LINEAR SYSTEMS OF GENERALIZED ORDINARY

DIFFERENTIAL EQUATIONS

M. ASHORDIA AND N. KEKELIA

Abstract. Necessary and sufficient conditions and effective sufficient condi-
tions are established for the so-called*ξ-exponentially asymptotic stability of*
the linear system

*dx(t) =**dA(t)**·**x(t) +**df*(t),

where *A* : [0,+∞[*→* R* ^{n×n}* and

*f*: [0,+∞[

*→*R

*are respectively matrix- and vector-functions with bounded variation components, on every closed interval from [0,+∞[ and*

^{n}*ξ*: [0,+∞[

*→*[0,+∞[ is a nondecreasing function such that lim

*t→+∞**ξ(t) = +∞.*

2000 Mathematics Subject Classification: 34B05.

Key words and phrases: *ξ-exponentially asymptotic stablility in the Lya-*
punov sense, linear generalized ordinary differential equations, Lebesgue–

Stiltjes integral.

Let the components of matrix-functions *A* : [0,+∞[*→* R* ^{n×n}* and vector-
functions

*f*: [0,+∞[

*→*R

*have bounded total variations on every closed seg- ment from [0,+∞[ .*

^{n}In this paper, sufficient (necessary and sufficient) conditions are given for
the so-called*ξ-exponentially asymptotic stability in the Lyapunov sense for the*
linear system of generalized ordinary differential equations

*dx(t) =dA(t)·x(t) +df*(t). (1)

The theory of generalized ordinary differential equations enables one to inves- tigate ordinary differential, difference and impulsive equations from the unified standpoint. Quite a number of questions of this theory have been studied suf- ficiently well ([1]–[3], [5], [6], [8], [10], [11]).

The stability theory has been investigated thoroughly for ordinary differential equations (see [4], [7] and the references therein). As to the questions of stability for impulsive equations and for generalized ordinary differential equations they are studied, e.g., in [3], [9], [10] (see also the references therein).

The following notation and definitions will be used in the paper:

R = ]*− ∞,*+∞[ , R_{+} = [0,+∞[ , [a, b] and ]a, b[ (a, b*∈*R) are, respectively,
closed and open intervals.

Re*z* is the real part of the complex number *z.*

ISSN 1072-947X / $8.00 / c*°*Heldermann Verlag *www.heldermann.de*

R* ^{n×m}* is the space of all real

*n×m-matrices*

*X*= (x

*)*

_{ij}

^{n,m}*with the norm*

_{i,j=1}*kXk*= max

*j=1,...,m*

X*n*
*i=1*

*|x*_{ij}*|,* *|X|*= (|x_{ij}*|)*^{n,m}_{i,j=1}*,*

R^{n×m}_{+} =^{n}*X* = (x*ij*)^{n,m}* _{i,j=1}* :

*x*

*ij*

*≥*0 (i= 1, . . . , n;

*j*= 1, . . . , m)

^{o}

*.*

The components of the matrix-function *X* are also denoted by [x]* _{ij}* (i =
1, . . . , n;

*j*= 1, . . . , m).

*O**n×m* (or *O) is the zero* *n×m-matrix.*

R* ^{n}* =R

*is the space of all real column*

^{n×1}*n-vectorsx*= (x

*)*

_{i}

^{n}*.*

_{i=1}If *X* *∈* R* ^{n×n}*, then

*X*

*and det(X) are, respectively, the matrix inverse to*

^{−1}*X*and the determinant of

*X.*

*I*

*is the identity*

_{n}*n×n-matrix; diag(λ*

_{1}

*, . . . , λ*

*) is the diagonal matrix with diagonal elements*

_{n}*λ*

_{1}

*, . . . , λ*

*.*

_{n}*r(H) is the spectral radius of the matrix* *H* *∈*R* ^{n×n}*.

+∞*∨*

0 (X) = sup

*b∈R*+

*∨**b*

0(X), where*∨*^{b}

0(X) is the sum of total variations on [0, b] of the
components *x** _{ij}* (i = 1, . . . , n;

*j*= 1, . . . , m) of the matrix-function

*X*: R

_{+}

*→*R

*;*

^{n×m}*V*(X)(t) = (v(x

*ij*)(t))

^{n,m}*, where*

_{i,j=1}*v(x*

*ij*)(0) = 0 and

*v(x*

*ij*)(t) =

*∨*

^{t}0(x*ij*)
for 0*< t <*+∞ (i= 1, . . . , n; *j* = 1, . . . , m).

*X(t−) and* *X(t+) are the left and the right limit of the matrix-function*
*X* :R+ *→*R* ^{n×m}*at the point

*t;d*1

*X(t) =X(t)−X(t−),d*2

*X(t) =*

*X(t+)−X(t).*

*BV** _{loc}*(R

_{+}

*,*R

*) is the set of all matrix-functions*

^{n×m}*X*:R

_{+}

*→*R

*of bounded total variation on every closed segment from R*

^{n×m}_{+}.

*L** _{loc}*(R

_{+}

*,*R

*) is the set of all matrix-functions*

^{n×m}*X*: R

_{+}

*→*R

*such that their components are measurable and integrable functions in the Lebesgue sense on every closed segment from R*

^{n×m}_{+}.

*C*e* _{loc}*(R

_{+}

*,*R

*) is the set of all matrix-functions*

^{n×m}*X*:R

_{+}

*→*R

*such that their components are absolutely continuous functions on every closed segment from R*

^{n×m}_{+}.

*s*0 :*BV**loc*(R+*,*R)*→BV**loc*(R+*,*R) is the operator defined by
*s*0(x)(t)*≡x(t)−* ^{X}

0<τ≤t

*d*1*x(τ)−* ^{X}

0≤τ <t

*d*2*x(τ).*

If *g* :R_{+} *→*R is a nondecreasing function *x*:R_{+}*→*R and 0*≤s < t <*+∞,
then

Z*t*

*s*

*x(τ)dg(τ*) =

Z

]s,t[

*x(τ*)*dg*_{1}(τ)*−*

Z

]s,t[

*x(τ*)*dg*_{2}(τ)

+ ^{X}

*s<τ**≤t*

*x(τ*)*d*1*g(τ*)*−* ^{X}

*s≤τ <t*

*x(τ*)*d*2*g(τ),*

where *g**j* :R+ *→*R (j = 1,2) are continuous nondecreasing functions such that
*g*1(t)*−g*2(t)*≡s*0(g)(t), and ^{R}

]s,t[

*x(τ*)*dg**j*(τ) is the Lebesgue–Stiltjes integral over

the open interval ]s, t[ with respect to the measure corresponding to the function
*g** _{j}* (j = 1,2) (if

*s*=

*t, then*

^{R}

^{t}*s* *x(τ)dg(τ*) = 0).

