Mem. Differential Equations Math. Phys. 64 (2015), 143–154
Short Communications
Malkhaz Ashordia and Goderdzi Ekhvaia
ON THE SOLVABILITY OF MULTIPOINT
BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF
NONLINEAR DIFFERENTIAL EQUATIONS WITH
FIXED POINTS OF IMPULSES ACTIONS
Abstract. The necessary and sufficient conditions and the effective sufficient conditions are given for the solvability of the multipoint boundary
value problems for systems of nonlinear impulsive equations.
ÒÄÆÉÖÌÄ. ÌÏÚÅÀÍÉËÉÀ ×ÉØÓÉÒÄÁÖËÉ ÉÌÐÖËÓÖÒÉ ØÌÄÃÄÁÄÁÉÓ ßÄÒÔÉËÄÁÉÀÍ ÀÒÀßÒ×ÉÅ ÃÉ×ÄÒÄÍÝÉÀËÖÒ ÂÀÍÔÏËÄÁÀÈÀ ÓÉÓÔÄÌÄÁÉÓÈÅÉÓ ÌÒÀÅÀËßÄÒÔÉËÏÅÀÍÉ ÓÀÓÀÆÙÅÒÏ ÀÌÏÝÀÍÄÁÉÓ ÀÌÏáÓÍÀÃÏÁÉÓ ÀÖÝÉËÄÁÄËÉ
ÃÀ ÓÀÊÌÀÒÉÓÉ ÃÀ Ä×ÄØÔÖÒÉ ÓÀÊÌÀÒÉÓÉ ÐÉÒÏÁÄÁÉ.
2000 Mathematics Subject Classification: 34K45.
Key words and phrases: Systems of nonlinear impulsive equations, multipoint boundary value problem, solvability, unique solvability, criterion,
effective conditions.
For the nonlinear impulsive system with fixed and finite number of impulses points
dx
0
= f (t, x) for a.e. t ∈ [a, b] \ {τk }m
(1)
k=1 ,
dt
x(τk +) − x(τk −) = Ik (x(τk )) (k = 1, . . . , m0 ),
(2)
we consider the multipoint boundary value problem of the Caucy–Nicoletti’s
type
xi (ti ) = φi (x1 , . . . , xn ) (i = 1, . . . , n),
(3)
where ti ∈ [a, b] (i = 1, . . . , n), a < τ1 < · · · < τm0 < b (we will assume
τ0 = a and τm0 +1 = b if necessary), f = (fk )nk=1 ∈ K([a, b] × Rn , Rn ),
Ik = (Iki )ni=1 : Rn → Rn (k = 1, . . . , m0 ; i = 1, . . . , n) are continuous
operators, and φi : BVs ([a, b], Rn ) (i = 1, . . . , n) are continuous functionals
which are nonlinear, in general.
In this paper, the necessary and sufficient conditions as well as the effective sufficient conditions are given for the solvability and unique solvability
of the boundary value problem (1), (2); (3). Analogous problem on the interval [−a, a] have been considered in [3], when the multipoint problem is
144
degenerated into the two-point one, i.e., when ti = −σi a (i = 1, . . . , n). The
general nonlinear boundary value problems for the impulsive system (1), (2)
is considered in [4], where the Conti–Opial type existence and uniqueness
theorems are prescribed for the problem.
We realize the results for the boundary condition
xi (ti ) = ci (i = 1, . . . , n),
(4)
i.e., when φi (x1 , . . . , xn ) ≡ ci (i = 1, . . . , n), where ci ∈ Rn (i = 1, . . . , n) are
the constant vectors.
Note, in addition, that analogous results are contained in [5, 6, 8, 9] for
the multipoint boundary value problems for systems of ordinary differential
equations.
Throughout the paper, the following notation and definitions will be used.
R = ] − ∞, +∞[ , R+ = [0, +∞[ ; [a, b] (a, b ∈ R) is a closed interval.
Rn×m is the space of all real n × m-matrices X = (xij )n,m
i,j=1 with the
norm
∥X∥ = max
j=1,...,m
n
∑
|xij |;
i=1
{
}
Rn×m
= (xij )n,m
+
i,j=1 : xij ≥ 0 (i = 1, . . . , n; j = 1, . . . , m) .
Rn = Rn×1 is the space of all real column n-vectors x = (xi )ni=1 ; Rn+ =
n×1
R+ .
diag(λ1 , . . . , λn ) is the diagonal matrix with diagonal elements λ1 , . . . , λn ;
δij is the Kronecker symbol, i.e., δij = 1 if i = j, and δij = 0 if i ̸= j
(i, j = 1, . . . , n).
