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Let us prove that the parameter
c
∗
∈
R
ρ
exists, and the solution
η
(
t, c
)
of Cauchy problem (17) satisfies the condition
η
(
t
∗
, c
∗
) =
η
∗
. If
b
i
(
t
)
,
d
i
(
t
)
∈
C
r
i
([0
, t
∗
])
,
i
= 1
, m
, and
q
(
z
1
, . . . , z
m
, η
)
∈
C
∞
(
R
n
)
, then the
vector-function
η
(
t, c
)
is differentiable using the parameter
c
, while the
matrix function
ν
=
∂η/∂c
satisfies the system of equations (13):
˙
ν
=
Kν
+
r
X
i
=1
A
i
+
∂p
∂z
i
d
(
i
−
1)
(
t
);
ν
(0) = 0
,
(18)
which is obtained as the result of system (17) differentiation with the
parameter
c
.
Now we introduce the mapping
Ψ :
R
ρ
→
R
ρ
, which assigns to
each parameter
c
∈
R
ρ
, the value
η
(
t
∗
, c
)
∈
R
ρ
of the
η
(
t, c
)
solution
of Cauchy problem (17) at a moment of time
t
∗
. Let us show that while
satisfying the theorem’s conditions, there exists the parameter
c
∗
, for which
the equality
Ψ(
c
∗
) =
η
∗
is fulfilled. For this, we will introduce the mapping
v
:
R
ρ
→
R
ρ
, functioning according to the rule
v
(
c
) =
c
−
M
−
1
(Ψ(
c
)
−
η
∗
)
.
The equality
Ψ(
c
∗
) =
η
∗
is equivalent to the fact that the parameter
c
∗
is a fixed point of the mapping
v
. To prove the existence of the fixed point
in the mapping
v
, we prove that the mapping
v
is compressing. The Jacobi
matrix of the mapping
v
has a form of
v
0
(
c
) =
E
−
M
−
1
Ψ
0
(
c
)
, where
E
is a
unity
ρ
×
ρ
-matrix;
Ψ
0
(
c
)
is the Jacobi matrix of the mapping
Ψ
. According
to the definition of the mapping
Ψ
,
Ψ
0
(
c
) =
ν
(
t
∗
)
, then
v
0
(
c
) =
E
−
M
−
1
ν
(
t
∗
)
.
Let us denote
D
(
t
) =
t
Z
0
d
(
τ
)
dτ
. The choice of the functions
d
(
t
)
in
formula (9) ensures that with
t
∈
[0
, t
∗
]
both the inequality
0
6
D
(
t
)
6
1
and
the equality
D
(
t
∗
) = 1
are fulfilled. Let us consider the matrix function
W
(
t
) =
D
(
t
)
E
+
r
−
1
X
i
=1
d
(
i
−
1)
(
t
)
N
i
−
M
−
1
ν
(
t
)
,
(19)
here
N
i
is
ρ
×
ρ
-matrices, which will be chosen later. From the equations
D
(0) = 0
,
d
(0) = 0
,
. . .
,
d
(
r
−
2)
(0) = 0
,
ν
(0) = 0
, it follows that
W
(0) = 0
,
and from the equations
D
(
t
∗
) = 1
,
d
(
t
∗
) = 0
, . . . , d
(
r
−
2)
(
t
∗
) = 0
—
W
(
t
∗
) =
E
−
M
−
1
ν
(
t
∗
)
. Having shown that
k
W
(
t
∗
)
k
6
γ <
1
, thereby
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5
23