Mem. Differential Equations Math. Phys. 47 (2009), 159–167
Short Communications
M. Ashordia
ON SOLVABILITY OF BOUNDARY VALUE PROBLEMS ON
AN INFINITY INTERVAL FOR NONLINEAR TWO
DIMENSIONAL GENERALIZED AND IMPULSIVE
DIFFERENTIAL SYSTEMS
Abstract. Sufficient conditions are given for the solvability of boundary value problems on an infinite interval for nonlinear two dimensional
generalized and impulsive differential systems.
!
"
2000 Mathematics Subject Classification: 34K10, 34K45.
Key words and phrases: Two dimensional generalized and impulsive differential systems, the Lebesgue–Stiltjes integral, infinite interval, boundary
value problem, sufficient conditions.
Let c ∈ R, aik : R+ → R (i, k = 1, 2) be nondecreasing continuous from
the left functions, and let fk : R+ × R2 → R be a vector-function belonging
to the Carathéodory class corresponding to the aik for every i, k ∈ {1, 2}.
In this paper we investigate the question of existence of solutions for the
two dimensional generalized differential system
dxi (t) = f1 (t, x1 (t), x2 (t)) · dai1 (t)+
+ f2 (t, x1 (t), x2 (t)) · dai2 (t) for t ∈ R+ (i = 1, 2), (1)
satisfying one of the following two conditions
o
n
x1 (0) = c, sup |x1 (t)| + |x2 (t)| : t ∈ R+ < ∞
and
o
n
sup |x1 (t)| + |x2 (t)| : t ∈ R+ < ∞.
(2)
(3)
We give sufficient conditions for the existence of solutions of the boundary
value problems (1), (2) and (1), (3). Analogous results are contained in [10],
[11], [13]–[17] for ordinary differential and functional differential systems.
Reported on the Tbilisi Seminar on Qualitative Theory of Differential Equations on
July 14, 2008.
160
The theory of generalized ordinary differential equations enables one to
investigate ordinary differential, impulsive and difference equations from a
common point of view (see, e.g., [1]–[9], [12], [23], and references therein).
We realize the obtained result for the following second order system of
impulsive equations
dxi
= fi (t, x1 , x2 ) for almost all t ∈ R+ \ {τ1 , τ2 , . . .} (i = 1, 2),
(4)
dt
xi(τk+)−xi (τk−) = αki Iki (x1(τk−), x2(τk−)) for k∈ {1, 2, . . .} (i = 1, 2), (5)
where 0 < τ1 < τ2 < . . . , τk → ∞ (k → ∞) (we will assume τ0 = 0 if
necessary), αki ∈ R (i = 1, 2; k = 1, 2, . . .), fi ∈ Kloc (R+ × R2 , R) (i = 1, 2),
and Iki : R2 → R (i = 1, 2; k = 1, 2, . . .) are continuous operators.
Throughout the paper the following notation and definitions will be used.
R = ] − ∞, +∞[ , R+ = [0, +∞[ ; [a, b] (a, b ∈ R) is a closed segment.
Rn×m is the set of all real n × m-matrices X = (xij )n,m
i,j=1 .
n
n×1
R = R
is the set of all real column n-vectors x = (xi )ni=1 , Rn+ =
n×1
R+ .
diag(λ1 , . . . , λn ) is the diagonal matrix with the diagonal elements
λ1 , . . . , λ n .
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 limits 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).
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
BVloc (R, Rn×m ) is the set of all matrix-functions X : R → Rn×m for
b
which ∨(X) < +∞) for every a, b ∈ R (a < b).
a
sj : BV([a, b], R) → BV([a, b], R) (j = 0, 1, 2) are the operators defined,
respectively, by
s1 (x)(t) =
X
a<τ ≤t
s1 (x)(a) = s2 (x)(a) = 0,
X
d1 x(τ ) and s2 (x)(t) =
d2 x(τ ) for a < t ≤ b
a≤τ <t
and
s0 (x)(t) = x(t) − s1 (x)(t) − s2 (x)(t) for t ∈ [a, b].
