LINEAR SYSTEMS OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

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1 Georgian Mathematical Journal Volume 8 21), Number 4, ON THE ξ-exponentially ASYMPTOTIC STABILITY OF LINEAR SYSTEMS OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS M. ASHORDIA AND N. KEKELIA Abstract. Necessary sufficient conditions effective sufficient conditions are established for the so-called ξ-exponentially asymptotic stability of the linear system dxt) = dat) xt) + dft), where A : [, + [ R n n f : [, + [ R n are respectively matrix vector-functions with bounded variation components, on every closed interval from [, + [ ξ : [, + [ [, + [ is a nondecreasing function such that lim t + ξt) = +. 2 Mathematics Subject Classification: 34B5. Key words phrases: ξ-exponentially asymptotic stablility in the Lyapunov sense, linear generalized ordinary differential equations, Lebesgue Stiltjes integral. Let the components of matrix-functions A : [, + [ R n n vectorfunctions f : [, + [ R n have bounded total variations on every closed segment from [, + [. In this paper, sufficient necessary 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 dxt) = dat) xt) + dft). 1) The theory of generalized ordinary differential equations enables one to investigate ordinary differential, difference impulsive equations from the unified stpoint. Quite a number of questions of this theory have been studied sufficiently well [1] [3], [5], [6], [8], [1], [11]). The stability theory has been investigated thoroughly for ordinary differential equations see [4], [7] the references therein). As to the questions of stability for impulsive equations for generalized ordinary differential equations they are studied, e.g., in [3], [9], [1] see also the references therein). The following notation definitions will be used in the paper: R = ], + [, R + = [, + [, [a, b] ]a, b[ a, b R) are, respectively, closed open intervals. Re z is the real part of the complex number z. ISSN X / $8. / c Heldermann Verlag

2 646 M. ASHORDIA AND N. KEKELIA R n m is the space of all real n m-matrices X = x ij ) n,m i,j=1 with the norm X = max n j=1,...,m i=1 x ij, X = x ij ) n,m i,j=1, R n m + = { X = x ij ) n,m i,j=1 : x ij i = 1,..., n; j = 1,..., m) }. 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 n 1 is the space of all real column n-vectors x = x i ) n i=1. If X R n n, then X 1 detx) are, respectively, the matrix inverse to X the determinant of X. I n is the identity n n-matrix; diagλ 1,..., λ n ) is the diagonal matrix with diagonal elements λ 1,..., λ n. rh) is the spectral radius of the matrix H R n n. + b X) = sup X), where b X) is the sum of total variations on [, b] of the b R + components x ij i = 1,..., n; j = 1,..., m) of the matrix-function X : R + R n m ; V X)t) = vx ij )t)) n,m i,j=1, where vx ij )) = vx ij )t) = t x ij ) for < t < + i = 1,..., n; j = 1,..., m). Xt ) Xt+) are the left the right limit of the matrix-function X : R + R n m at the point t; d 1 Xt) = Xt) Xt ), d 2 Xt) = Xt+) Xt). BV loc R +, R n m ) is the set of all matrix-functions X : R + R n m of bounded total variation on every closed segment from R +. L loc R +, R n m ) is the set of all matrix-functions X : R + R n m such that their components are measurable integrable functions in the Lebesgue sense on every closed segment from R +. C loc R +, R n m ) is the set of all matrix-functions X : R + R n m such that their components are absolutely continuous functions on every closed segment from R +. s : BV loc R +, R) BV loc R +, R) is the operator defined by s x)t) xt) d 1 xτ) d 2 xτ). <τ t τ<t If g : R + R is a nondecreasing function x : R + R s < t < +, then t s xτ) dgτ) = ]s,t[ s<τ t xτ) dg 1 τ) ]s,t[ xτ) d 1 gτ) xτ) dg 2 τ) s τ<t xτ) d 2 gτ), where g j : R + R j = 1, 2) are continuous nondecreasing functions such that g 1 t) g 2 t) s g)t), xτ) dg j τ) is the Lebesgue Stiltjes integral over ]s,t[

