Mem. Differential Equations Math. Phys. 37 (2006), 154–157
M. Ashordia and G. Ekhvaia
CRITERIA OF CORRECTNESS OF LINEAR BOUNDARY VALUE
PROBLEMS FOR SYSTEMS OF IMPULSIVE EQUATIONS WITH
FINITE AND FIXED POINTS OF IMPULSES ACTIONS
(Reported on October 3, 2005)
Let P ∈ L([a, b]; Rn×n ), p ∈ L([a, b]; Rn ), Qj ∈ Rn×n (j = 1, . . . , m), qj ∈ Rn (j =
1, . . . , m), a = τ0 < τ1 < · · · < τm ≤ τm+1 = b, c0 ∈ Rn , and ` : BVC([a, b]; τ1 , . . . , τm ; Rn ) →
Rn be a linear bounded operator such that the impulsive system
dx
= P (t)x + p(t),
(1)
dt
x(τj +) − x(τj −) = Qj x(τj ) + qj (j = 1, . . . , m)
(2)
has a unique solution x0 satisfying the boundary condition `(x) = c0 .
Consider sequences of matrix- and vector-functions Pk ∈ L([a, b]; Rn×n ) (k = 1, 2, . . .)
and pk ∈ L([a, b]; Rn ) (k = 1, 2, . . .), sequences of constant matrices Qkj ∈ Rn×n
(j = 1, . . . , m; k = 1, 2, . . .) and constant vectors qkj ∈ Rn (j = 1, . . . , m; k =
1, 2, . . .) and c0k ∈ Rn (k = 1, 2, . . .) and a sequence of linear bounded operators
`k : BVC([a, b]; τ1 , . . . , τm ; Rn ) → Rn (k = 1, 2, . . .).
In this paper necessary and sufficient conditions as well as effective sufficient conditions
are established for a sequence of boundary value problems
dx
= Pk (t)x + pk (t),
(3)
dt
x(τj +) − x(τj −) = Qkj x(τj ) + qkj (j = 1, . . . , m),
(4)
`k (x) = c0k
(5)
(k = 1, 2, . . .) to have a unique solution xk for sufficiently large k and
lim xk (t) = x0 (t)
k→∞
(6)
uniformly on [a, b].
Analogous questions are investigated e.g. in [1], [2], [5], [6] (see the references therein,
too) for systems of ordinary differential equations and in [3], [4] for systems of generalized
ordinary differential equations.
Throughout the paper, the following notation and definitions will be used.
R = ] − ∞, ∞[ . Rn×l is the space of all real n × l-matrices X = (xij )n,l
i,j=1 with the
n
P
norm kXk = max
|xij |. On×l is the zero n × l-matrix.
j=1,...,l i=1
det(X) is the determinant of a matrix X ∈ Rn×n . In is the identity n × n-matrix.
δij is the Kronecker symbol, i.e. δii = 1 and δij = 0 for i 6= j.
Rn = Rn×1 is the space of all real column n-vectors x = (xi )n
i=1 .
BVC([a, b]; τ1 , . . . , τm ; Rn×l ) is the Banach space of all continuous on the intervals
n×l
[a, τ1 ], ]τk , τk+1 ] (k = 1, . . . ,m) matrix-functions
of bounded variation X : [a, b] → R
with the norm kXkS = sup kX(t)k : t ∈ [a, b] .
2000 Mathematics Subject Classification. 34B37.
Key words and phrases. Linear impulsive systems, linear boundary value problems,
criteria of correctness.
155
L([a, b]; Rn×l ) is the set of all measurable and Lebesgue integrable on [a, b] matrixfunctions.