A matrix-function is said to be nondecreasing if each of its components is nondecreasing.

If *G* = (g* _{ik}*)

^{`,n}*: R*

_{i,k=1}_{+}

*→*R

*is a nondecreasing matrix-function,*

^{`×n}*X*= (x

*)*

_{ik}

^{n,m}*:R*

_{i,k=1}_{+}

*→*R

*, then*

^{n×m}Z*t*

*s*

*dG(τ*)*·X(τ*) =

µX_{n}

*k=1*

Z*t*

*s*

*x** _{kj}*(τ)

*dg*

*(τ)*

_{ik}¶_{`,m}

*i,j=1*

for 0*≤s* *≤t <*+∞,
*S*_{0}(G)(t)*≡*^{³}*s*_{0}(g* _{ik}*)(t)

^{´}

^{`,n}*i,k=1**.*

If *G** _{j}* : R

_{+}

*→*R

*(j = 1,2) are nondecreasing matrix-functions,*

^{`×n}*G(t)*

*≡*

*G*

_{1}(t)

*−G*

_{2}(t) and

*X*:R

_{+}

*→*R

*, then*

^{n×m}Z*t*

*s*

*dG(τ*)*·X(τ*) =

Z*t*

*s*

*dG*1(τ)*·X(τ)−*

Z*t*

*s*

*dG*2(τ)*·X(τ*) for 0*≤s≤t <*+∞.

*A* and *B* : *BV** _{loc}*(R

_{+}

*,*R

*)*

^{n×n}*×BV*

*(R*

_{loc}_{+}

*,*R

*)*

^{n×m}*→*

*BV*

*(R*

_{loc}_{+}

*,*R

*) are the operators defined, respectively, by*

^{n×m}*A(X, Y*)(t) = *Y*(t) + ^{X}

0<τ≤t

*d*_{1}*X(τ*)*·*^{³}*I*_{n}*−d*_{1}*X(τ)*^{´}^{−1}*d*_{1}*Y*(τ)

*−* ^{X}

0≤τ <t

*d*_{2}*X(τ*)*·*^{³}*I** _{n}*+

*d*

_{2}

*X(τ*)

^{´}

^{−1}*d*

_{2}

*Y*(τ) for

*t∈*R

_{+}and

*B(X, Y*)(t) = *X(t)Y*(t)*−X(0)Y*(0)*−*

Z*t*

0

*dX(τ)·Y*(τ) for *t* *∈*R_{+}*.*
*L*:*BV*_{loc}^{2} (R_{+}*,*R* ^{n×n}*)

*→BV*

*(R*

_{loc}_{+}

*,*R

*) is an operator given by*

^{n×n}*L(X, Y*)(t) =

Z*t*

0

*d*^{³}*X(τ) +B(X, Y*)(τ)^{´}*·X** ^{−1}*(τ) for

*t∈*R

_{+}

*.*We will use the following properties of these operators (see [2]):

*B*^{³}*X,B(Y, Z*)^{´}(t)*≡ B(XY, Z*)(t),
*B*

µ

*X,*

Z

0

*dY*(s)*·Z(s)*

¶

(t)*≡*

Z*t*

0

*dB(X, Y*)(s)*·Z(s).*

Under a solution of the system (1) we understand a vector-function *x* *∈*
*BV** _{loc}*(R

_{+}

*,*R

*) such that*

^{n}*x(t) =* *x(s) +*

Z*t*

*s*

*dA(τ*)*·x(τ*) +*f*(t)*−f(s) (0≤s≤t <*+∞).

Note that the linear system of ordinary differential equations
*dx*

*dt* =*P*(t)x+*q(t) (t∈*R_{+}), (2)

where *P* *∈L** _{loc}*(R

_{+}

*,*R

*) and*

^{n×n}*q∈L*

*(R*

_{loc}_{+}

*,*R

*), can be rewritten in form (1) if we set*

^{n}*A(t)≡*

Z*t*

0

*P*(τ)*dτ, f*(t)*≡*

Z*t*

0

*q(τ)dτ.*

We assume that *A∈BV** _{loc}*(R

_{+}

*,*R

*),*

^{n×n}*f*

*∈BV*

*(R*

_{loc}_{+}

*,*R

*),*

^{n}*A(0) =*

*O*

*and det*

_{n×n}^{³}

*I*

*+ (−1)*

_{n}

^{j}*d*

_{j}*A(t)*

^{´}

*6= 0 for*

*t∈*R

_{+}(j = 1,2).

These conditions guarantee the unique solvability of the Cauchy problem for system (1) (see [11]).

Definition 1. Let*ξ* :R_{+} *→*R_{+} be a nondecreasing function such that

*t→+∞*lim *ξ(t) = +∞.* (3)

Then the solution*x*_{0} of system (1) is called*ξ-exponentially asymptotic stable if*
there exists a positive number *η*such that for every*ε >*0 there exists a positive
number *δ*=*δ(ε) such that an arbitrary solution* *x*of system (1), satisfying the
inequality *kx(t*_{0})*−x*_{0}(t_{0})k*< δ* for some *t*_{0} *∈*R_{+}, admits the estimate

*kx(t)−x*_{0}(t)k*< ε*exp^{³}*−η*^{³}*ξ(t)−ξ(t*_{0})^{´´} for *t≥t*_{0}*.*

Stability, uniform stability, asymptotic stability and exponentially asymptotic
stability are defined just in the same way as for systems of ordinary differential
equations, i.e., when *A(t)* *≡* diag(t, . . . , t) (see, e.g., [4] or [7]). Note that the
exponentially asymptotic stability is a particular case of the *ξ-exponentially*
asymptotic stablility if we assume *ξ(t)≡t.*

Definition 2. System (1) is called stable in this or another sense if every solution of this system is stable in the same sense.