3 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 647 the open interval ]s, t[ with respect to the measure corresponding to the function g j j = 1, 2) if s = t, then t xτ) dgτ) = ). s A matrix-function is said to be nondecreasing if each of its components is nondecreasing. If G = g ik ) l,n i,k=1 : R + R l n is a nondecreasing matrix-function, X = x ik ) n,m i,k=1 : R + R n m, then s n dgτ) Xτ) = k=1 s S G)t) s g ik )t) ) l,n i,k=1. ) l,m x kj τ) dg ik τ) i,j=1 for s t < +, If G j : R + R l n j = 1, 2) are nondecreasing matrix-functions, Gt) G 1 t) G 2 t) X : R + R n m, then s dgτ) Xτ) = s dg 1 τ) Xτ) s dg 2 τ) Xτ) for s t < +. A B : BV loc R +, R n n ) BV loc R +, R n m ) BV loc R +, R n m ) are the operators defined, respectively, by AX, Y )t) = Y t) τ<t <τ t BX, Y )t) = Xt)Y t) X)Y ) d 1 Xτ) I n d 1 Xτ) ) 1 d1 Y τ) d 2 Xτ) I n + d 2 Xτ) ) 1 d2 Y τ) for t R + dxτ) Y τ) for t R +. L : BV 2 locr +, R n n ) BV loc R +, R n n ) is an operator given by LX, Y )t) = d Xτ) + BX, Y )τ) ) X 1 τ) for t R +. We will use the following properties of these operators see [2]): B X, B X, BY, Z) ) t) BXY, Z)t), ) dy s) Zs) t) dbx, Y )s) Zs).

4 648 M. ASHORDIA AND N. KEKELIA Under a solution of the system 1) we underst a vector-function x BV loc R +, R n ) such that xt) = xs) + t s daτ) xτ) + ft) fs) s t < + ). Note that the linear system of ordinary differential equations dx dt = P t)x + qt) t R +), 2) where P L loc R +, R n n ) q L loc R +, R n ), can be rewritten in form 1) if we set At) P τ) dτ, ft) qτ) dτ. We assume that A BV loc R +, R n n ), f BV loc R +, R n ), A) = O n n det I n + 1) j d j At) ) 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 lim ξt) = +. 3) t + Then the solution x of system 1) is called ξ-exponentially asymptotic stable if there exists a positive number η such that for every ε > there exists a positive number δ = δε) such that an arbitrary solution x of system 1), satisfying the inequality xt ) x t ) < δ for some t R +, admits the estimate xt) x t) < ε exp η ξt) ξt ) )) for t t. Stability, uniform stability, asymptotic stability exponentially asymptotic stability are defined just in the same way as for systems of ordinary differential equations, i.e., when At) diagt,..., 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 ξt) 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 uniformly stable) if only if its corresponding homogeneous system dxt) = dat) xt) 1 ) is ξ-exponentially asymptotically stable uniformly stable).

5 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 649 Proposition 2. System 1 ) is ξ-exponentially asymptotically stable uniformly stable) if only if its zero solution is ξ-exponentially asymptotically stable uniformly stable). Proposition 3. System 1 ) is ξ-exponentially asymptotically stable uniformly stable) if only if there exist positive numbers ρ η such that Ut, s) ρ exp η ξt) ξs) )) for t s Ut, s) ρ for t s ), where U is the Cauchy matrix of system 1 ). The proofs of these propositions are analogous to those for ordinary differential 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 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 the following definition is natural. Definition 3. The matrix-function A is called ξ-exponentially asymptotically stable uniformly stable) if the system 1 ) is ξ-exponentially asymptotically stable uniformly stable). Theorem 1. Let the matrix-function A BV loc R +, R n n ) be ξ-exponentially asymptotically stable, det I n + 1) j d j A t) ) for t R + j = 1, 2) 4) νξ)t) lim t + t AA, A A ) =, 5) where ξ : R + R + is a nondecreasing function satisfying condition 3), νξ)t) = sup { τ t : ξτ) ξt+) + 1 }. 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 BV loc R +, R n n ) satisfy condition 4). Let, moreover, the following conditions hold: a) the Cauchy matrix U of the system satisfies the inequality dxt) = da t) xt) 6) U t, t ) Ω exp ξt) + ξt ) ) t t ) 7)