θ is the function defined by θ(t) = 1 for t ≥ 0, and θ(t) = −1 for t < 0.
b
∨(X) is the total variation of the matrix-function X : [a, b] → Rn×m , i.e.,
a
the sum of total variations of the latter’s components.
X(t−) and X(t+) are the left and the right limit of the matrix-function
X : [a, b] → Rn×m at the point t (we will assume X(t) = X(a) for t ≤ a
and X(t) = X(b) for t ≥ b, if necessary);
d1 X(t) = X(t)
{ − X(t−), d2 X(t)} = X(t+) − X(t);
∥X∥s = sup ∥X(t)∥ : t ∈ [a, b] .
BV([a, b], Rn×m ) is the set of all matrix-functions of bounded variation
b
X : [a, b] → Rn×m (i.e., such that ∨(X) < +∞);
a
BVs ([a, b], Rn ) is the normed space (BV([a, b], Rn ), ∥ · ∥s );
e
C([a,
b], D), where D ⊂ Rn×m , is the set of all absolutely continuous
matrix-functions X : [a, b] → D;
eloc ([a, b] \ {τk }m , D) is the set of all matrix-functions X : [a, b] → D
C
k=1
whose restrictions to an arbitrary closed interval [c, d] from [a, b] \ {τk }m
k=1
e
belong to C([c,
d], D).
145
If B1 and B2 are normed spaces, then an operator g : B1 → B2 (nonlinear, in general) is positive homogeneous if
g(λx) = λg(x)
for every λ ∈ R+ and x ∈ B1 .
An operator φ : BV([a, b], Rn ) → Rn is called nondecreasing if for every
x, y ∈ BV([a, b], Rn ) such that x(t) ≤ y(t) for t ∈ [a, b] the inequality
φ(x)(t) ≤ φ(y)(t) holds for t ∈ [a, b].
If α ∈ BV([a, b], R) has no more than a finite number of points of discontinuity, and m ∈ {1, 2}, then Dαm = {tαm1 , . . . , tαmnαm } (tαm1 < · · · <
tαmnαm ) is the set of all points from [a, b] for which dm α(t) ̸= 0.
µαm = max{dm α(t) : t ∈ Dαm } (m = 1, 2).
If β ∈ BV([a, b], R), then
}
{
∑
dj β(τ ) : l = 1, . . . , nαm
ναmβj = max dj β(tαml )+
tαm l+1−m <τ <tαm l+2−m
(j, m = 1, 2); here tα20 = a − 1, tα1nα1 +1 = b + 1.
By ν(t) (a < t ≤ b) we denote a number of points τk (k = 1, . . . , m0 )
belonging to [a, t[ .
A matrix-function is said to be continuous, nondecreasing, integrable,
etc., if each of its components is such.
L([a, b], D), where D ⊂ Rn×m , is the set of all measurable and integrable
matrix-functions X : [a, b] → D.
If D1 ⊂ Rn and D2 ⊂ Rn×m , then K([a, b]×D1 , D2 ) is the Carathéodory
class, i.e., the set of all mappings F = (fkj )n,m
k,j=1 : [a, b] × D1 → D2 such
that for each i ∈ {1, . . . , l}, j ∈ {1, . . . , m} and k ∈ {1, . . . , n}:
a) the function fkj ( · , x) : [a, b] → D2 is measurable for every x ∈ D1 ;
b) the function fkj (t, · ) : D1 → D2 is continuous for almost every
t ∈ [a, b], and
{
}
sup |fkj ( · , x)| : x ∈ D0 ∈ L([a, b], R; gik )
for every compact D0 ⊂ D1 .
By a solution of the impulsive system (1), (2) we understand a continuous
eloc ([a, b] \ {τk }m0 , Rn ) ∩ BVs ([a, b], Rn )
from the left vector-function x ∈ C
k=1
0
satisfying both the system (1) for a.e. t ∈ [a, b] \ {τk }m
k=1 and the relation
(2) for every k ∈ {1, . . . , m0 }.
Quite a number of issues on the theory of systems of differential equations with impulsive effect (both linear and nonlinear) have been studied sufficiently well (for a survey of the results on impulsive systems see
e.g. [1, 2, 7, 10, 11, 13, 14], and references therein). But the above-mentioned
works do not contain the results analogous to those obtained in [5, 6, 8, 9]
for ordinary differential equations.