We will use the following propositions.

Proposition 1. *System* (1) *is* *ξ-exponentially asymptotically stable* (unifor-
*mly stable)* *if and only if its corresponding homogeneous system*

*dx(t) =* *dA(t)·x(t)* (1_{0})

*is* *ξ-exponentially asymptotically stable* (uniformly stable).

Proposition 2. *System*(1_{0})*isξ-exponentially asymptotically stable*(unifor-
*mly stable)if and only if its zero solution isξ-exponentially asymptotically stable*
(uniformly stable).

Proposition 3. *System*(1_{0})*isξ-exponentially asymptotically stable*(unifor-
*mly stable)* *if and only if there exist positive numbers* *ρ* *and* *η* *such that*

*kU*(t, s)k ≤*ρ*exp^{³}*−η*^{³}*ξ(t)−ξ(s)*^{´´} *for* *t≥s* *≥*0

³*kU*(t, s)k ≤*ρ* *for* *t* *≥s≥*0^{´}*,*
*where* *U* *is the Cauchy matrix of system* (1_{0}).

The proofs of these propositions are analogous to those for ordinary differen- tial equations.

Therefore, the *ξ-exponentially asymptotic stability (uniform stability) is not*
the property of a solution of system (1). It is the common property of all
solutions and a vector-function *f* does not influence on this property. Hence
the *ξ-exponentially asymptotic stability (uniform stability) is the property of*
the matrix-function *A* and the following definition is natural.

Definition 3. The matrix-function *A* is called *ξ-exponentially asymptoti-*
cally stable (uniformly stable) if the system (10) is *ξ-exponentially asymptoti-*
cally stable (uniformly stable).

Theorem 1. *Let the matrix-function* *A*_{0} *∈* *BV** _{loc}*(R

_{+}

*,*R

*)*

^{n×n}*be*

*ξ-exponenti-*

*ally asymptotically stable,*

det^{³}*I** _{n}*+ (−1)

^{j}*d*

_{j}*A*

_{0}(t)

^{´}

*6= 0*

*for*

*t*

*∈*R

_{+}(j = 1,2) (4)

*and*

*t→+∞*lim

*ν(ξ)(t)*_

*t*

*A(A*_{0}*, A−A*_{0}) = 0, (5)

*where* *ξ*:R+ *→*R+ *is a nondecreasing function satisfying condition* (3),
*ν(ξ)(t) = sup*^{n}*τ* *≥t* : *ξ(τ)≤ξ(t+) + 1*^{o}*.*

*Then the matrix-function* *A* *is* *ξ-exponentially asymptotically stable as well.*

To prove the theorem we will use the following lemma.

Lemma 1. *Let the matrix-function* *A*_{0} *∈* *BV** _{loc}*(R

_{+}

*,*R

*)*

^{n×n}*satisfy condition*(4). Let, moreover, the following conditions hold:

(a) *the Cauchy matrix* *U*_{0} *of the system*

*dx(t) =dA*_{0}(t)*·x(t)* (6)

*satisfies the inequality*

*|U*_{0}(t, t_{0})| ≤Ω exp^{³}*−ξ(t) +ξ(t*_{0})^{´} (t*≥t*_{0}) (7)

*for some* *t*_{0} *∈* R_{+}*, where* Ω *∈* R^{n×n}_{+} *, and* *ξ* *is a function from* *BV** _{loc}*(R

_{+}

*,*R)

*satisfying*(3);

(b) *there exists a matrix* *H* *∈*R^{n×n}_{+} *such that*

*r(H)<*1 (8)

*and*

Z*t*

*t*0

exp^{³}*ξ(t)−ξ(τ*)^{´}*|U*0(t, τ)|*dV*^{³}*A(A*0*, A−A*0)^{´}(τ)*< H* *for* *t≥t*0*.* (9)
*Then an arbitrary solution* *x* *of system* (1) *admits an estimate*

*|x(t)| ≤R|x(t*0)|exp^{³}*−ξ(t) +ξ(t*0)^{´} *for* *t≥t*0*,* (10)
*where* *R*= (I*n**−H)** ^{−1}*Ω.

*Proof.* Let *A*= (a* _{ik}*)

^{n}*,*

_{i,k=1}*A*

_{0}= (a

_{0ik})

^{n}*,*

_{i,k=1}*U*

_{0}= (u

_{0ik})

^{n}*,*

_{i,k=1}*H*= (h

*)*

_{ik}

^{n}*, and*

_{i,k=1}*x*= (x

*)*

_{i}

^{n}*be an arbitrary solution of system (1*

_{i=1}_{0}).

According to the variation of constants formula and properties of the Cauchy
matrix *U*0 (see [11]) we have

*x(t) =U*_{0}(t, t_{0})x(t_{0}) +

Z*t*

*t*0

*U*_{0}(t, s)*d*^{³}*A(s)−A*_{0}(s)^{´}*·x(s)*

*−* ^{X}

*t*0*<s≤t*

*d*_{1}*U*_{0}(t, s)*·d*_{1}^{³}*A(s)−A*_{0}(s)^{´}*·x(s)*

+ ^{X}

*t*0*≤s<t*

*d*_{2}*U*_{0}(t, s)*·d*_{2}^{³}*A(s)−A*_{0}(s)^{´}*·x(s)*

=*U*_{0}(t, t_{0})x(t_{0}) +

Z*t*

*t*0

*U*_{0}(t, s)*d*^{³}*A(s)−A*_{0}(s)^{´}*·x(s)*

+ ^{X}

*t*0*<s≤t*

*U*0(t, s)*d*1*A(s)·*^{³}*I**n**−d*1*A*0(s)^{´}^{−1}*d*1

³*A(s)−A*0(s)^{´}*·x(s)*

*−* ^{X}

*t*0*≤s<t*

*U*_{0}(t, s)*d*_{2}*A(s)·*^{³}*I** _{n}*+

*d*

_{2}

*A*

_{0}(s)

^{´}

^{−1}*d*

_{2}

^{³}

*A(s)−A*

_{0}(s)

