Mem. Differential Equations Math. Phys. 43 (2008), 143–152
Short Communications
Malkhaz Ashordia
ON THE SOLVABILITY OF A MULTIPOINT BOUNDARY
VALUE PROBLEM FOR SYSTEMS OF NONLINEAR
GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS
Abstract. Necessary and sufficient conditions and effective sufficient conditions are given for the existence of solutions of the multipoint boundary
value problem for a system of nonlinear generalized ordinary differential
equations.
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2000 Mathematics Subject Classification: 34K10.
Key words and phrases: Systems of nonlinear generalized ordinary
differential equations, the Lebesgue–Stiltjes integral, multipoint boundary
value problem, solvability criterion, effective conditions.
Let σ1 , . . . , σn ∈ {−1, 1}; for m ∈ {1, 2} and i, k ∈ {1, . . . , n}, amik :
[−a, a] → R be nondecreasing functions continuous at the points −a and a;
A=
aik (t)
n
(aik )i,k=1 ,
≡ a1ik (t) − a2ik (t),
Am = (amik )ni,k=1 (m = 1, 2);
f = (fk )nk=1 : [−a, a] × Rn → Rn be a vector-function belonging to the
Carathéodory class corresponding to the matrix-function A, and ϕi :
BVs ([−a, a], Rn ) → R (i = 1, . . . , n) be continuous functionals which are
nonlinear in general.
For the system of generalized ordinary differential equations
dx(t) = dA(t) · f (t, x(t)),
(1)
where x = (xi )ni=1 , consider the multipoint boundary value problem
xi (−σi a) = ϕi (x1 , . . . , xn ) (i = 1, . . . , n).
(2)
In this paper necessary and sufficient conditions as well effective sufficient
conditions are given for the existence of solutions of the boundary value
Reported on the Tbilisi Seminar on Qualitative Theory of Differential Equations on
October 16, 2008.
144
problem (1), (2). Analogous results are contained in [1]–[4] for multipoint
boundary value problems for systems of ordinary differential equations.
The theory of generalized ordinary differential equations enables one to
investigate ordinary differential, impulsive and difference equations from a
common point of view (see [5]–[16]).
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 space of all real n × m-matrices X = (xij )n,m
i,j=1 with the
norm
kXk = max
j=1,...,m
n
X
|xij |;
i=1
o
n
n×m
R+
= (xij )n,m
i,j=1 : xij ≥ 0 (i = 1, . . . , n; j = 1, . . . , m) .
n
=
Rn = Rn×1 is the space of all real column n-vectors x = (xi )ni=1 ; R+
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 6= j
(i, j = 1, . . . , n).
b
∨(X) is the total variation of the matrix-function X : [a, b] → R n×m ,
a
i.e., 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);
kXks = sup kX(t)k : 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 ), k · ks );
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].
145
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].
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
and if a > b, then we assume
Zb
x(t) dg(t) = −
a
Za
x(t) dg(t).
b
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 =
146
(xkj )n,m
k,j=1 : [a, b] → D such that xkj ∈ L([a, b], R; gik ) (i = 1, . . . , l; k =
1, . . . , n; j = 1, . . . , m);
Zt
dG(τ ) · X(τ ) =
s
X
n Zt
xkj (τ )dgik (τ )
for a ≤ s ≤ t ≤ b,
i,j=1
k=1 s
Sj (G)(t) ≡ sj (gik )(t)
l,m
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 ⊂ D1 .
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
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 ∈ BV([−a, a], Rn ) is said to be a solution of the
system (1) if
x(t) = x(s) +
Zt
dA(τ ) · f (τ, x(τ )) for − a ≤ s ≤ t ≤ a.
s
By a solution of the system of generalized ordinary differential inequalities
dx(t) ≤ dA(t) · f (t, x(t)) (≥)
147
we mean a vector-function x ∈ BV([−a, a], Rn ) such that
x(t) ≤ x(s) +
Zt
dA(τ ) · f (τ, x(τ )) (≥) for − a ≤ s ≤ t ≤ a.
s
If s ∈ R and β ∈ BV[a, b], 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 [6], [8]) that