146
Using the theory of the so-called generalized ordinary differential equations (see e.g. [1, 2, 12] and references therein), we extend these results to
the systems of impulsive equations.
To establish the results dealing with the boundary value problems for the
impulsive system (1), (2) we use the following concept.
It is easy to show that the vector-function x is a solution of the impulsive
system (1), (2) if and only if it is a solution of the following system of
generalized ordinary differential equations (see e.g. [8, 9, 13] and references
therein):
dx(t) = dA(t) · f (t, x(t)),
where
A(t) ≡ diag(a11 (t), . . . , ann (t)),
{
t
for a ≤ t ≤ τ1 ,
aii (t) =
t + k for τk < t ≤ τk+1 (k = 1, . . . , m0 ; i = 1, . . . , n);
f (τk , x) ≡ Ik (x) (k = 1, . . . , m0 ).
It is evident that the matrix-function A is continuous from the left,
0
d2 A(t) = 0 if t ∈ [a, b] \ {τk }m
k=1 and d2 A(τk ) = 1 (k = 1, . . . , m0 ).
Definition. Let t1 , . . . , tn ∈ [a, b] and a < τ1 < · · · < τm0 < b. We say that
n
0
the triplet (P, {Hk }m
k=1 , φ0 ) consisting of a matrix-function P = (pil )i,l=1 ∈
n×n
n
L([a, b], R
), a finite sequence of constant matrices Hk = (hkil )i,l=1 ∈
Rn×n (k = 1, . . . , m0 ) and a positive homogeneous nondecreasing continuous operator φ0 = (φ0i )ni=1 : BVs ([a, b], Rn+ ) → Rn+ belongs to the set
U (t1 , . . . , tn ; τ1 , . . . , τm0 ) if pil (t) ≥ 0 for a.e. t ∈ [a, b] (i ̸= l; i, l = 1, . . . , n),
hkil ≥ 0 (i ̸= l; i, l = 1, . . . , n; k = 1, . . . , m0 ), and the system
x′i (t) sgn(t − ti ) ≤
n
∑
0
pil (t)xl (t) for a.e. t ∈ [a, b] \ {τk }m
k=1 (i = 1, . . . , n),
l=1
n
∑
(xi (τk +)−xi (τk −)) sgn(τk −ti ) ≤ hkil xl (τk ) (i = 1, . . . , n; k = 1, . . . , m0 )
l=1
has no nontrivial nonnegative solution satisfying the condition
xi (ti ) ≤ φ0i (x1 , . . . , xn ) (i = 1, . . . , n).
The set analogous to U (t1 , . . . , tn ; τ1 , . . . , τm0 ) has been introduced by
I. Kiguradze for ordinary differential equations (see [5, 6]).
Theorem 1. The problem (1), (2); (3) is solvable if and only if there exist
eloc ([a, b] \
continuous from the left vector-functions αm = (αmi )ni=1 ∈ C
m0
n
n
{τk }k=1 , R ) ∩ BVs ([a, b], R ) (m = 1, 2) such that the conditions
α1 (t) ≤ α2 (t) for t ∈ [a, b]
147
and
( (
)
)
′
(−1)j fi t, x1 , . . . , xi−1 , αji (t), xi+1 , . . . , xn − αji
(t) sgn(t − ti ) ≤ 0
for almost every t ∈ [a, b], t ̸= ti , t ∈
/ {τ1 , . . . , τm0 },
α1 (t) ≤ (xl )nl=1 ≤ α2 (t) (j = 1, 2; i = 1, . . . , n),
(
)
(−1)m xi + Iki (x1 , . . . , xn ) − αmi (τk +) ≤ 0 for τk ≥ ti ,
α1 (τk ) ≤ (xl )nl=1 ≤ α2 (τk ) (m = 1, 2; i = 1, . . . , n; k = 1, . . . , m0 )
hold, and the inequalities
α1i (ti ) ≤ φi (xl , . . . , xn ) ≤ α2i (ti ) (i = 1, . . . , n)
{
eloc ([a, b] \ {τk }m0 , Rn ) ∩ BVs ([a, b], Rn ),
are fulfilled on the set (xl )nl=1 ∈ C
k=1
}
α1 (t) ≤ (xl )nl=1 ≤ α2 (t) for t ∈ [a, b] .