^{´}

*·x(s).*

Therefore

*x(t) =* *U*_{0}(t, t_{0})x(t_{0}) +

Z*t*

*t*0

*U*(t, τ)*dA(A*_{0}*, A−A*_{0})(τ)*·x(τ) for* *t≥t*_{0}*.* (11)
Let

*y** _{k}*(t) = max

^{n}exp

^{³}

*ξ(τ*)

*−ξ(t*

_{0})

^{´}

*· |x*

*(τ)|:*

_{k}*t*

_{0}

*≤τ*

*≤t*

^{o}

*,*

*y(t) =*

^{³}

*y*

*k*(t)

^{´}

^{n}*k=1**.*

Then

¯¯

¯¯

¯
X*n*
*j,k=1*

Z*t*

*t*0

*u*_{0ij}(t, τ)x* _{k}*(τ)

*d(b*

*)(τ)*

_{jk}¯¯

¯¯

¯*≤*

X*n*
*j,k=1*

Z*t*

*t*0

*|u*_{0ij}(t, τ)| |x* _{k}*(τ)|

*dv(b*

*)(τ)*

_{jk}*≤*

X*n*
*k,j=1*

Z*t*

*t*0

exp^{³}*−ξ(τ*) +*ξ(t*_{0})^{´}*|u*_{0ij}(t, τ)|*dv(b** _{jk}*)(τ)

*·y*

*(t) for*

_{k}*t≥t*

_{0}

*,*(i= 1, . . . , n),

where *b** _{jk}*(t)

*≡ A(a*

_{0jk}

*, a*

_{jk}*−a*

_{0jk})(t) (j, k = 1, . . . , n). From this and (11) we have

exp^{³}*ξ(t)−ξ(t*_{0})^{´}*· |x** _{i}*(t)| ≤

X*n*
*k=1*

exp^{³}*ξ(t)−ξ(t*_{0})^{´}*|u*_{0ik}(t, t_{0})| |x* _{k}*(t

_{0})|

+

X*n*
*k,j=1*

Z*t*

*t*0

exp^{³}*ξ(t)−ξ(τ*)^{´}*|u*0ij(t, τ)|*dv(b**jk*)(τ)*·y**k*(t)
for *t≥t*_{0}*,* (i= 1, . . . , n).

By this, (7) and (9) we obtain

*y(t)≤*Ω|x(t_{0})|+*Hy(t) for* *t≥t*_{0}*.*
Hence

(I_{n}*−H)y(t)≤*Ω|x(t_{0})| for *t≥t*_{0}*.* (12)
On the other hand, by (8) the matrix *I**n**−H* is nonsingular and the matrix
(I_{n}*−H)** ^{−1}* is nonnegative since

*H*is a nonnegative matrix. From this, (12) and the definition of

*y*we have

*y(t)≤*(I*n**−H)** ^{−1}*Ω|x(t0)| for

*t*

*≥t*0

and

*|x(t)| ≤*(I_{n}*−H)** ^{−1}*Ω|x(t

_{0})|exp

^{³}

*−ξ(t) +ξ(t*

_{0})

^{´}for

*t≥t*

_{0}

*.*Therefore estimate (10) is proved.

*Proof of Theorem* 1. By the *ξ-exponentially asymptotic stability of the matrix-*
function *A*_{0} and Proposition 3 there exist positive numbers *η* and *ρ*_{0} such that
the Cauchy matrix *U*_{0} of system (6) satisfies the estimate

*|U*0(t, τ)| ≤*R*0exp^{³}*−*2η^{³}*ξ(t)−ξ(τ*)^{´´} for *t≥τ* *≥*0, (13)
where *R*_{0} is an *n×n* matrix whose every component equals *ρ*_{0}.

Let

*ε* = (4nρ_{0})^{−1}^{³}exp(η)*−*1^{´}exp(−2η). (14)

By (5) there exists *t*^{∗}*∈*R_{+} such that

*ν(ξ)(t)*_

*t*

*A(A*_{0}*, A−A*_{0})*< ε* for *t≥t*^{∗}*.* (15)
On the other hand, by (13) we have

Z*t*

*t*0

exp^{³}*η*^{³}*ξ(t)−ξ(τ*)^{´´}*|U*_{0}(t, τ)|*dV*(B)(τ)*≤ J*(t) (t *≥t*_{0}) (16)
for every *t*_{0} *≥*0, where *B(t)≡ A(A*_{0}*, A−A*_{0})(t) and

*J*(t)*≡R*_{0}

Z*t*

*t*0

exp^{³}*−η*^{³}*ξ(t)−ξ(τ)*^{´´}*dV*(B)(τ).

Let *k(t) be the integer part of* *ξ(t)−ξ(t*_{0}) for every *t≥t*_{0},

*T** _{i}* =

^{n}

*τ*

*≥t*

_{0}:

*ξ(t*

_{0}) +

*i≤ξ(τ*)

*< ξ*(t

_{0}) +

*i*+ 1

^{o}(i= 0, . . . , k(t)), where

*k*

*=*

_{i}*k(t*

*) (i= 0, . . . , k(t)), the points*

_{i}*t*

_{0}

*, t*

_{1}

*, . . . , t*

*are defined by*

_{k(t)}*t*_{0} = sup*T*_{0}*, t** _{i}* =

*t** _{i−1}* if

*T*

*=∅*

_{i}sup*T** _{i}* if

*T*

_{i}*6=*∅ (i= 1, . . . , k(t)).

Let us show that

*t*_{i}*≤ν(ξ)(t** _{i−1}*) (i= 1, . . . , k(t)). (17)
If

*T*

*=∅, then (17) is evident.*

_{i}Let now *T*_{i}*6=*∅. It is sufficient to show that

*T*_{i}*⊂Q*_{i}*,* (18)

where

*Q** _{i}* =

^{n}

*τ*:

*ξ(τ*)

*< ξ(t*

*+) + 1*

_{i−1}^{o}

*.*It is easy to verify that

*ξ(t** _{i−1}*+)

*≥ξ(t*

_{0}) +

*i.*(19)

Indeed, otherwise there exists *δ >*0 such that

*ξ(t** _{i−1}*+

*s)< ξ*(t

_{0}) +

*i*for 0

*≤s≤δ.*

On the other hand, by the definition of *t** _{i−1}* we have

*ξ(t*

_{0}) +

*i−*1

*≤ξ(t*

_{i−1}*−)*and therefore

*ξ(t*_{0}) +*i−*1*≤ξ(t** _{i−1}*+

*s)< ξ*(t

_{0}) +

*i*for 0

*≤s≤δ.*

But this contradicts the definition of *t** _{i−1}*.

