Georgian Mathematical Journal 1(1994), No. 2, 115-126

ON THE STABILITY OF SOLUTIONS OF LINEAR BOUNDARY VALUE PROBLEMS FOR A SYSTEM OF

ORDINARY DIFFERENTIAL EQUATIONS

M. ASHORDIA

Abstract. Linear boundary value problems for a system of ordinary differential equations are considered. The stability of the solution with respect to small perturbations of coefficients and boundary values is investigated.

Let P_{0} : [a, b] → ℝ^{n×n} and q_{0} : [a.b] → ℝ^{n} be integrable matrix- and vector-functions, respectively, c_{0} ∈ ℝ^{n}, and let l_{0} : C([a, b]; ℝ^{n}) → ℝ^{n} be a linear continuous operator such that the boundary value problem

\frac{dx}{dt} = P_{0}(t)x + q_{0}(t),  (1)
l_{0}(x) = c_{0}  (2)

has the unique solution x_{0}. Consider the sequences of integrable matrixand vector-functions, P_{k} : [a, b] → ℝ^{n×n} (k = 1, 2, …), and q_{k} : [a, b] → ℝ^{n} (k = 1, 2, …), respectively, the sequence of constant vectors c_{k} ∈ ℝ^{n} (k = 1, 2, …) and the sequence of linear continuous operators l_{k} : C([a, b]; ℝ^{n}) → ℝ^{n} (k = 1, 2, …). In [1,2], sufficient conditions are given for the problem

\frac{dx}{dt} = P_{k}(t)x + q_{k}(t),  (3)
l_{k}(x) = c_{k}  (4)

to have a unique solution x_{k} for any sufficiently large k and

lim_{k→+∞} x_{k}(t) = x_{0}(t) uniformly on [a, b].  (5)

In the present paper, necessary and sufficient conditions are established for the sequence of boundary value problems of the form (3),(4) to have the above-mentioned property.

1991 Mathematics Subject Classification. 34B05.

Throughout the paper the following notations and definitions will be used:

ℝ =] − ∞, +∞[;

ℝ^{n} is a space of real column n-vectors x = (x_{i})_{i=1}^{n} with the norm

∥x∥ = ∑_{i=1}^{n} |x_{i}|;

ℝ^{n×n} is a space of real n × n matrices X = (x_{ij})_{i,j=1}^{n} with the norm

∥X∥ = ∑_{i,j=1}^{n} |x_{ij}|;

if X = (x_{ij})_{i,j=1}^{n} ∈ ℝ^{n×n}, then diag X is a diagonal matrix with diagonal components x_{11}, …, x_{nn}; X^{−1} is an inverse matrix to X; E is an identity n × n matrix;

C([a, b]; ℝ^{n}) is a space of continuous vector-functions x : [a, b] → ℝ^{n} with the norm

∥x∥_{c} = max{∥x(t)∥ : a ≤ t ≤ b};

C̃([a, b]; ℝ^{n}) and C̃([a, b]; ℝ^{n×n}) are the sets of absolutely continuous vector- and matrix- functions, respectively;

L([a, b]; ℝ^{n}) and L([a, b]; ℝ^{n×n}) are the sets of vector- and matrix-functions x : [a, b] → ℝ^{n} and X : [a, b] → ℝ^{n×n}, respectively, whose components are Lebesgue-integrable;

∥l∥ is the norm of the linear continuous operator l : C([a, b]; ℝ^{n}) → ℝ^{n}. The vector-function x : [a, b] → ℝ^{n} is said to be a solution of the problem

(1),(2) if it belongs to C̃([a, b]; ℝ^{n}) and satisfies the condition (2) and the system (1) a.e. on [a, b].

Definition 1. We shall say that the sequence (P_{k}, q_{k}, l_{k}) (k = 1, 2, …) belongs to S(P_{0}, q_{0}, l_{0}) if for every c_{0} ∈ ℝ^{n} and c_{k} ∈ ℝ^{n} (k = 1, 2, …) satisfying the condition

lim_{k→+∞} c_{k} = c_{0}  (6)

the problem (3),(4) has the unique solution x_{k} for any sufficiently large k and (5) holds.

Along with (1),(2) and (3),(4) we shall consider the corresponding homogeneous problems

\frac{dx}{dt} = P_{0}(t)x,  (1_{0})
l_{0}(x) = 0  (2_{0})

and

\frac{dx}{dt} = P_{k}(t)x,  (3_{0})
l_{k}(x) = 0.  (4_{0})

Theorem 1. Let

lim_{k→+∞} l_{k}(y) = l_{0}(y) for y ∈ C̃([a, b]; ℝ^{n})  (7)

and

lim_{k→+∞} sup ∥l_{k}∥ < +∞.  (8)