Theorem 2. Let the conditions
fi (t, x1 , . . . , xn ) sgn[(t − ti )xi ] ≤
n
∑
pil (t)|xl | + qi (t)
l=1
for almost every t ∈ [a, b], t ̸= ti , t ∈
/ {τ1 , . . . , τm0 } (i = 1, . . . , n), (5)
where qi ∈ L([a, b], R+ ) (i = 1, . . . , n), and
Iki (x1 , . . . , xn )θ(τk − ti ) sgn xi ≤
≤
n
∑
hkil |xl |+qi (τk ) (k = 1, . . . , m0 ; i = 1, . . . , n) (6)
l=1
be fulfilled on Rn , and the inequalities
(
)
|φi (x1 , . . . , xn )| ≤ φ0i |x1 |, . . . , |xn | + γ (i = 1, . . . , n)
eloc ([a, b] \ {τk }m0 , Rn ) ∩ BVs ([a, b], Rn ), where γ ∈
be fulfilled on the set C
k=1
R+ . Let, moreover,
)
(
}m0
{
(7)
(pil )ni,l=1 , (hkil )ni,l=1 k=1 ; (φ0i )ni=1 ∈ U (t1 , . . . , tn ; τ1 , . . . , τm0 ),
where qi ∈ L([a, b], R+ ) (i = 1, . . . , n), γ ∈ R+ . Then the problem (1), (2); (3)
is solvable.
Corollary 1. Let the conditions (5) and (6) be fulfilled on Rn , where pii ∈
Lµ ([a, b], R), pil ∈ Lµ ([a, b], R+ ) (i ̸= l; i, l = 1, . . . , n), qi ∈ L([a, b], R+ )
(i = 1, . . . , n), hkil = αkl pil (τk ) (k = 1, . . . , m0 ; i, l = 1, . . . , n), 1 ≤ µ ≤
+∞, αik ∈ R+ (i = 1, . . . , n; k = 1, . . . , m0 ). Let, moreover, the inequalities
m0
n (
[∑
] ν1 )
∑
φi (x1 , . . . , xn ) ≤
+γ
γ1il ∥xl ∥Lν + γ2il
|xl (τl )|ν
k=l
be fulfilled on the set
and let
k=1
eloc ([a, b]\{τk }m0 , Rn )∩BVs ([a, b], Rn )
C
k=1
r(H0 ) < 1,
(i = 1, . . . , n)
(8)
148
where γ1ik , γ2ik ∈ R+ (i, k = 1, . . . , n), µ1 + ν2 = 1, γ ∈ R+ , and the
2n × 2n-matrix H0 = (H0jm )2j,m=1 is defined by
(
)n
[2
] ν1
1
ν
H011 = (b − a) γ1ik +
(b − a) ∥pik ∥Lµ
,
π
i,k=1
(
m0
m0
[
] ν1 ( ∑
) 1 )n
∑
1
µ µ
ν
αli
|pik (τl )|
H012 = (b − a) γ2ik + (b − a)
,
H021 =
H022 =
(( ∑
m0
l=1
(( ∑
m0
αli
αli
l=1
) ν1
) ν1
l=1
γ2ik +
(1
4
i,k=1
l=1
)n
m0
[
] ν1
∑
γ1ik + (b − a)
αli ∥pik ∥Lµ
l=1
,
i,k=1
0
) µ1
π ) ν1 ( ∑
|pik (τl )|µ
4nk +2
m
µi µk sin−2
)n
;
i,k=1
l=1
here µi = max{αli : l = 1, . . . , m0 }, and nk = nαk 2 is the quantity of nonzero
numbers from the sequence α1k , . . . , αm0 k . Then the problem (1), (2); (3) is
solvable.
Remark 1. The 2n × 2n-matrix H appearing in Corollary 1 can be replaced
by the n × n-matrix
(
{[
m0
(∑
) ν1 ]
1
γ1ik +
max (b − a) ν +
αli
+
([
2
l=1
m0
m0
] ν1 )
(∑
) ν1 ]
] ν2 [
[
∑
1
αli
γ2ik +
(b−a) + (b−a) αli
∥pik ∥Lµ , (b−a) ν +
π
l=1
l=1
)
([
)
m
m
1
0
0
]
(
(
) µ1 } n
∑
∑
1
1
π
ν
−2
µ
+ (b−a) αli + µi µk sin
)ν
|plk (τl )|
4
4nk + 2
l=1
l=1
.
i,k=1
Remark 2. In the Corollary 1, as a matrix-function C = (cil )ni,l=1 we take
cil (t) ≡ pil t + αil (t) (i, l = 1, . . . , n).