Let *τ* *∈T** _{i}*. Then from (19) and the inequality

*ξ(τ*)

*< ξ*(t

_{0}) +

*i*+ 1 it follows that

*ξ(τ)< ξ(t*

*+) + 1,*

_{i−1}*τ*

_{i}*∈Q*

*. Hence (17) is proved.*

_{i}Let *t*_{0} *≥t** ^{∗}*. Then according to (15) and (17) we get

*J*(t)

*≤R*

_{0}exp

^{³}

*−η*

^{³}

*ξ(t)−ξ(t*

_{0})

^{´´}

1+k(t)X

*i=1*
*t**i*

Z

*t**i−1*

exp^{³}*η*^{³}*ξ(τ*)*−ξ(t*_{0})^{´´}*dV*(B)(τ)

=*R*0exp^{³}*−η*^{³}*ξ(t)−ξ(t*0)^{´´}

Ã _{1+k(t)}
X

*i=1, i=1+k**i*

*t**i*

Z

*t**i−1*

exp^{³}*η*^{³}*ξ(τ*)*−ξ(t*0)^{´´}*dV*(B)(τ)

+

1+k(t)X

*i=1, i6=1+k**i*

*t**i*

Z

*t**i−1*

exp^{³}*η*^{³}*ξ(τ)−ξ(t*_{0})^{´´}*dV*(B)(τ)

!

*≤R*_{0}exp^{³}*−η*^{³}*ξ(t)−ξ(t*_{0})^{´´}

Ã _{1+k(t)}
X

*i=1, i=1+k**i*

exp(η i)^{h}*V*(B)(t* _{i}*)

*−V*(B)(t

*)*

_{i−1}^{i}

+

1+k(t)X

*i=1, i6=1+k**i*

exp(η i)^{h}*V*(B)(t* _{i}*)

*−V*(B)(t

*)*

_{i−1}^{i}+

1+k(t)X

*i=1, i6=1+k**i*

exp^{³}(1 +*k** _{i}*)η

^{´}

*d*

_{1}

*B(t*

*)*

_{i}!

*≤εR*_{0}exp^{³}*−η*^{³}*ξ(t)−ξ(t*_{0})^{´´µ}

1+k(t)X

*i=1*

exp(η i) +

1+k(t)X

*i=1, i6=1+k**i*

exp^{³}(1 +*k** _{i}*)η

^{´¶}

*≤*2εR_{0}exp^{³}*−η*^{³}*ξ(t)−ξ(t*_{0})^{´´}

1+k(t)X

*i=1*

exp(η i)

= 2εR_{0}exp^{³}*−η*^{³}*ξ(t)−ξ(t*_{0})^{´´}exp(η)

µ

exp^{³}(1 +*k(t))η*^{´}*−*1^{¶³}exp(η)*−*1^{´}^{−1}

*≤*2εR_{0}exp^{³}*−ηk(t)*^{´}exp^{³}(2 +*k(t))η*^{´³}exp(η)*−*1^{´}

= 2εR_{0}exp(2η)^{³}exp(η)*−*1^{´}^{−1}*.*

From (14), (16) and (20) it follows that inequality (9) holds for *t*_{0} *≥* *t** ^{∗}*, where

*H*

*∈*R

*is the matrix whose every component equals*

^{n×n}_{2n}

^{1}. On the other hand, it can be easily shown that

*r(H)<* 1
2*.*

Consequently, by Lemma 1 an arbitrary solution *x* of the system (10) admits
an estimate

*kx(t)k ≤ρ*exp^{³}*−η*^{³}*ξ(t)−ξ(t*_{0})^{´´} for *t≥t*_{0} *≥t*^{∗}*,*
where *ρ >*0 is a constant independent of *t*_{0}.

Note that a similar theorem is proved in [7] for the case of ordinary differential equations.

Corollary 1. *Let the components* *a** _{ik}* (i, k= 1, . . . , n)

*of the matrix-function*

*A*

*satisfy the conditions*

1 + (−1)^{j}*d*_{j}*a** _{ii}*(t)

*6= 0*

*for*

*t∈*R

_{+}(i= 1, . . . , n;

*j*= 1,2), (20)

*t→+∞*lim

*ν(ξ)(t)*_

*t*

*A(a*_{ii}*, a** _{ik}*) = 0 (i, k = 1, . . . , n), (21)

*and*

*a** _{ii}*(t)

*−a*

*(τ)*

_{ii}*≤ −η*

^{³}

*ξ(t)−ξ(τ*)

^{´}

*for*

*t≥τ*

*≥*0 (i= 1, . . . , n), (22)

*where*

*η >*0,

*ξ*: R

_{+}

*→*R

_{+}

*is a nondecreasing function satisfying condition*(3), and

*ν(ξ) :*R

_{+}

*→*R

_{+}

*is the function defined as in Theorem*1. Then the

*matrix-function*

*A*

*is*

*ξ-exponentially asymptotically stable.*

*Proof.* Corollary 1 follows from Theorem 1 if we assume that
*A*_{0}(t)*≡*diag^{³}*a*_{11}(t), . . . , a* _{nn}*(t)

^{´}

*.*Indeed, by the definition of the operator

*A*we have

h*A(A*0*, A−A*0)(t)^{i}

*ik* =*a**ik*(t) + ^{X}

0<τ≤t

*d*1*a**ii*(τ)

1*−d*_{1}*a** _{ii}*(τ)

*d*1

*a*

*ik*(τ)

*−* ^{X}

0≤τ <t

*d*_{2}*a** _{ii}*(τ)

1 +*d*_{2}*a** _{ii}*(τ)

*d*2

*a*

*ik*(τ) =

*A(a*

*ii*

*, a*

*ik*)(t) for

*t*

*∈*R

_{+}(i

*6=k;*

*i, k*= 1, . . . , n)

and h

*A(A*0*, A−A*0)(t)^{i}

*ii* = 0 for *t∈*R+ (i= 1, . . . , n).