6 65 M. ASHORDIA AND N. KEKELIA for some t R +, where Ω R n n +, ξ is a function from BV loc R +, R) satisfying 3); b) there exists a matrix H R n n + such that t rh) < 1 8) ) exp ξt) ξτ) U t, τ) dv AA, A A ) ) τ) < H for t t. 9) Then an arbitrary solution x of system 1) admits an estimate where R = I n H) 1 Ω. xt) R xt ) exp ξt) + ξt ) ) for t t, 1) Proof. Let A = a ik ) n i,k=1, A = a ik ) n i,k=1, U = u ik ) n i,k=1, H = h ik ) n i,k=1, x = x i ) n i=1 be an arbitrary solution of system 1 ). According to the variation of constants formula properties of the Cauchy matrix U see [11]) we have xt) = U t, t )xt ) + Therefore Let t <s t t s<t t U t, s) d As) A s) ) xs) d 1 U t, s) d 1 As) A s) ) xs) d 2 U t, s) d 2 As) A s) ) xs) = U t, t )xt ) + t <s t t s<t xt) = U t, t )xt ) + t U t, s) d As) A s) ) xs) U t, s) d 1 As) I n d 1 A s) ) 1 d1 As) A s) ) xs) U t, s) d 2 As) I n + d 2 A s) ) 1 d2 As) A s) ) xs). t Ut, τ) daa, A A )τ) xτ) for t t. 11) y k t) = max { exp ξτ) ξt ) ) x k τ) : t τ t }, yt) = y k t) ) n k=1.

7 Then GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 651 n n u ij t, τ)x k τ) db jk )τ) u ij t, τ) x k τ) dvb jk )τ) j,k=1 t j,k=1 t n t exp ξτ) + ξt ) ) u ij t, τ) dvb jk )τ) y k t) k,j=1 t for t t, i = 1,..., n), where b jk t) Aa jk, a jk a jk )t) j, k = 1,..., n). From this 11) we have exp ξt) ξt ) ) n x i t) exp ξt) ξt ) ) u ik t, t ) x k t ) k=1 n t ) + exp ξt) ξτ) uij t, τ) dvb jk )τ) y k t) k,j=1 By this, 7) 9) we obtain Hence t for t t, i = 1,..., n). yt) Ω xt ) + Hyt) for t t. I n H)yt) Ω xt ) for t t. 12) On the other h, by 8) the matrix I n H is nonsingular the matrix I n H) 1 is nonnegative since H is a nonnegative matrix. From this, 12) the definition of y we have yt) I n H) 1 Ω xt ) for t t xt) I n H) 1 Ω xt ) exp ξt) + ξt ) ) for t t. Therefore estimate 1) is proved. Proof of Theorem 1. By the ξ-exponentially asymptotic stability of the matrixfunction A Proposition 3 there exist positive numbers η ρ such that the Cauchy matrix U of system 6) satisfies the estimate U t, τ) R exp 2η ξt) ξτ) )) for t τ, 13) where R is an n n matrix whose every component equals ρ. Let ε = 4nρ ) 1 expη) 1 ) exp 2η). 14)