A : BVloc (R, R) × BVloc (R, R) → BVloc (R, R) is the operator defined by
A(x, y)(0) = 0,
161
X
A(x, y)(t) = y(t) +
d1 x(τ ) · 1 − d1 x(τ )
0<τ ≤t
X
−
d2 x(τ ) · 1 + d2 x(τ )
0≤τ <t
X
A(x, y)(t) = y(t) −
+
X
−1
d1 x(τ ) · 1 − d1 x(τ )
t<τ ≤0
d2 x(τ ) · 1 + d2 x(τ )
t≤τ <0
−1
d2 y(τ ) for t > 0,
−1
−1
d1 y(τ )−
d1 y(τ )+
d2 y(τ ) for t < 0
for every x ∈ BVloc (R, R) such that for every x ∈ BVloc (R, R) such that
1 + (−1)j dj x(t) 6= 0 for t ∈ R (j = 1, 2).
If g : [a, b] → R is a nondecreasing function, x : [a, b] → R and a ≤ s <
t ≤ b, then
Zt
x(τ ) dg(τ ) =
s
Z
x(τ ) ds0 (g)(τ ) +
where
x(τ )d1 g(τ ) +
s<τ ≤t
]s,t[
R
X
X
x(τ )d2 g(τ ),
s≤τ <t
x(τ ) ds0 (g)(τ ) is the Lebesgue–Stieltjes integral over the open
]s,t[
interval ]s, t[ with respect to the measure µ0 (s0 (g)) corresponding to the
function s0 (g).
If a = b, then we assume
Zb
x(t) dg(t) = 0.
a
If g(t) ≡ g1 (t) − g2 (t), where g1 and g2 are nondecreasing functions, then
Zt
x(τ ) dg(τ ) =
s
Zt
s
x(τ ) dg1 (τ ) −
Zt
x(τ ) dg2 (τ ) for s ≤ t.
s
L([a, b], R; g) is the set of all functions x : [a, b] → R measurable and
integrable with respect to the measures µ(gi ) (i = 1, 2), i.e., such that
Zb
|x(t)| dgi (t) < +∞ (i = 1, 2).
a
A matrix-function is said to be continuous, nondecreasing, integrable,
etc., if each of its components is such.
l×n
If G = (gik )l,n
is a nondecreasing matrix-function
i,k=1 : [a, b] → R
n×m
and D ⊂ R
, then L([a, b], D; G) is the set of all matrix-functions X =
(xkj )n,m
k,j=1 : [a, b] → D such that xkj ∈ L([a, b], R; gik ) (i = 1, . . . , l; k =
162
1, . . . , n; j = 1, . . . , m);
Zt
dG(τ ) · X(τ ) =
X
n Zt
xkj (τ )dgik (τ )
k=1 s
s
Sj (G)(t) ≡ sj (gik )(t)
l,m
for a ≤ s ≤ t ≤ b,
i,j=1
l,n
i,k=1
(j = 0, 1, 2).
If D1 ⊂ Rn and D2 ⊂ Rn×m , then K([a, b] × D1 , D2 ; G) 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 µ(gik )-measurable for every x ∈ D1 ;
b) the function fkj
(t, ·) : D1 → D2 is
continuous for µ(gik )-almost every
t ∈ [a, b], and sup |fkj (·, x)| : x ∈ D0 ∈ L([a, b], R; gik ) for every compact
D0 ⊂ D 1 .
If Gj : [a, b] → Rl×n (j = 1, 2) are nondecreasing matrix-functions,
G = G1 − G2 and X : [a, b] → Rn×m , then
Zt
dG(τ ) · X(τ ) =
s
Zt
dG1 (τ ) · X(τ ) −
s
Zt
dG2 (τ ) · X(τ ) for s ≤ t,
s
Sk (G) = Sk (G1 ) − Sk (G2 ) (k = 0, 1, 2),
L([a, b], D; G) =
2
\
L([a, b], D; Gj ),
2
\
K([a, b] × D1 , D2 ; Gj ).
j=1
K([a, b] × D1 , D2 ; G) =
j=1
Lloc (R, D; G) is the set of all matrix-functions X : R → D such that
its restriction on [a, b] belongs to L([a, b], D; G) for every a and b from R
(a < b).
K([a, b] × D1 , D2 ; G) is the set of all matrix-functions F = (fkj )n,m
k,j=1 :
R × D1 → D2 such that its restriction on [a, b] belongs to K([a, b], D; G) for
every a and b from R (a < b).
If G(t) ≡ diag(t, . . . , t), then we omit G in the notation containing G.
The inequalities between the vectors and between the matrices are understood componentwise.