Therefore, by (21) and (22) the matrix-function *A*is*ξ-exponentially asymptot-*
ically stable.

Corollary 2. *Let the matrix-function* *P* *∈L** _{loc}*(R

_{+}

*,*R

*)*

^{n×n}*be*

*ξ-exponentially*

*asymptotically stable and*

*t→+∞*lim

*ξ(t)+1*_

*t*

(A*−A*_{0}) = 0 *for* *t∈*R_{+}*,*
*where* *A*0(t) *≡* ^{R}^{t}

0 *P*(τ)*dτ,* *ξ* : R+ *→* R+ *is a continuous nondecreasing func-*
*tion satisfying condition* (3). Then the matrix-function *A* *is* *ξ-exponentially*
*asymptotically stable as well.*

*Proof.* Corollary 2 immediately follows from Theorem 1 if we observe that
*A(A*_{0}*, A−A*_{0})(t) =*A(t)−A*_{0}(t) (t *∈*R_{+})

in this case and, moreover,

*ν(ξ)(t) =* *ξ(t) + 1 (t* *∈*R_{+})
because *ξ* is a nondecreasing continuous function.

Theorem 2. *The matrix-function* *A* *is* *ξ-exponentially asymptotically stable*
*if and only if there exist a positive number* *η* *and a nonsingular matrix-function*
*H* *∈BV** _{loc}*(R

_{+}

*,*R

*)*

^{n×n}*such that*

sup^{n}*kH** ^{−1}*(t)H(s)k:

*t*

*≥s≥*0

^{o}

*<*+∞ (23)

*and*

+∞_

0

*B** _{η}*(H, A)

*<*+∞, (24)

*where*

*B** _{η}*(H, A)(t)

*≡*

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*d*

"

exp^{³}*ηξ(τ*)^{´}*H(τ*)

+ exp^{³}*ηξ(τ*)^{´}*H(τ*)A(τ)*−*

Z*τ*

0

*d*^{³}exp^{³}*ηξ(s)*^{´}*H(s)*^{´}*·A(s)*

#

*.* (25)

*Proof.* Let*U* and *U** ^{∗}* be the Cauchy matrices of systems (1

_{0}) and

*dy(t) =*

*dA*

*(t)*

^{∗}*·y(t),*

respectively, where *A** ^{∗}*(t) =

*L(exp(ηξ(·))H, A)(t). Then by the definition of the*operator

*L*and by the equality

*U(t, s) = exp*^{³}*−η*^{³}*ξ(t)−ξ(s)*^{´´}*H** ^{−1}*(t)U

*(t, s)H(s) for*

^{∗}*t, s∈*R

_{+}we obtain that

exp^{³}*η*^{³}*ξ(τ)−ξ(s)*^{´´}*U*(t, s) =*H** ^{−1}*(t)H(s)
+H

*(t)*

^{−1}Z*t*

*s*

exp^{³}*η*^{³}*ξ(τ)−ξ(s)*^{´´}*dB** _{η}*(H, A)(τ)

*·U*(τ, s) for

*t, s*

*∈*R

_{+}

*.*Hence

*W*(t, s) =*H** ^{−1}*(t)H(s) +

*H*

*(t)d*

^{−1}_{1}

*B*

*(H, A)(t)*

_{η}*·W*(t, s) +

*H*

*(t)*

^{−1}Z*t*

*s*

*dG(τ*)*·W*(τ, s) for *t, s∈*R_{+}*,* (26)

where

*W*(t, s) = exp^{³}*η*^{³}*ξ(t)−ξ(s)*^{´´}*U(t, s), G(t) =* *B** _{η}*(H, A)(t−).

On the other hand by (23), (24) and by the equalities

det^{³}*I** _{n}*+ (−1)

^{j}*d*

_{j}*A*

*(t)*

^{∗}^{´}= exp

^{³}(−1)

^{j}*nη d*

_{j}*ξ(t)*

^{´}det

^{³}

*H(t) + (−1)*

^{j}*d*

_{j}*H(t)*

^{´}

*×*det^{³}*I** _{n}*+ (−1)

^{j}*d*

_{j}*A(t)*

^{´}det

^{³}

*H*

*(t)*

^{−1}^{´}for

*t∈*R

_{+}(j = 1,2) and

*I** _{n}*+ (−1)

^{j}*H*

*(t)*

^{−1}*d*

_{j}*B*

*(H, A)(t)*

_{η}=*H** ^{−1}*(t)

^{³}

*I*

*+ (−1)*

_{n}

^{j}*d*

_{j}*A*

*(t)*

^{∗}^{´}

*H(t) for*

*t∈*R

_{+}(j = 1,2) there exists a positive number

*r*

_{0}such that

det^{³}*I** _{n}*+ (−1)

^{j}*H*

*(t)*

^{−1}*d*

_{j}*B*

*(H, A)(t)*

_{η}^{´}

*6= 0 for*

*t*

*∈*R

_{+}(j = 1,2) (27) and °

°°

°

³*I** _{n}*+ (−1)

^{j}*H*

*(t)*

^{−1}*d*

_{j}*B*

*(H, A)(t)*

_{η}^{´}

^{−1}°°

°° *< r*_{0} for *t* *∈*R_{+} (j = 1,2). (28)
From (26), by (23), (27) and (28) we get

*kW*(t, s)k ≤*r*_{0}

Ã

*ρ*+*ρ*_{1}

Z*t*

*s*

*kW*(τ, s)k*dkV*(G)(τ)k

!

for *t* *≥s≥*0,
where

*ρ*= sup^{n}*kH** ^{−1}*(t)H(s)k:

*t*

*≥s*

^{o}

*, ρ*

_{1}=

*ρkH*

*(0)k.*

^{−1}Hence, according to the Gronwall inequality ([11])

*kW*(t, s)k ≤*M <* +∞ for *t≥s* *≥*0,
where

*M* =*r*_{0}exp

µ

*r*_{0}*ρ*_{1}

+∞_

0

*B** _{η}*(H, A)

¶

*.*
Therefore

*kU*(t, s)k ≤*M*exp^{³}*−η*^{³}*ξ(t)−ξ(s)*^{´´} for *t≥s* *≥*0,
i.e., the matrix-function *A* is *ξ-exponentially asymptotically stable.*

Let us show the necessity. Let the matrix-function *A* is *ξ-exponentially*
asymptotically stable. Then there exist positive numbers *η* and *ρ* such that

*kZ(t)Z** ^{−1}*(s)k ≤

*ρ*exp

^{³}

*−η*

^{³}

*ξ(t)−ξ(s)*

^{´´}for

*t≥s*

*≥*0, (29) where

*Z*(Z(0) =

*I*

*n*) is the fundamental matrix of system (10).