C([a, b]; Rn×l ) is the set of all continuous on [a, b] matrix-functions.
e
C([a,
b]; Rn×l ) is the set of all absolutely continuous on [a, b] matrix-functions.
n×l ) is the set of all matrix-functions restrictions of which on every
e
C([a,
b]\{τj }m
j=1 ; R
e
closed interval [c, d] from [a, b] \ {τj }m belong to C([c,
d]; Rn×l ).
j=1
On the set C([a, b]; Rn×l ) × Rn×l × · · · × Rn×l ×L([a, b]; Rl×k ) we introduce the ope{z
}
|
m
rator
B0 (Φ, G1 , . . . , Gm , X)(t) ≡
Z
t
Φ(s)X(s) ds +
a
m
X
Gj
j=0, τj ∈[a,t[
Z
t
X(s) ds,
tj
where G0 = On×n .
Under a solution of the system (1), (2) we understand a continuous from the left
n×l ) ∩ BVC([a, b]; τ , . . . , τ ; Rn ) satisfying the
e
vector-function x ∈ C([a,
b] \ {τj }m
m
1
j=1 ; R
system (1) for a.e. t ∈ [a, b] and the equality (2) for every j ∈ {1, . . . , n}.
We assume everywhere that det(In + Qj ) 6= 0 (j = 1, . . . , m).
Note that this condition guarantees the unique solvability of the system (1), (2) under
the Cauchy condition x(t0 ) = c0 .
m
Definition 1. We say that a sequence (Pk , pk , {Qkj }m
j=1 , {qkj }j=1 , `k ) (k = 1, 2, . . .)
m , `) if for every c ∈ Rn and c ∈ Rn (k =
belongs to the set S(P, p, {Qj }m
,
{q
}
0
j j=1
k
j=1
1, 2, . . .) satisfying the condition lim ck = c0 the problem (3)–(5) has a unique solution
k→∞
xk for any sufficiently large k and the condition (6) holds uniformly on [a, b].
Theorem 1. Let
lim `k (y) = `(y) for y ∈ BVC([a, b]; τ1 , . . . , τm ; Rn ).
(7)
∞
(8)
k→∞
Then
m
(Pk , pk , {Qkj }m
j=1 , {qkj }j=1 , `k )
k=1
m
∈ S(P, p, {Qj }m
j=1 , {qj }j=1 , `)
e
if and only if there exist sequences of matrix-functions Φ, Φk ∈ C([a,
b]; Rn×n ) (k =
1, 2, . . .) and constant matrices Gj , Gkj ∈ Rn×n , G0 = Gk0 = On×n (j = 0, . . . , m;
k = 1, 2, . . .) such that
lim sup
k→∞
m Z
X
j=0
τj+1
τj
j
X
0
Qkj Pk (t) dt < ∞,
Φk (t) + Φk (t) +
(9)
i=0
j
n
o
X
inf det Φ(t) +
Gi : t ∈ ]τj , τj+1 ] > 0 (j = 0, . . . , m),
(10)
i=0
lim Gkj = Gj (j = 1, . . . , m),
k→∞
lim Qkj = Qj ,
k→∞
lim qkj = qj (j = 1, . . . , m),
k→∞
(11)
(12)
and the conditions
lim Φk (t) = Φ(t),
(13)
lim B0 (Φk , Gk1 , . . . , Gkm , Pk )(t) = B0 (Φ, G1 , . . . , Gm , P )(t),
(14)
lim B0 (Φk , Gk1 , . . . , Gkm , pk )(t) = B0 (Φ, G1 , . . . , Gm , p)(t)
(15)
k→∞
k→∞
k→∞
are fulfilled uniformly on [a, b].
156
Remark 1. The conditions (14) and (15) are fulfilled uniformly on [a, b] if and only if
the conditions
Z t
Z t
j
j
X
X
lim
Φk (s) +
Gki Pk (s) ds =
Φ(s) +
Gi P (s) ds,
k→∞
lim
τj
k→∞
Z
τj
i=0
t
τj
Φk (s) +
j
X
Z
Gki pk (s) ds =
i=0
t
τj
i=0
Φ(s) +
j
X
i=0
Gi p(s) ds,
respectively, are fulfilled uniformly on [τj , τj+1 ] for every j ∈ {0, . . . , m}.