Corollary 2. Let the conditions (5) and (6) be fulfilled on Rn , where pii ∈
Lµ ([a, b], R), pil ∈ Lµ ([a, b], R+ ) (i ̸= l; i, l = 1, . . . , n), qi ∈ L([a, b], R+ )
(i = 1, . . . , n), hkil = αkl pil (τk ) (k = 1, . . . , m0 ; i, l = 1, . . . , n), 1 ≤ µ ≤
+∞, αik ∈ R+ (i = 1, . . . , n; k = 1, . . . , m0 ). Let, moreover, the inequality
(8) hold, where the 2n × 2n-matrix H0 = (H0jm )2j,m=1 is defined by
([
)n
] ν1
2
µ
H011 =
(b − a) ∥hik ∥L
,
π
i,k=1
([
m0
m0
) 1 )n
] ν1 ( ∑
∑
µ µ
|hik (τl )|
,
H012 = (b − a)
αli
l=1
H021
l=1
i,k=1
l=1
([
)n
m0
] ν1
∑
= (b − a)
αli ∥hik ∥Lµ
i,k=1
,
149
H022 =
((
0
) ν1 ( ∑
) µ1
1
π
µi µk sin−2
|hik (τl )|µ
4
4nk + 2
m
l=1
)n
;
i,k=1
here µ1 + ν2 = 1, µi = max{αli : l = 1, . . . , m0 }, and nk = nαk 2 is the
quantity of nonzero numbers from the sequence α1k , . . . , αm0 k . Then the
problem (1), (2); (4) is solvable.
Remark 3. The 2n × 2n-matrix H0 appearing in Corollary 2 can be replaced
by the n × n-matrix
(
{([
m0
] ν2 [
] ν1 )
∑
2
max
(b − a) + (b − a)
αli
∥pik ∥Lµ ,
π
l=1
)
([
m0
m0
] ν1 ( 1
) ν1 )( ∑
)1 } n
∑
π
−2
µ µ
(b−a)
αli +
µi µk sin
|plk (τl )|
.
4
4nk + 2
l=1
l=1
i,k=1
By Remark 3, the Corollary 2 has the following form for pil (t) ≡ pil =
const (i, l = 1, . . . , n) and µ = +∞.
Corollary 3. Let the conditions (6) and
fi (t, x1 , . . . , xn ) sgn[(t − ti )xi ] ≤
n
∑
pil |xl | + qi (t)
l=1
0
for almost every t ∈ [a, b] \ {τk }m
k=1 (i = 1, . . . , n) (9)
be fulfilled on Rn , where pii ∈ R, pil ≥ 0 (i ̸= l; i, l = 1, . . . , n), hkil = αk pil
(k = 1, . . . , m0 ; i, l = 1, . . . , n), αk ≥ 0 (i = 1, . . . , n; k = 1, . . . , m0 ),
qi ∈ L([a, b], R+ ) (i = 1, . . . , n). Let, moreover, the inequality (8) hold,
where H0 = ρ0 H, H = (hi l)ni,l=1 , and
{
}
m0
(
) 21
∑
2
1
π
−1
ρ0 = (b − a)
αl
+ max
(b − a), µα sin
,
π
2
4nα + 2
l=1
µα = max{αl : l = 1, . . . , m0 }, nα is the quantity of nonzero numbers from
the sequence α1 , . . . , αm0 . Then the problem (1), (2); (4) is solvable.
Corollary 4. Let the conditions (5) and (6) be fulfilled on Rn , the inequalities
φi (x1 , . . . , xn ) ≤ µi |xi (si )| + γ (i = 1, . . . , n)
(10)
eloc ([a, b] \ {τk }m0 , Rn ) ∩ BVs ([a, b], Rn ) and let
be fulfilled on the set C
k=1
µi γi (si , ti ) < 1 (i = 1, . . . , n),
(11)
where pil (t) ≡ hil βi (t) + βil (t), qi ∈ L([a, b], R+ ) (i = 1, . . . , n), hkil =
hil βki + βkil (i, l = 1, . . . , n; k = 1, . . . , m0 ), hii < 0, hil ≥ 0 (i ̸= l;
i, l = 1, . . . , n); µi ∈ R+ , si ∈ [a, b], si ̸= ti (i = 1, . . . , n); γ ∈ R+ ,
βii ∈ L([a, b], R+ ) (i = 1, . . . , n); βil , βi ∈ L([a, b]; R) (i ̸= l; i, l = 1, . . . , n)
are such that βil (t) ≥ 0 (i ̸= l) and βi (t) ≥ 0 for a.e. t ∈ [a, ti [ ∪ ]ti , b];
150
βkii ∈ R+ (i = 1, . . . , n; k = 1, . . . , m0 ); βkil , βki ∈ R (i ̸= l; i, l = 1, . . . , n;
k = 1, . . . , m0 ) are such that βkil ≥ 0 and βki ≥ 0 if τk ̸= ti , βki ≤ 0 if τk =
ti , and 0 ≤ βki < |ηi |−1 if τk > ti , ηi < 0; γi (t, t) = 1, γi (t, s) = γi−1 (s, t)
for t < s and
( ∫t
) ∏
γi (t, s) = exp ηi βi (τ ) dτ
(1+ηi βki ) for t > s (i = 1, . . . , n).