Y

exp(s
(β)(t)
−
s
(β)(s))
(1 − d1 β(τ ))−1 ×

0
0



s<τ ≤t

Y



×
(1
+
d
β(τ
))
for t > s,

2



s≤τ <t

Y
γβ (t, s) = exp(s0 (β)(t) − s0 (β)(s))
(1 − d1 β(τ ))×



t<τ
≤s

Y



×
(1 + d2 β(τ ))−1 for t < s,




t≤τ <s



1
for t = s.
(3)
Definition 1. Let σ1 , . . . , σn ∈ {−1, 1}. We say that the pair
((cil )ni,l=1 ; ϕ0i )ni=1 ) consisting of a matrix-function (cil )ni,l=1∈ BV([a, b],Rn×n )
n
and a positive homogeneous nondecreasing operator (ϕ0i )ni=1 : BVs ([a, b],R+
)
n
σ1 ,...,σn
→ R+ belongs to the set U
if the functions cil (i 6= l; i, l = 1, . . . , n)
are nondecreasing on [a, b] and continuous at the point ti = −σi a,
dj cii (t) ≥ 0 for t ∈ [−a, a] (j = 1, 2; i = 1, . . . , n)
and the problem
σi dxi (t) ≤
n
X
xl (t)dcil (t) for t ∈ [−a, a] \ {−σi a} (i = 1, . . . , n),
l=1
(−1)j dj xi (−σi a) ≤ xi (−σi a)dj cii (−σi a) (j = 1, 2; i = 1, . . . , n);
xi (−σi a) ≤ ϕ0i |x1 |, . . . , |xn | (i = 1, . . . , n)
has no nontrivial non-negative solution.
The set U σ1 ,...,σn has been introduced by I. Kiguradze for ordinary differential equations (see [1], [2]).
Theorem 1. The problem (1), (2) is solvable if and only if there exist vector-functions αm = (αmi )ni=1 ∈ BV ([−a, a], Rn ) (m = 1, 2) and
148
matrix-functions (βmik )ni,k=1 : [−a, a] → Rn×n (m = 1, 2) such that βmik ∈
L([−a, a], R; ajik ) (m, j = 1, 2; i, k = 1, . . . , n),
αmi (t) ≡ αmi (−σi a) +
n Zt
X
k=1
βmik (τ ) da1ik (τ ) −
Zt
β3−mik (τ ) da2ik (τ )
−σi a
−σi a
(m = 1, 2; i = 1, . . . , n),
α1 (t) ≤ α2 (t) for t ∈ [−a, a],
(4)
m
(−1) σi fk (t, x1 , . . . , xi−1 , αji (t), xi+1 , . . . , xn ) − βmik (t) ≤ 0
for µ(a1+|m−j|ik )-almost every t ∈ [−a, a],
α1 (t) ≤ (xl )nl=1 ≤ α2 (t) (m, j = 1, 2; i, k = 1, . . . , n),
(−1)
m
xi −(−1)
j
n
X
j
fk (t, x1 , . . . , xn )dj aik (t)−αmi (t)−(−1) dj αmi (t) ≤
k=1
≤ 0 for t ∈ [−a, a], α1 (t) ≤ (xl )nl=1 ≤ α2 (t),
(−1)j σi > 0 (m, j = 1, 2; i = 1, . . . , n)
(5)
and the inequalities
α1i (−σi a) ≤ ϕi (xl , . . . , xn ) ≤ α2i (−σi a) (i = 1, . . . , n)
(6)
are fulfilled on the set {(xl )nl=1 ∈ BV([a, b], Rn ), α1 (t) ≤ (xl )nl=1 ≤ α2 (t) for
t ∈ [−a, a]}.
Corollary 1. Let the matrix-function A(t) = (aik )ni,k=1 be nondecreasing on [−a, a]. Then the problem (1), (2) is solvable if and only if there
exist vector-functions αm = (αmi )ni=1 ∈ BV([−a, a], Rn ) (m = 1, 2) and
matrix-functions (βmik )ni,k=1 : [−a, a] → Rn×n (m = 1, 2) such that βmik ∈
L([−a, a], R; aik ) (m = 1, 2; i, k = 1, . . . , n),
αmi (t) ≡ αmi (−σi a) +
n Zt
X
l=1
βmik (τ )daik (τ )
−σi a
(m = 1, 2; i, k = 1, . . . , n),
the conditions (4)–(6) hold, and the inequalities
(−1)m σi fk (t, x1 , . . . , xi−1 , αji (t), xi+1 , . . . , xn ) − βjik (t) ≤ 0
(j = 1, 2; i, k = 1, . . . , n)
are fulfilled for µ(aik )-almost every t ∈ [−a, a] and α1 (t) ≤ (xl )nl=1 ≤ α2 (t).
Theorem 2. Let the condition
(−1)m+1 σi fk (t, x1 , . . . , xn ) sgn xi ≤
n
X
pmikl (t)|xl | + qk (t)
l=1
for µ(amik )-almost every t ∈ [−a, a] (m = 1, 2; i, k = 1, . . . , n)
(7)
149
be fulfilled on Rn , and let the inequalities
|ϕi (x1 , . . . , xn )| ≤ ϕ0i |x1 |, . . . , |xn | + ζi (i = 1, . . . , n)
be fulfilled on BV ([−a, a], Rn ), where (pmikl )nk,l=1 ∈ L([−a, a], Rn×n ; Am )
n
(m = 1, 2; i = 1, . . . , n), qk = (qki )ni=1 ∈ L([−a, a], R+
; Am ) (m = 1, 2), ζi ∈
R+ (i = 1, . . . , n). Let, moreover, there exist a matrix-function (cil )ni,l=1 ∈
BV ([−a, a], Rn×n ) such that
(cil )ni,l=1 ; (ϕ0i )ni=1 ∈ U σ1 ,...,σn
and
2 X
n Z
X
t
pmikl (τ )damik (τ ) ≤ cil (t) − cil (s)
m=1 k=1 s
for
− a ≤ s < t ≤ a (i, l = 1, . . . , n).