Let

*H(t)≡*exp^{³}*−ηξ(t)*^{´}*Z** ^{−1}*(t).

Then according to (25), (29) and the equality
*Z** ^{−1}*(t) =

*I*

_{n}*−Z*

*(t)A(t) +*

^{−1}Z*t*

0

*dZ** ^{−1}*(τ)

*·A(τ*) for

*t∈*R

_{+}(30) (see [11]) we have

*kH** ^{−1}*(t)H(s)k=

*kZ(t)Z*

*(s)kexp*

^{−1}^{³}

*η*

^{³}

*ξ(t)−ξ(s)*

^{´´}

*≤ρ*for

*t≥s≥*0 and

*B** _{η}*(H, A)(t) =

*B*

_{η}^{³}exp(−ηξ)Z

^{−1}*, A*

^{´}(t) = 0 for

*t∈*R

_{+}

*.*Therefore conditions (23) and (24) are fulfilled.

*Remark* 1. If in Theorem 2 the function *ξ* : R_{+} *→* R_{+} is continuous, then
condition (24) can be rewritten as

°°

°°

°

+∞Z

0

*dV*^{³}*I(H, A) +η*diag(ξ, . . . , ξ)^{´}(t)*· |H(t)|*

°°

°°

°*<*+∞.

Corollary 3. *Let the matrix-function* *Q* *∈* *BV** _{loc}*(R

_{+}

*,*R

*)*

^{n×n}*be uniformly*

*stable and*

det^{³}*I** _{n}*+ (−1)

^{j}*d*

_{j}*Q(t)*

^{´}

*6= 0*

*for*

*t∈*R

_{+}(j = 1,2). (31)

*Let, moreover, there exist a positive number*

*η*

*such that*

°°

°°

°

+∞Z

0

*|Z** ^{−1}*(t)|

*dV*

^{³}

*G*

*η*(ξ, Q, A)

^{´}(t)

°°

°°

°*<*+∞ (32)
*where* *Z* (Z(0) = *I** _{n}*)

*is the fundamental matrix of the system*

*dz(t) =dQ(t)·z(t),* (33)

*and*

*G** _{η}*(ξ, Q, A)(t)

*≡ A(Q, A−Q)(t) +ηs*

_{0}(ξ)(t)

*·I*

_{n}+ ^{X}

0<τ≤t

exp^{³}*−ηξ(τ)*^{´}*d*1exp^{³}*ηξ(τ)*^{´}*·*^{³}*I**n**−d*1*Q(τ*)^{´}^{−1}^{³}*I**n**−d*1*A(τ)*^{´}

+ ^{X}

0≤τ <t

exp^{³}*−ηξ(τ)*^{´}*d*_{2}exp^{³}*ηξ(τ)*^{´}*·*^{³}*I** _{n}*+

*d*

_{2}

*Q(τ)*

^{´}

^{−1}^{³}

*I*

*+*

_{n}*d*

_{2}

*A(τ*)

^{´}

*.*(34)

*Then the matrix-function*

*A*

*is*

*ξ-exponentially asymptotically stable.*

*Proof.* Let *B** _{η}*(H, A) be the matrix-function defined by (25), where

*H(t)*

*≡*

*Z*

*(t). Using the formula of integration by parts ([11]), the properties of the operator*

^{−1}*B*given above and equality (30), we conclude that

*B** _{η}*(H, A)(t)

=

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*d*

µ

exp(ηξ(τ))Z* ^{−1}*(τ) +

*B*

^{³}exp(ηξ)Z

^{−1}*, A*

^{´}(τ)

¶

=

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*d*

µ

exp(ηξ(τ))Z* ^{−1}*(τ)

¶

+

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*dB*

µ

exp(ηξ)I_{n}*,B*^{³}exp(ηξ)Z^{−1}*, A*^{´}(τ)

¶

=

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*d*^{³}exp(ηξ(τ))Z* ^{−1}*(τ)

^{´}

+

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*dB*^{³}exp(ηξ)I_{n}*,B(Z*^{−1}*, A)*^{´}(τ) for *t* *∈*R_{+}; (35)

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*d*^{³}exp(ηξ(τ))Z* ^{−1}*(τ)

^{´}

=

Z*t*

0

*Z** ^{−1}*(τ)

*d*

^{³}

*ηs*0(ξ)(τ)I

*n*

*− A(Q, Q)(τ*)

^{´}

+ ^{X}

0<s≤τ

exp^{³}*−ηξ(s)*^{´}*d*_{1}exp^{³}*ηξ(s)*^{´}*·*^{³}*I*_{n}*−d*_{1}*Q(s)*^{´}^{−1}

+ ^{X}

0≤s<τ

exp^{³}*−ηξ(s)*^{´}*d*_{2}exp^{³}*ηξ(s)*^{´}*·*^{³}*I** _{n}*+

*d*

_{2}

*Q(s)*

^{´}

*for*

^{−1}*t*

*∈*R

_{+}; (36)

*B(Z*^{−1}*, A)(t)≡*

Z*t*

0

*Z** ^{−1}*(τ)

*dA(τ*)

*−*

^{X}

0<τ≤t

*d*1*Z** ^{−1}*(τ)

*·d*1

*A(τ*)

+ ^{X}

0≤τ <t

*d*_{2}*Z** ^{−1}*(τ)

*·d*

_{2}

*A(τ*) =

Z*t*

0

*Z** ^{−1}*(τ)

*dA(Q, A−Q)(τ*) for

*t∈*R

_{+}

*,*(37)

*B*

^{³}exp(ηξ)I

_{n}*,B(Z*

^{−1}*, A)*

^{´}(t)