s≤τk <t
s
Let, moreover, the condition
gii < 1 (i = 1, . . . , n)
(12)
hold and the real part of every characteristic value of the matrix (ξil )ni,l=1
be negative, where
(
)
ξil = hil δil + (1 − δil )hi − hii gil (i, l = 1, . . . , n),
(
)−1
{
}
gil = µi 1−µi γi (si , ti )
γil (τi )+max γil (a), γil (b) (i, l = 1, . . . , n),
γil (t) = αil (t) − αil (ti ) for t ≤ ti ,
γil (t) = αil (t) − αil (ti ) for t > ti if ti ̸∈ {τ1 , . . . , τm0 },
γil (t) = αil (t) − αil (ti ) − (1 − δil )βki
for t > ti if ti ∈ {τ1 , . . . , τm0 } (i, l = 1, . . . , n),
hi = 1 if 0 ≤ µi ≤ 1,
)−1
hi = 1 + (µi − 1) 1 − µi γi (si , ti )
if µi > 1 (i = 1, . . . , n),
(
∫t
αil (t) ≡
βil (τ ) dτ +
a
∑
βkil (i, l = 1, . . . , n).
a≤τk <t
Then the problem (1), (2); (4) is solvable.
Corollary 4 has the following form if we assume therein that βi (1 ≡ 1,
βil (t) ≡ 0, βki = 0, hkil = βkil (i, l = 1, . . . , n; k = 1, . . . , m0 , and pil (t) ≡
pil = const, where pil = hil (i, l = 1, . . . , n).
Corollary 5. Let the conditions (6) and (9) be fulfilled on Rn , the inequalieloc ([a, b] \ {τk }m0 , Rn ) ∩ BVs ([a, b], Rn ) and
ties (10) be fulfilled on the set C
k=1
let the condition (11) hold, where pii < 0, pil ≥ 0 (i ̸= l; i, l = 1, . . . , n);
hkii ∈ R+ , hkil ∈ R (k = 1, . . . , m0 ; i ̸= l; i, l = 1, . . . , n), qi ∈ L([a, b], R+ )
(i = 1, . . . , n), µi and γ ∈ R+ , si ∈ [a, b] and si ̸= ti (i = 1, . . . , n),
γi (si , ti ) = exp(ηi |si − ti |) (i = 1, . . . , n), ηi < 0 (i = 1, . . . , n), and
qi ∈ L([a, b], R+ ) (i = 1, . . . , n). Let, moreover, the condition (12) hold
and the real part of every characteristic value of the matrix (ξil )ni,l=1 be
negative, where
(
)
ξil = pil δil + (1 − δil )hi − pii gil (i, l = 1, . . . , n),
gil = µi (1 − µi γi (si , ti ))−1 γil (si ) + max{γil (a), γil (b)} (i, l = 1, . . . , n),
151
γil (t) ≡ |αil (t) − αil (ti )| (i, l = 1, . . . , n),
hi = 1 for 0 ≤ µi ≤ 1,
)−1
hi = 1 + (µi − 1) 1 − µi γi (si , ti )
for µi > 1 (i = 1, . . . , n),
∑
αil (a) = 0, αil (t) =
hkil for t ∈]a, b] (i, l = 1, . . . , n).
(
a<τk <t
Then the problem (1), (2), (3) is solvable.
Corollary 4 has the following form if we assume therein that βi (1 ≡ 1,
βil (t) ≡ 0, βki = 1, βkil = 0 (i, l = 1, . . . , n; k = 1, . . . , m0 , and pil (t) ≡
pil = const, where pil = hil (i, l = 1, . . . , n).