8 652 M. ASHORDIA AND N. KEKELIA By 5) there exists t R + such that νξ)t) On the other h, by 13) we have t t AA, A A ) < ε for t t. 15) )) exp η ξt) ξτ) U t, τ) dv B)τ) J t) t t ) 16) for every t, where Bt) AA, A A )t) J t) R t )) exp η ξt) ξτ) dv B)τ). Let kt) be the integer part of ξt) ξt ) for every t t, T i = { τ t : ξt ) + i ξτ) < ξt ) + i + 1 } i =,..., kt)), where k i = kt i ) i =,..., kt)), the points t, t 1,..., t kt) are defined by t i 1 if T i = t = sup T, t i = i = 1,..., kt)). sup T i if T i Let us show that t i νξ)t i 1 ) i = 1,..., kt)). 17) If T i =, then 17) is evident. Let now T i. It is sufficient to show that where It is easy to verify that T i Q i, 18) Q i = { τ : ξτ) < ξt i 1 +) + 1 }. Indeed, otherwise there exists δ > such that ξt i 1 +) ξt ) + i. 19) ξt i 1 + s) < ξt ) + i for s δ. On the other h, by the definition of t i 1 we have therefore ξt ) + i 1 ξt i 1 ) ξt ) + i 1 ξt i 1 + s) < ξt ) + i for s δ. But this contradicts the definition of t i 1.

9 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 653 Let τ T i. Then from 19) the inequality ξτ) < ξt ) + i + 1 it follows that ξτ) < ξt i 1 +) + 1, τ i Q i. Hence 17) is proved. Let t t. Then according to 15) 17) we get J t) R exp η )) 1+kt) ξt) ξt ) i=1 = R exp η ξt) ξt ) )) 1+kt) + 1+kt) t i i=1, i 1+k i t i 1 R exp η ξt) ξt ) )) 1+kt) t i t i 1 t i i=1, i=1+k i t i 1 exp η ξτ) ξt ) )) dv B)τ) exp η ξτ) ξt ) )) dv B)τ) exp η ξτ) ξt ) )) dv B)τ) 1+kt) i=1, i=1+k i expη i) [ V B)t i ) V B)t i 1 ) ] + expη i) [ V B)t i ) V B)t i 1 ) ] i=1, i 1+k i 1+kt) εr exp η ξt) ξt ) + exp 1 + k i )η ) d 1 Bt i ) i=1, i 1+k i )) 1+kt) i=1 2εR exp η ξt) ξt ) ) 1+kt) expη i) + exp 1 + k i )η )) i=1, i 1+k i )) 1+kt) i=1 expη i) = 2εR exp η ξt) ξt ) )) expη) exp 1 + kt))η ) ) ) 1 1 expη) 1 2εR exp ηkt) ) exp 2 + kt))η ) expη) 1 ) = 2εR exp2η) expη) 1 ) 1. From 14), 16) 2) it follows that inequality 9) holds for t t, where H R n n is the matrix whose every component equals 1. On the other h, 2n it can be easily shown that rh) < 1 2. Consequently, by Lemma 1 an arbitrary solution x of the system 1 ) admits an estimate xt) ρ exp η ξt) ξt ) )) for t t t, where ρ > is a constant independent of t. )

10 654 M. ASHORDIA AND N. KEKELIA 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) for t R + i = 1,..., n; j = 1, 2), 2) νξ)t) lim t + t Aa ii, a ik ) = i, k = 1,..., n), 21) a ii t) a ii τ) η ξt) ξτ) ) for t τ i = 1,..., n), 22) where η >, ξ : R + R + is a nondecreasing function satisfying condition 3), νξ) : 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 t) diag a 11 t),..., a nn t) ). Indeed, by the definition of the operator A we have [ AA, A A )t) ] ik = a ikt) τ<t <τ t d 1 a ii τ) 1 d 1 a ii τ) d 1a ik τ) d 2 a ii τ) 1 + d 2 a ii τ) d 2a ik τ) = Aa ii, a ik )t) for t R + i k; i, k = 1,..., n) [ AA, A A )t) ] ii = for t R + i = 1,..., n). Therefore, by 21) 22) the matrix-function A is ξ-exponentially asymptotically stable. Corollary 2. Let the matrix-function P L loc R +, R n n ) be ξ-exponentially asymptotically stable where A t) t ξt)+1 lim t + t P τ) dτ, ξ : R + R + is a continuous nondecreasing function satisfying condition 3). asymptotically stable as well. A A ) = for t R +, Then the matrix-function A is ξ-exponentially