A vector-function x = (xi )21 ∈ BVloc (R+ , R2 ) is said to be a solution of
the system (1) if
xi (t) = xi (s) +
Zt
f1 (τ, x1 (τ ), x2 (τ )) · dai1 (τ )+
s
+
Zt
s
f2 (τ, x1 (τ ), x2 (τ )) · dai2 (t) for 0 ≤ s ≤ t (s, t ∈ R) (i = 1, 2).
163
If s ∈ R and β ∈ BVloc (R, R) are such that
1 + (−1)j dj β(t) 6= 0 for (−1)j (t − s) < 0 (j = 1, 2),
then by γβ (·, s) we denote the unique solution of the Cauchy problem
dγ(t) = γ(t) dβ(t), γ(s) = 1.
It is known (see [9], [12]) that
γβ (t, s) = exp ξβ (t) − ξβ (s)
×
Y
Y
s<τ ≤t
sgn 1 − d1 β(τ ) ×
sgn 1 + d2 β(τ )
s≤τ <t
for t > s,
γβ (t, s) = γβ−1 (s, t) for t < s,
where
ξβ (t) = s0 (β)(t) − s0 (β)(0)−
X
X
−
ln 1 − d1 β(τ ) +
ln 1 + d2 β(τ ) for t > 0,
0<τ ≤t
0≤τ <t
ξβ (t) = s0 (β)(t) − s0 (β)(0)+
X
X
+
ln 1 − d1 β(τ ) −
sgn 1 + d2 β(τ ) for t < 0.
t<τ ≤0
t≤τ <0
Remark 1. Let β ∈ BV([a, b], R) be such that
1 + (−1)j dj β(t) > 0 for t ∈ [a, b] (j = 1, 2).
Let, moreover, one of the functions β, ξβ and A(β, β) be nondecreasing
(nonincreasing). Then the other two functions will be nondecreasing (nonincreasing) as well.
Let δ > 0. We introduce the operators
n
o
ν1δ (ξ)(t) = sup τ ≥ t : ξ(τ ) ≤ ξ(t+) + δ
and
n
o
ν−1δ (η)(t) = inf τ ≤ t : η(τ ) ≤ η(t−) + δ ,
respectively, on the set of all nondecresing functions ξ : R → R and on the
set of all nonincreasing functions η : R → R.
e
C([a,
b], D), where D ⊂ Rn×m , is the set of all absolutely continuous
matrix-functions X : [a, b] → D;
eloc (R+ \ {τ1 , τ2 , . . .}, D) is the set of all matrix-functions X : R+ → D
C
whose restriction to an arbitrary closed interval [a, b] from R+ \ {τ1 , τ2 , . . .}
e
belongs to C([a,
b], D).
L([a, b], D) is the set of all matrix-functions X : [a, b] → D, measurable
and integrable.
164
Lloc (R+ \ {τ1 , τ2 , . . .}, D) is the set of all matrix-functions X : R+ → D
whose restriction to an arbitrary closed interval [a, b] from R+ \ {τ1 , τ2 , . . .}
e
belongs to C([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 all t ∈ [a, b], and
sup{|fkj (· , x)| : x ∈ D0 } ∈ L([a, b], R; gik ) for every compact D0 ⊂ D1 .
Kloc (R+ × D1 , D2 ) is the set of all mappings F : R+ × D1 → D2 whose
restriction to an arbitrary closed interval [a, b] from R+ \{τ1 , τ2 , . . .} belongs
to K([a, b] × D1 , D2 ).
By a solution of the impulsive system (3), (4) we understand a continuous
eloc (R+ \ {τ1 , τ2 , . . .}) ∩ BVloc s(R+ , Rn )
from the left vector-function x ∈ C
0
satisfying both the system (1) for a.a. t ∈ R+ \ {τk }m
k=1 and the relation (2)
for every k ∈ {1, 2, . . .}.