Corollary 1. Let the conditions (7) and (12) hold. Let, moreover, there exist matrixe
functions Φ, Φk ∈ C([a,
b]; Rn×n ) (k = 1, 2, . . .) such that the conditions (9) and
n
o
inf det Φ(t) + (1 − δ0j )jIn : t ∈ ]τj , τj+1 ] > 0 (j = 0, . . . , m)
hold and the conditions (13),
Z t
Z
lim
Φk (s) + (1 − δ0j )jIn Pk (s) ds =
k→∞
τj
t
Φ(s) + (1 − δ0j )jIn P (s) ds
τj
and
lim
k→∞
Z
t
τj
Φk (s) + (1 − δ0j )jIn pk (s) ds =
Z
t
Φ(s) + (1 − δ0j )jIn p(s) ds
τj
be fulfilled uniformly on [τj , τj+1 ] for every j ∈ {0, . . . , m}. Then the condition (8) holds.
Corollary 2. Let the conditions (7) and (12) hold. Let, moreover, there exist matrixe
functions Φ, Φk ∈ C([a,
b]; Rn×n ) (k = 1, 2, . . .) such that
Z b
n
o
0
Φ (t) + Φk (t)Pk (t) dt < ∞, inf det(Φ(t)) : t ∈ [a, b] > 0
lim sup
k
k→∞
a
and the conditions (13) and
Z
Z t
Φk (s)Pk (s) ds =
lim
k→∞
a
t
Φ(s)P (s) ds,
a
lim
k→∞
Z
t
Φk (s)pk (s) ds =
a
Z
t
Φ(s)p(s) ds
a
are fulfilled uniformly on [a, b]. Then the condition (8) holds.
Corollary 3. Let the conditions (7), (11) and (12) hold. Let, moreover, there exist
constant matrices Gj , Gkj ∈ Rn×n , G0 = Gk0 = On×n (j = 0, . . . , m; k = 1, 2, . . .) such
that
j
m Z τj+1 X
X
lim sup
Qki Pk (t) dt < ∞,
(16)
In +
k→∞
j=0
τj
det In +
i=0
j
X
i=1
Gi 6= 0 (j = 1, . . . , m)
and the conditions
lim
Z
lim
Z
k→∞
k→∞
t
τj
t
τj
In +
j
X
Z
Gki Pk (s) ds =
j
X
Z
i=0
In +
i=0
Gki pk (s) ds =
t
τj
t
τj
In +
j
X
i=0
Gi P (s) ds,
In +
j
X
Gi p(s) ds
i=0
are fulfilled uniformly on [τj , τj+1 ] for every j ∈ {0, . . . , m}. Then the condition (8)
holds.
157
Corollary 4. Let the conditions (7), (12) and (16) hold and the conditions
Z t
Z t
Z t
Z t
lim
Pk (s) ds =
P (s) ds,
lim
pk (s) ds =
p(s) ds
k→∞
a
k→∞
a
a
(17)
a
be fulfilled uniformly on [a, b]. Then the condition (8) holds.
Corollary 5. Let the conditions (7), (12), and (16) hold and the condition (17) be
fulfilled uniformly on [a, b]. Then the condition (8) holds.
Remark 2. In Theorem 1 and Corollaries 1–5 we can assume without loss of generality
that Φ(t) ≡ In and Gj = On×n (j = 1, . . . , m) everywhere they appear. So that the
condition (10) in Theorem 1 as well as the analogous conditions in the corollaries are
valid automatically.
These results follow from analogous results for a system of so-called generalized ordinary differential equations contained in [4] because the system (1), (2) is its particular
case.
References
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Authors’ Addresses:
M. Ashordia
I. Vekua Institute of Applied Mathematics
I. Javakhishvili Tbilisi State University
2, University St., Tbilisi 0143
Georgia
E-mail: ashord@rmi.acnet.ge
M. Ashordia and G. Ekhvaia
Sukhumi Branch of I. Javakhishvili Tbilisi State University
12, Jikia St., Tbilisi 0186
Georgia