11 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 655 Proof. Corollary 2 immediately follows from Theorem 1 if we observe that in this case, moreover, AA, A A )t) = At) A t) t R + ) νξ)t) = ξt) + 1 t R + ) because ξ is a nondecreasing continuous function. Theorem 2. The matrix-function A is ξ-exponentially asymptotically stable if only if there exist a positive number η a nonsingular matrix-function H BV loc R +, R n n ) such that where sup { H 1 t)hs) : t s } < + 23) B η H, A)t) + + exp ηξτ) ) Hτ)Aτ) B η H, A) < +, 24) exp ηξτ) ) d τ [ exp ηξτ) ) Hτ) d exp ηξs) ) Hs) ) As) Proof. Let U U be the Cauchy matrices of systems 1 ) dyt) = da t) yt), ]. 25) respectively, where A t) = Lexpηξ ))H, A)t). Then by the definition of the operator L by the equality Ut, s) = exp η ξt) ξs) )) H 1 t)u t, s)hs) for t, s R + we obtain that Hence t +H 1 t) s exp η ξτ) ξs) )) Ut, s) = H 1 t)hs) exp η ξτ) ξs) )) db η H, A)τ) Uτ, s) for t, s R +. W t, s) = H 1 t)hs) + H 1 t)d 1 B η H, A)t) W t, s) + H 1 t) dgτ) W τ, s) for t, s R +, 26) s

12 656 M. ASHORDIA AND N. KEKELIA where W t, s) = exp η ξt) ξs) )) Ut, s), Gt) = B η H, A)t ). On the other h by 23), 24) by the equalities det I n + 1) j d j A t) ) = exp 1) j nη d j ξt) ) det Ht) + 1) j d j Ht) ) det I n + 1) j d j At) ) det H 1 t) ) for t R + j = 1, 2) I n + 1) j H 1 t) d j B η H, A)t) = H 1 t) I n + 1) j d j A t) ) Ht) for t R + j = 1, 2) there exists a positive number r such that det I n + 1) j H 1 t) d j B η H, A)t) ) for t R + j = 1, 2) 27) In + 1) j H 1 t) d j B η H, A)t) ) 1 < r for t R + j = 1, 2). 28) From 26), by 23), 27) 28) we get where W t, s) r ρ + ρ 1 s W τ, s) d V G)τ) ρ = sup { H 1 t)hs) : t s }, Hence, according to the Gronwall inequality [11]) where Therefore ) for t s, ρ 1 = ρ H 1 ). W t, s) M < + for t s, M = r exp r + ) ρ 1 B η H, A). Ut, s) M exp η ξt) ξs) )) for t s, 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 ρ such that Zt)Z 1 s) ρ exp η ξt) ξs) )) for t s, 29) where Z Z) = I n ) is the fundamental matrix of system 1 ). Let Ht) exp ηξt) ) Z 1 t).

13 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 657 Then according to 25), 29) the equality see [11]) we have Z 1 t) = I n Z 1 t)at) + dz 1 τ) Aτ) for t R + 3) H 1 t)hs) = Zt)Z 1 s) exp η ξt) ξs) )) ρ for t s B η H, A)t) = B η exp ηξ)z 1, A ) t) = for t R +. Therefore conditions 23) 24) are fulfilled. Remark 1. If in Theorem 2 the function ξ : R + R + is continuous, then condition 24) can be rewritten as + dv IH, A) + η diagξ,..., ξ) ) t) Ht) < +. Corollary 3. Let the matrix-function Q BV loc R +, R n n ) be uniformly stable det I n + 1) j d j Qt) ) for t R + j = 1, 2). 31) Let, moreover, there exist a positive number η such that + Z 1 t) dv G η ξ, Q, A) ) t) < + 32) where Z Z) = I n ) is the fundamental matrix of the system <τ t τ<t dzt) = dqt) zt), 33) G η ξ, Q, A)t) AQ, A Q)t) + ηs ξ)t) I n exp ηξτ) ) d 1 exp ηξτ) ) I n d 1 Qτ) ) 1 I n d 1 Aτ) ) exp ηξτ) ) d 2 exp ηξτ) ) I n + d 2 Qτ) ) 1 I n + d 2 Aτ) ). 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 Ht) Z 1 t). Using the formula of integration by parts [11]), the properties of the operator B given above equality 3), we conclude that B η H, A)t)