=

Z*t*

0

*Z** ^{−1}*(τ)

*dB*

^{³}exp(ηξ)I

*n*

*,A(Q, A)*

^{´}(τ) for

*t∈*R+ (38)

and

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*dB*^{³}exp(ηξ)I_{n}*,A(Q, A)*^{´}(τ)

=*A(Q, A)(t)−* ^{X}

0<τ≤t

exp^{³}*−ηξ(τ*)^{´}*d*_{1}exp^{³}*ηξ(τ*)^{´}*·*^{³}*I*_{n}*−d*_{1}*Q(τ)*^{´}^{−1}*d*_{1}*A(τ*)

+ ^{X}

0≤τ <t

exp^{³}*−ηξ(τ*)^{´}*d*_{2}exp^{³}*ηξ(τ*)^{´}*·*^{³}*I** _{n}*+

*d*

_{2}

*Q(τ*)

^{´}

^{−1}*d*

_{2}

*A(τ)*(39) for

*t∈*R+

*.*

From (35), by (36)–(39) we get
*B** _{η}*(H, A)(t) =

Z*t*

0

exp^{³}*−ηξ(τ)*^{´}*d*^{³}exp(ηξ(τ))*·Z** ^{−1}*(τ)

^{´}

+

Z*t*

0

*Z** ^{−1}*(τ)

*d*

ÃZ_{τ}

0

exp^{³}*−ηξ(s)*^{´}*dB*^{³}exp(ηξ)I_{n}*,A(Q, A)*^{´}(s)

!

=

Z*t*

0

*Z** ^{−1}*(τ)

*dG*

*(ξ, Q, A)(τ) for*

_{η}*t∈*R

_{+}and

+∞_

0

*B** _{η}*(H, A)

*≤*

°°

°°

°

+∞Z

0

*|Z** ^{−1}*(t)|

*dV*

^{³}

*G*

*(ξ, Q, A)*

_{η}^{´}(t)

°°

°°

°*.*

Therefore from (32) and the fact that the matrix-function *Q* is *ξ-exponen-*
tially asymptotically stable, it follows that the conditions of Theorem 2 are
fulfilled.

*Remark* 2. In Corollary 3 if the function *ξ*:R_{+} *→*R_{+} is continuous, then
*G** _{η}*(ξ, Q, A)(t) =

*A(Q, A−Q)(t) +ηξ(t)I*

*for*

_{n}*t*

*∈*R

_{+}

*.*

Corollary 4. *Let the matrix-function* *Q* *∈BV**loc*(R+*,*R* ^{n×n}*), satisfying con-

*dition*(31), be

*ξ-exponentially asymptotically stable and*

+∞_

0

*B(Z*^{−1}*, A−Q)<*+∞, (40)
*where* *Z* (Z(0) = *I** _{n}*)

*is the fundamental matrix of system*(33). Then the

*matrix-function*

*A*

*is*

*ξ-exponentially asymptotically stable as well.*

*Proof.* Since *Q* is *ξ-exponentially asymptotically stable there exists a positive*
number *η* such that the estimate (29) holds.

Let now *B** _{η}*(H, A) be the matrix-function defined by (25), where

*H(t)≡*exp

^{³}

*−ηξ(t)*

^{´}

*Z*

*(t).*

^{−1}Using equality (30) for the matrix-function *Q* we conclude that
*Z** ^{−1}*(t) =

*I*

*+*

_{n}*B(Z*

^{−1}*,−Q)(t) for*

*t*

*∈*R

_{+}and

*B**η*(H, A)(t) =

Z*t*

0

exp^{³}*−ηξ(τ*)^{´}*dB(Z*^{−1}*, A−Q)(τ*) for *t∈*R+*.*

By this and (40), condition (24) holds. Therefore, the conditions of Theorem 2 are fulfilled.

*Remark* 3. By the equality
*B(Z*^{−1}*, A−Q)(t) =*

Z*t*

0

*Z** ^{−1}*(τ)

*d*

^{³}

*A(τ*)

*−Q(τ*)

^{´}for

*t∈*R+

the condition

°°

°

+∞Z

0

*|Z** ^{−1}*(t)|

*dV*

^{³}

*A(Q, A−Q)*

^{´}(t)

°°

°°

° *<*+∞

guarantees the fulfilment of condition (40) in Corollary 4. On the other hand,

*η→0+*lim *G**η*(ξ, Q, A)(t) =*A(Q, A−Q)(t) for* *t∈*R+*,*

where*G**η*(ξ, Q, A)(t) is defined by (34). Consequently, Corollary 3 is true in the
limit case (η = 0), too, if we require the *ξ-exponentially asymptotic stability of*
*Q* instead of the uniform stability.

Corollary 5. *Let* *Q∈BV** _{loc}*(R

_{+}

*,*R

*)*

^{n×n}*be a continuous matrix-function sat-*

*isfying the Lappo-Danilevskiˇi condition*

Z*t*

0

*Q(τ*)*dQ(τ) =*

Z*t*

0

*dQ(τ)·Q(τ)* *for* *t∈*R+*.*

*Let, moreover, there exist a nonnegative number* *η* *such that*

°°

°°

°

+∞Z

0

¯¯

¯exp(−Q(t))^{¯}^{¯}¯*dV*(A*−Q*+*ηξI** _{n}*)(t)

°°

°°

° *<*+∞,

*where* *ξ*:R_{+} *→*R_{+} *is a continuous function satisfying condition* (3). Then:

(a) *the uniform stability of the matrix-function* *Q* *guarantees the* *ξ-exponen-*
*tially asymptotic stability of the matrix-function* *A* *for* *η >*0;

(b) *the* *ξ-exponentially asymptotic stability of* *Q* *guarantees the* *ξ-exponenti-*
*ally asymptotic stability of* *A* *for* *η* = 0.

*Proof.* The corollary follows immediatelly from Corollaries 3 and 4 and Remark
3 if we note that

*Z(t) = exp(Q(t)) for* *t* *∈*R_{+}
and in this case

*G** _{η}*(ξ, Q, A)(t) =

*A(t)−Q(t) +ηξ(t)I*

*for*

_{n}*t∈*R

_{+}

*.*