Corollary 6. Let τk ̸= ti (i = 1, . . . , n; k = 1, . . . , m0 ), the conditions (9)
and
[
]
Iki (x1 , . . . , xn ) sgn (τk − ti )xi ≤
≤
n
∑
pil |xl | + qi (τk ) (k = 1, . . . , m0 ; i = 1, . . . , n)
l=1
eloc ([a, b] \
be fulfilled on Rn , the inequalities (10) be fulfilled on the set C
m0
n
n
{τk }k=1 , R ) ∩ BVs ([a, b], R ) and let the condition (11) hold, where pii <
0, pil ≥ 0 (i ̸= l; i, l = 1, . . . , n); µi and γ ∈ R+ , qi ∈ L([a, b], R+ )
(i = 1, . . . , n), si ∈ [a, b] and si ̸= ti (i = 1, . . . , n), qi ∈ L([a, b], R+ )
(i = 1, . . . , n), γi (si , ti ) = (1 + ηi )ν(si )−ν(ti ) exp(ηi |si − ti |) (i = 1, . . . , n),
and −1 < ηi < 0 (i = 1, . . . , n). Let, moreover, the condition (12) hold and
the real part of every characteristic value of the matrix (ξil )ni,l=1 be negative,
where
(
)
ξil = pil δil + (1 − δil )hi − pii gil (i, l = 1, . . . , n),
gil = µi (1 − µi γi (si , ti ))−1 γil (si ) + max{γil (a), γil (b)} (i, l = 1, . . . , n),
γil (t) = 0 for t ≤ ti , γil (t) = 0 for t > ti if ti ̸∈ {τ1 , . . . , τm0 },
γil (t) = δil − 1 for t > ti if ti ∈ {τ1 , . . . , τm0 } (i, l = 1, . . . , n),
hi = 1 for 0 ≤ µi ≤ 1,
(
)−1
hi = 1 + (µi − 1) 1 − µi γi (si , ti )
for µi > 1 (i = 1, . . . , n).
Then the problem (1), (2), (3) is solvable.
Theorem 3. Let the conditions
[
]
fi (t, x1 , . . . , xn ) − fi (t, y1 , . . . , yn ) sgn[(t − ti )(xi − yi )] ≤
≤
n
∑
0
pil (t)|xl −yl | for almost every t ∈ [a, b] \ {τk }m
k=1 (i = 1, . . . , n) (13)
l=1
152
and
[
Iki (x1 , . . . , xn ) − Iki (y1 , . . . , yn )]θ(τk − ti ) sgn(xi − yi )] ≤
≤
n
∑
hkil |xl − yl | (k = 1, . . . , m0 ; i = 1, . . . , n) (14)
l=1
n
be fulfilled on R , the inequalities
φi (x1 , . . . , xn ) − φi (y1 , . . . , yn ) ≤
(
)
≤ φ0i |x− y1 |, . . . , |xn − yn | (i = 1, . . . , n)
eloc ([a, b]\{τk }m0 , Rn )∩BVs ([a, b], Rn ), where pil , hkil
be fulfilled on the set C
k=1
and φ0i (i, l = 1, . . . , n; k = 1, . . . ; m0 ) are such that the condition (7) holds.
Then the problem (1), (2), (3) has one and only one solution.
Corollary 7. Let the conditions (13) and (14) be fulfilled on Rn , where
pii ∈ Lµ ([a, b], R), pil ∈ Lµ ([a, b], R+ ) (i ̸= l; i, l = 1, . . . , n), qi ∈ L([a, b], R+ )
(i = 1, . . . , n), hkil = αkl pil (τk ) (k = 1, . . . , m0 ; i, l = 1, . . . , n), 1 ≤ µ ≤
+∞, αik ∈ R+ (i = 1, . . . , n; k = 1, . . . , m0 ). Let, moreover, the conditions
(8) and
φi (x1 , . . . , xn ) − φi (y1 , . . . , yn ) ≤
m0
n (
[∑
] ν1 )
∑
≤
γ1il ∥xl − yl ∥Lν + γ2il
|xl (τl − yl |ν
k=l
k=1
eloc ([a, b]\{τk }m0 , Rn )∩BVs ([a, b], Rn ) (i = 1, . . . , n),
be fulfilled on the set C
k=1
where γ1ik , γ2ik ∈ R+ (i, k = 1, . . . , n), µ1 + ν2 = 1, γ ∈ R+ , and H0 =
(H0jm )2j,m=1 is the 2n×2n-matrix defined in Corollary 1. Then the problem
(1), (2), (3) has one and only one solution.