14 658 M. ASHORDIA AND N. KEKELIA = exp ηξτ) ) d expηξτ))z 1 τ) + B expηξ)z 1, A ) ) τ) + + s<τ τ<t = exp ηξτ) ) ) d expηξτ))z 1 τ) exp ηξτ) ) db = expηξ)i n, B expηξ)z 1, A ) ) τ) exp ηξτ) ) d expηξτ))z 1 τ) ) exp ηξτ) ) db expηξ)i n, BZ 1, A) ) τ) for t R + ; 35) <s τ = exp ηξτ) ) d expηξτ))z 1 τ) ) Z 1 τ) d ηs ξ)τ)i n AQ, Q)τ) ) exp ηξs) ) d 1 exp ηξs) ) I n d 1 Qs) ) 1 exp ηξs) ) d 2 exp ηξs) ) I n + d 2 Qs) ) 1 for t R+ ; 36) BZ 1, A)t) t d 2 Z 1 τ) d 2 Aτ) = = Z 1 τ) daτ) t <τ t d 1 Z 1 τ) d 1 Aτ) Z 1 τ) daq, A Q)τ) for t R +, 37) B expηξ)i n, BZ 1, A) ) t) Z 1 τ) db expηξ)i n, AQ, A) ) τ) for t R + 38) exp ηξτ) ) db expηξ)i n, AQ, A) ) τ) = AQ, A)t) exp ηξτ) ) d 1 exp ηξτ) ) I n d 1 Qτ) ) 1 d1 Aτ) <τ t

15 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 659 τ<t From 35), by 36) 39) we get + exp ηξτ) ) d 2 exp ηξτ) ) I n + d 2 Qτ) ) 1 d2 Aτ) 39) B η H, A)t) = Z 1 τ) d + = τ for t R +. exp ηξτ) ) d expηξτ)) Z 1 τ) ) exp ηξs) ) db expηξ)i n, AQ, A) ) s) Z 1 τ) dg η ξ, Q, A)τ) for t R + B η H, A) + Z 1 t) dv G η ξ, Q, A) ) t). Therefore from 32) the fact that the matrix-function Q is ξ-exponentially 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) = AQ, A Q)t) + ηξt)i n for t R +. Corollary 4. Let the matrix-function Q BV loc R +, R n n ), satisfying condition 31), be ξ-exponentially asymptotically stable + BZ 1, A Q) < +, 4) where Z Z) = I n ) is the fundamental matrix of system 33). matrix-function A is ξ-exponentially asymptotically stable as well. ) Then the 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 Ht) exp ηξt) ) Z 1 t). Using equality 3) for the matrix-function Q we conclude that B η H, A)t) = Z 1 t) = I n + BZ 1, Q)t) for t R + exp ηξτ) ) dbz 1, A Q)τ) for t R +.