Then the problem (1), (2) is solvable.
Corollary 2. Let there exist m, m1 ∈ {1, 2} such that m + m1 = 3 and
the conditions (7) and
(−1)m1 +1 σi fk (t, x1 , . . . , xn ) sgn xi ≤
n
X
ηil |xl | + qk (t)
l=1
for µ(am1 ik )-almost every t ∈ [−a, a] (i, k = 1, . . . , n)
are fulfilled on Rn , the inequalities
|ϕi (x1 , . . . , xn )| ≤ µi |xi (si )| + ζi (i = 1, . . . , n)
(8)
n
be fulfilled on BVs ([a, b], R ), and let
0 ≤ dj αi (t) < |ηii |−1 for (−1)j (t + σi a) > 0 (j = 1, 2; i = 1, . . . , n) (9)
and
µi γi (si , −σi a) < 1 (i = 1, . . . , n),
(10)
n×n
where (pmikl )nk,l=1 ∈ L([−a, a], R+
; Am ) (i = 1, . . . , n), ηil ∈ R+ (i 6= l;
n
i, l = 1, . . . , n), ηii < 0 (i = 1, . . . , n), qk = (qki )nk=1 ∈ L([−a, a], R+
; Am )
(m = 1, 2), ζi ∈ R+ (i = 1, . . . , n), µi ∈ R+ and si ∈ [−a, a], si 6= −σi a
(i = 1, . . . , n),
αi (t) ≡
n
X
am1 ik (t) (i = 1, . . . , n),
k=1
γi (t, s) ≡ γai (t, s) (i = 1, . . . , n),
ai (t) ≡ ηii σi αi (t) − αi (−σi a) (i = 1, . . . , n),
and the functions γai (i = 1, . . . , n) are defined according to (3). Let, moreover,
gii < 1 (i = 1, . . . , n)
150
and the real part of every characteristic value of the matrix (ξ il )ni,l=1 be
negative, where
ξil = ηil δil + (1 − δil )hi − ηii gil (i, l = 1, . . . , n),
−1
gil = µi 1 − µi γi (si , −σi a)
γil (si )+
+ max γil (−a), γil (a) (i, l = 1, . . . , n),
γil (−σi a) = 0, γil (t) = βil (t) − βil (−σi a) − (1 − δil )dj βil (−σi a)
for (−1)j (t + σi a) > 0 (j = 1, 2; i, l = 1, . . . , n),
βil (t) ≡
n Z
X
t
pmikl (τ )damik (τ ) (i = 1, . . . , n),
k=1−a
hi = 1 for µi ≤ 1 and
hi = 1 + (µi − 1)(1 − µi γi (si , −σi a))−1 for µi > 1 (i = 1, . . . , n).
Then the problem (1), (2) is solvable.
Remark 1. In Corollary 2 as the matrix-function C = (cil )ni,l=1 we take
cil (−σi a) = 0 (i, l = 1, . . . , n),
cil (t) = ηil αi (t) − αi (−σi a) − (−1)j dj αi (−σi a) +
+βil (t) − βil (−σi a) − (−1)j dj βil (−σi a)
for (−1)j (t + σi a) > 0 (j = 1, 2; i, l = 1, . . . , n).
If the matrix-function A = (aik )ni,k=1 : [−a, a] → Rn×n is nondecreasing,
then Corollary 2 has the following form.
Corollary 3. Let the matrix-function A = (aik )ni,k=1 : [−a, a] → Rn×n
be nondecreasing, the conditions (8)–(10) hold, the condition
σi fk (t, x1 , . . . , xn ) sgn xi ≤
n
X
ηil |xl | + qk (t)
l=1
for µ( aik )-almost every t ∈ [−a, a] (i, k = 1, . . . , n)
(11)
be fulfilled on Rn and let the
n real part of every characteristic value of the
matrix ηil (δil + (1 − δil )hi ) i,l=1 be negative, where
αi (t) ≡
n
X
aik (t) (i = 1, . . . , n),
k=1
and the functions γi (t, s) (i = 1, . . . , n) and ai (t) (i = 1, . . . , n) and the
numbers hi (i = 1, . . . , n) are defined as in Corollary 2. Then the problem
(1), (2) is solvable.
Corollary 4. Let the matrix-function A = (aik )ni,k=1 : [−a, a] → Rn×n
be nondecreasing and continuous from the left, the conditions (8), (10), (11)
151
and
0 ≤ d2 αi (t) < |ηii |−1 for t ∈] − a, a[ (i = 1, . . . , n)
hold and let the real part of every characteristic value of the matrix ηil (δil +
n
(1−δil )hi ) i,l=1 be negative, where the functions αi (t) (i = 1, . . . , n), γi (t, s)
(i = 1, . . . , n) and ai (t) (i = 1, . . . , n) and the numbers hi (i = 1, . . . , n) are
defined as in Corollary 3. Then the problem (1), (2) is solvable.
Acknowledgement
This work is supported by the Georgian National Science Foundation
(Grant No. GNSF/ST06/3-002).
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(Received 27.06.2007)
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