Theorem 1. Let
0 ≤ d2 (ai1 (t) + ai2 (t)) < |ηii |−1 for t ∈ R+ (i = 1, 2),
1 + σi d2 aii (t) > 0 for t ∈ R+ (i = 1, 2),
σi fk (t, x1 , x2 ) sgn xi ≤ ηi1 |x1 | + ηi2 |x2 | + qk (t)
for µ(aik )-almost all t ∈ R+ and x1 , x2 ∈ R (i, k = 1, 2),
and let the real part of every eigenvalue of the matrix (ηil )2i,l=1 be negative,
where σ1 = 1, σ2 = −1 (σ1 = σ2 = −1), η12 , η21 ∈ R; ηii < 0 (i = 1, 2),
qk ∈ Lloc (R+ , R; a1k ) ∩ Lloc (R+ , R; a2k ) (k = 1, 2). Let, moreover,
σi lim inf ξσi aii (t) − ξσi aii (0) > δ > 0 (i = 1, 2)
t→∞
for some δ > 0,
νZi (t)
sup
|qk (τ )| ds0 (aik )(τ )+
t
+
X
(1 + σi d2 aii (t))−1 |qk (τ )|d2 aik (τ ) : t ∈ R+
t<τ ≤νi (t)
< ∞ (i, k = 1, 2),
+∞
Z
sil = γσi aii (t, s) dA(σi aii , ail )(s) < ∞ (i 6= l; i, l = 1, 2)
0
and
s1 s2 < 1,
where νi (t) ≡ νσi δ (−ξσi aii )(t) (i = 1, 2). Then the problem (1), (2) ((1), (3))
is solvable.
165
Consider now the impulsive system (4), (5).
Quite a number of issues of the theory of systems of differential equations with impulsive effect (both linear and nonlinear) have been studied
sufficiently well (for survey of the results on impulsive systems see, e.g., [6],
[8], [18]–[22], and references therein).
It is easy to show that the vector-function x = (xi )2i=1 is a solution of
the impulsive system (4), (5) if and only if it is a solution of the system (1),
where a12 (t) = a21 (t) ≡ 0,
X
aii (t) ≡ t +
αki (i = 1, 2),
k: 0≤τk <t
fi (τk , x1 , x2 ) = Iki (x1 , x2 ) for x1 , x2 ∈ R (i = 1, 2; k = 1, 2, . . .).
It is evident that aii (i = 1, 2) are nondecreasing if αki ≥ 0, d2 aii (τk ) =
αki and d2 aii (t) = 0 it t 6= τk (i = 1, 2; k = 1, 2, . . .). Moreover, they are
continuous from the left. In this case
X
ξσi aii ≡ σi t +
ln |1 + σi αki | (i = 1, 2).
(6)
k: 0≤τk <t
Theorem 2. Let
0 ≤ αki < |ηii |−1 (i = 1, 2; k = 1, 2, . . .),
(7)
1 + σi αki > 0 (i = 1, 2; k = 1, 2, . . .),
(8)
σi fi (t, x1 , x2 ) sgn xi ≤ ηi1 |x1 | + ηi2 |x2 | + qi (t)
for almost all t ∈ R+ and x1 , x2 ∈ R (i = 1, 2; k = 1, 2, . . .),
σi Iki (x1 , x2 ) sgn xi ≤ ηi1 |x1 | + ηi2 |x2 | + qki
for x1 , x2 ∈ R (i = 1, 2; k = 1, 2, . . .),
and let the real part of every eigenvalue of the matrix (ηil )2i,l=1 be negative,
where σ1 = 1, σ2 = −1 (σ1 = σ2 = −1), η12 , η21 ∈ R, ηii < 0 (i = 1, 2),
qi ∈ Lloc (R+ , R) (i = 1, 2). Let, moreover,
X
lim inf t + σi
ln(1 + σi αki ) > δ > 0 (i = 1, 2)
(9)
t→∞
k: 0≤τk <t
for some δ > 0,
νZi (t)
sup
|qi (τ )|d(τ ) +
t
X
(1 + σi αki )
−1
|qki | : t ∈ R+
k: 0≤τk <νi (t)
<∞
(i = 1, 2),
where the functions νi (t) ≡ νσi δ (−ξσi aii )(t) (i = 1, 2) are defined according
to (6). Then the problem (4), (5); (2) ((4), (5); (3)) is solvable.
Remark 2. By condition (7), the conditions (8) and (9) are fulfilled if
σi = 1.
166
Acknowledgement
This work is supported by the Georgian National Science Foundation
(Grant No. GNSF/ST06/3-002)
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(Received 17.07.2008)
Author’s address:
A. Razmadze Mathematical Institute
1, M. Aleksidze St., Tbilisi 0193
Georgia
Sukhumi State University
12, Jikia St., Tbilisi 0186
Georgia
E-mail: ashord@rmi.acnet.ge