16 66 M. ASHORDIA AND N. KEKELIA By this 4), condition 24) holds. Therefore, the conditions of Theorem 2 are fulfilled. Remark 3. By the equality the condition BZ 1, A Q)t) = t Z 1 τ) d Aτ) Qτ) ) for t R + + Z 1 t) dv AQ, A Q) ) t) < + guarantees the fulfilment of condition 4) in Corollary 4. On the other h, lim G ηξ, Q, A)t) = AQ, A Q)t) for t R +, η + where G η ξ, Q, A)t) is defined by 34). Consequently, Corollary 3 is true in the limit case η = ), 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 satisfying the Lappo-Danilevskiǐ condition Qτ) dqτ) = dqτ) Qτ) for t R +. Let, moreover, there exist a nonnegative number η such that + exp Qt)) 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-exponentially asymptotic stability of the matrix-function A for η > ; b) the ξ-exponentially asymptotic stability of Q guarantees the ξ-exponentially asymptotic stability of A for η =. Proof. The corollary follows immediatelly from Corollaries 3 4 Remark 3 if we note that in this case Zt) = expqt)) for t R + G η ξ, Q, A)t) = At) Qt) + ηξt)i n for t R +.

17 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 661 Corollary 6. Let there exist a nonnegative number η such that the components a ik i, k = 1,..., n) of the matrix-function A satisfy conditions 2), s a ii )t) s a ii )τ) ln 1 d 1 a ii s) ln 1 + d 2 a ii s) τ<s t τ s<t η s ξ)t) s ξ)τ) ) µ ξt) ξτ) ) for t τ 41) 1) j + t<+ i = 1,..., n), zi 1 t) exp 1) j d j ξτ) ) ) 1 < + 42) j = 1, 2; i = 1,..., n) z 1 i t) dvg ik )t) < + i k; i, k = 1,..., n), 43) where µ = if η >, µ > if η =, z i t) exp s a ii )t) + ηs ξ)t) ) <τ t 1 d1 a ii τ) ) 1 g ik t) s a ik )t) <τ t τ<t τ<t 1 + d2 a ii τ) ) i = 1,..., n), exp η d 1 ξτ) ) d 1 a ik τ) 1 d 1 a ii τ) ) 1 exp η d 2 ξτ) ) d 2 a ik τ) 1 + d 2 a ii τ) ) 1 i k; i, k = 1,..., n), ξ : R + R + is a nondecreasing function satisfying condition 3). Then the matrix-function A is ξ-exponentially asymptotically stable. Proof. For η > the corollary follows from Corollary 3 if we assume that Qt) diag a 11 t) + ηs ξ)t),..., a nn t) + ηs ξ)t) ). Indeed, by the definition of the operator A we have [ AQ, A Q)t) ]ik = a ikt) d 1 a ii τ) 1 d 1 a ii τ) ) 1 d1 a ik τ) τ<t <τ t d 2 a ii τ) 1 + d 2 a ii τ) ) 1 d2 a ik τ) for t R + i k; i, k = 1,..., n), [ AQ, A Q)t) ] ii = ηs ξ)t) for t R + i = 1,..., n). From these relations, using 34) we obtain [ Gη ξ, Q, A)t) ] = a ikt) d 1 a ii τ) 1 d 1 a ii τ) ) 1 d1 a ik τ) ik <τ t

18 662 M. ASHORDIA AND N. KEKELIA <τ t τ<t τ<t d 2 a ii τ) 1 + d 2 a ii τ) ) 1 d2 a ik τ) d 1 a ik τ) 1 d 1 a ii τ) ) 1 1 exp ηd 1 ξτ) )) d 2 a ik τ) 1 + d 2 a ii τ) ) 1 1 exp ηd 2 ξτ) )) = g ik t) for t R + i k; i, k = 1,..., n) [ Gη ξ, Q, A)t) ] ii = τ<t <τ t 1 exp ηd1 ξτ) )) 1 exp ηd2 ξτ) )) for t R + i = 1,..., n). On the other h, the matrix-function Zt) = diagz 1 t),..., z n t)) is the fundamental matrix of the system 33), satisfying the condition Z) = I n. Therefore, by 41) 43) the conditions of Corollary 3 are valid. For η = the corollary follows from Corollary 4 Remark 3. Remark 4. If, in addition to 2), the components a ik i, k = 1,..., n) of the matrix-function A satisfy the condition 1) j d j a ik t) 1 + 1) j d j a ii t) ) 1 for t R+ i k; i, k = 1,..., n; j = 1, 2), then we can assume without loss of generality that η > µ = in Corollary 6. Corollary 7. Let there exist a nonnegative number η such that + t n l 1 exp t Re λ l ) dvb ik )t) < + l = 1,..., m; i, k = 1,..., n), where b ik t) a ik t) p ik t+ηξt) i, k = 1,..., n), ξ : R + R + is a nondecreasing function satisfying condition 3), λ 1,..., λ m λ i λ j for i j) are the characteristic values of the matrix P = p ik ) n i,k=1 with multiplicities n 1,..., n m, respectively. Then: a) if η >, Re λ l l = 1,..., m),, in addition, n l = 1 for Re λ l =, then A is ξ-exponentially asymptotically stable; b) if η = Re λ l < l = 1,..., m), then A is exponentially asymptotically stable; c) if η = P is ξ-exponentially asymptotically stable, then A is ξ- exponentially asymptotically stable as well. Proof. The corollary immediatelly follows from Corollary 5 if we assume Qt) P t derive the matrix-function exp P t) by the stard way using the Jordan canonical form of the matrix P.