Corollary 8. Let the conditions (13) and (14) be fulfilled on Rn , where pii ∈
Lµ ([a, b], R), pil ∈ Lµ ([a, b], R+ ) (i ̸= l; i, l = 1, . . . , n), hkil = αkl pil (τk )
(k = 1, . . . , m0 ; i, l = 1, . . . , n), 1 ≤ µ ≤ +∞, αik ∈ R+ (i = 1, . . . , n; k =
1, . . . , m0 ). Let, moreover, the inequality (8) hold, where H0 = (H0jm )2j,m=1
is the 2n×2n-matrix is defined in Corollary 2. Then the problem (1), (2), (3)
has one and only one solution.
Corollary 9. Let the conditions (14) and
[
]
fi (t, x1 , . . . , xn ) − fi (t, y1 , . . . , yn ) sgn[(t − ti )(xi − yi )] ≤
≤
n
∑
0
pil |xl − yl | for almost every t ∈ [a, b] \ {τk }m
k=1 (i = 1, . . . , n) (15)
l=1
be fulfilled on Rn , where pii ∈ R, pil ≥ 0 (i ̸= l; i, l = 1, . . . , n), hkil = αk pil
(k = 1, . . . , m0 ; i, l = 1, . . . , n), αk ≥ 0 (i = 1, . . . , n; k = 1, . . . , m0 ).
Let, moreover, the inequality (8) hold, where the matrix H0 is defined as in
Corollary 3. Then the problem (1), (2), (4) has one and only one solution.
153
Corollary 10. Let the conditions (14) and (15) be fulfilled on Rn and let
the condition (11) hold, where pii < 0, pil ≥ 0 (i ̸= l; i, l = 1, . . . , n);
hkii ∈ R+ , hkil ∈ R (k = 1, . . . , m0 ; i ̸= l; i, l = 1, . . . , n), si ∈ [a, b] and
si ̸= ti (i = 1, . . . , n), γi (si , ti ) = exp(ηi |si − ti |) (i = 1, . . . , n), ηi < 0
(i = 1, . . . , n), and µi ∈ R+ ) (i = 1, . . . , n). Let, moreover, the condition
(12) hold and the real part of every characteristic value of the matrix (ξil )ni,l=1
be negative, where the matrix (ξil )ni,l=1 is defined as in Corollary 5. Then
the system (1), (2) has one and only one solution under the condition
xi (ti ) = λi xi (si ) + βi (i = 1, . . . , n)
for every γi ∈ [−µi , µi ] and βi ∈ R (i = 1, . . . , n).
Corollary 11. Let τk ̸= ti (i = 1, . . . , n; k = 1, . . . , m0 ), the conditions
(15) and
[
]
Iki (x1 , . . . , xn ) − Iki (y1 , . . . , yn ) sgn[(τk − ti )(xi − yi )] ≤
≤
n
∑
pil |xl − yl | (k = 1, . . . , m0 ; i = 1, . . . , n)
l=1
be fulfilled on R and let the condition (11) hold, where pii < 0, pil ≥ 0
(i ̸= l; i, l = 1, . . . , n); µi R+ (i = 1, . . . , n), si ∈ [a, b] and si ̸= ti (i =
1, . . . , n), γi (si , ti ) = (1 + ηi )ν(si )−ν(ti ) exp(ηi |si − ti |) (i = 1, . . . , n), and
−1 < ηi < 0 (i = 1, . . . , n). Let, moreover, the condition (12) hold and the
real part of every characteristic value of the matrix (ξil )ni,l=1 be negative,
where the matrix (ξil )ni,l=1 is defined as in Corollary 6. Then the statement
of Corollary 10 is true.
n
Remarks 1–3 given above are true for the uniqueness case, as well.
Acknowledgement
This work is supported by the Georgian National Science Foundation
(Grant No. FR/182/5-101/11).
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(Received 12.01.2014)
Authors’ addresses:
Malkhaz Ashordia
1. A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State
University, 6 Tamarashvili St., Tbilisi 0177, Georgia.
2. Sokhumi State University, 12 Politkovskaia St., Tbilisi 0186, Georgia.
E-mail: ashord@rmi.ge
Goderdzi Ekhvaia
Sokhumi State University, 12 Politkovskaia St., Tbilisi 0186, Georgia.
E-mail: goderdzi.ekhvaia@mail.ru