19 GENERALIZED ODE. ξ-exponentially ASYMPTOTIC STABILITY 663 Consider now system 2), where P L loc R +, R n n ), q L loc R +, R n ). Theorem 2 Corollary 6 have the following form for this system. Theorem 2. Let ξ : R + R + be an absolutely continuous nondecreasing function satisfying condition 3). Then the matrix-function P L loc R +, R n n ) is ξ-exponentially asymptotically stable if only if there exist a positive number η nonsingular matrix-function H C loc R +, R n n ) such that sup { H 1 t)hs) : t s } < + + H t) + Ht) P t) + ηξ t)i n ) dt < +. Corollary 6. Let there exist a positive number η such that the components p ik L loc R +, R) i, k = 1,..., n) of the matrix-function P satisfy the conditions + t exp t p ii t) η for t t i = 1,..., n) pii τ) + η ) dτ ) p ik t) dt < + i k; i, k = 1,..., n) for some t R +. Then the matrix-function P is exponentially asymptotically stable. Acknowledgement This work was supported by a research grant in the framework of the Bilateral S&T Cooperation between the Hellenic Republic Georgia. References 1. M. Ashordia, On the stability of solutions of the multipoint boundary value problem for the system of generalized ordinary differential equations. Mem. Differential Equations Math. Phys ), M. Ashordia, Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations. Czechoslovak Math. J )1996), P. C. Das R. R. Sharma, Existence stability of measure differential equations. Czechoslovak Math. J. 2297)1972), B. P. Demidovich, Lectures on mathematical theory of stability. Russian) Nauka, Moscow, J. Groh, A nonlinear Volterra Stiltjes integral equation Gronwall inequality in one dimension. Illinois J. Math ), No. 2, T. H. Hildebrant, On systems of linear differentio-stiltjes integral equations. Illinois J. Math ),

20 664 M. ASHORDIA AND N. KEKELIA 7. I. T. Kiguradze, Initial boundary value problem for systems of ordinary differential equations, vol. I. Linear Theory. Russian) Metsniereba, Tbilisi, J. Kurzweil, Generalized ordinary differential equations continuous dependence on a parameter. Czechoslovak Math. J. 782)1957), A. M. Samoǐlenko N. A. Perestiuk, Differential equation with impulse action. Russian) Visha Shkola, Kiev, Š. Schwabik, Generalized ordinary differential equations. World Scientific, Singapore, Š. Schwabik, M. Tvrdˇy, O. Vejvoda, Differential integral equations: boundary value problems adjoint. Academia, Praha, Authors addresses: Received ) M. Ashordia I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 3843 Georgia Sukhumi Branch of Tbilisi State University 12, Jikia St., Tbilisi 3886 Georgia N. Kekelia Sukhumi Branch of Tbilisi State University 12, Jikia St., Tbilisi 3886 Georgia