Then

((P_{k}, q_{k}, l_{k}))_{k=1}^{+∞} ∈ S(P_{0}, q_{0}, l_{0})  (9)

if and only if there exist sequences of matrix- and vector-functions, Φ_{k} ∈ C̃([a, b]; ℝ^{n×n}) (k = 1, 2, …) and φ_{k} ∈ C̃([a, b]; ℝ^{n}) (k = 1, 2, …), respectively, such that

lim_{k→+∞} sup ∫_{a}^{b} ∥P_{k}^{∗}(τ)∥ dτ < +∞  (10)

and

lim_{k→+∞} Φ_{k}(t) = 0,  (11)
lim_{k→+∞} φ_{k}(t) = 0,  (12)
lim_{k→+∞} ∫_{a}^{t} P_{k}^{∗}(τ) dτ = ∫_{a}^{t} P_{0}(τ) dτ,  (13)
lim_{k→+∞} ∫_{a}^{t} q_{k}^{∗}(τ) dτ = ∫_{a}^{t} q_{0}(τ) dτ  (14)

uniformly on [a, b], where

P_{k}^{∗}(t) ≡ [E − Φ_{k}(t)] · P_{k}(t) − Φ′_{k}(t),  (15)
q_{k}^{∗}(t) ≡ [E − Φ_{k}(t)][P_{k}(t)φ_{k}(t) + q_{k}(t) − φ′_{k}(t)].  (16)

Theorem 1′. Let (7)and (8) be satisfied. The (9) holds if and only if there exist sequences of matrix- and vector-functions, Φ_{k} ∈ C̃([a, b]; ℝ^{n×n}) (k = 1, 2, …) and φ_{k} ∈ C̃([a, b]; ℝ^{n}) (k = 1, 2, …), respectively, such that

lim_{k→+∞} sup ∫_{a}^{b} ∥P_{k}^{∗}(τ) − diag P_{k}^{∗}(τ)∥ dτ < +∞  (17)

and the conditions (11)–(13) and

lim_{k→+∞} ∫_{a}^{t} exp ( − ∫_{a}^{τ} diag P_{k}^{∗}(s)ds) · q_{k}^{∗}(τ) dτ =

= ∫_{a}^{t} exp ( − ∫_{a}^{τ} diag P_{0}(s)ds) · q_{0}(τ) dτ  (18)

are fulfilled uniformly on [a, b], where

P_{k}^{∗}(t) ≡ [P_{k}(t) − Φ_{k}(t)P_{k}(t) − Φ′_{k}(t)] · [E − Φ_{k}(t)]^{−1}  (19)

and q_{k}^{∗}(t) is the vector-function defined by (16).

Before proving this theorems, we shall give a theorem from [1] and some of its generalizations.

Theorem 2_{0}. Let the conditions (6)–(8),

lim_{k→+∞} sup ∫_{a}^{b} ∥P_{k}(τ)∥ dτ < +∞  (20)

hold and let the following conditions

lim_{k→+∞} ∫_{a}^{t} P_{k}(τ) dτ = ∫_{a}^{t} P_{0}(τ) dτ,  (21)
lim_{k→+∞} ∫_{a}^{t} q_{k}(τ) dτ = ∫_{a}^{t} q_{0}(τ) dτ  (22)

hold uniformly on [a, b]. Then (9) is satisfied^{1}.

Theorem 2. Let there exist sequences of matrix- and vector-functions, Φ_{k} ∈ C̃([a, b]; ℝ^{n×n}) (k = 1, 2, …) and φ_{k} ∈ C̃([a, b]; ℝ^{n}) (k = 1, 2, …), respectively, such that the conditions (10),

lim_{k→+∞}[c_{k} − l_{k}(φ_{k})] = c_{0}  (23)

hold and let the conditions (11),(13),(14) be fulfilled uniformly on [a, b], where P_{k}^{∗}(t) and q_{k}^{∗}(t) are the matrix- and vector-functions defined by (15) and (16), respectively. Let, moreover, conditions (7),(8) hold. Then for any sufficiently large k the problem (3),(4) has the unique solution x_{k} and

lim_{k→+∞} ∥x_{k} − φ_{k} − x_{0}∥_{c} = 0.

1See [1], Theorem 1.2.

Proof. The transformation z = x − φ_{k} reduces the problem (3),(4) to

\frac{dz}{dt} = P_{k}(t)z + r_{k}(t),  (24)
l_{k}(z) = c_{k1},  (25)

where r_{k}(t) ≡ P_{k}(t)φ_{k}(t) + q_{k}(t) − φ′_{k}(t), c_{k1} = c_{k} − l_{k}(φ_{k}) (k = 1, 2, …).

Let us show that for any sufficiently large k the homogeneous problem (3_{0}),(4_{0}) has only trivial solution.

Suppose this proposition is invalid. It can be assumed without loss of generality that for every natural k the problem (3_{0}),(4_{0}) has the solution x_{k} for which

∥x_{k}∥_{c} = 1.  (26)

Moreover, it is evident that the vector-function x_{k} is the solution of the system

\frac{dx}{dt} = P_{k}^{∗}(t)x + [Φ_{k}(t) · x_{k}(t)]′.  (27)

According to (11) and (26)

lim_{k→+∞}[Φ_{k}(t) · x_{k}(t)] = 0

uniformly on [a, b]. Therefore the conditions of Theorem 2_{0} are fulfilled for the sequence of problems (27),(4_{0}). Hence

lim_{k→+∞} ∥x_{k}∥_{c} = 0,

which contradicts (26). This proves that the problem (3_{0}),(4_{0}) has only trivial solution.

From this fact it follows that for any sufficiently large k the problem (24),(25) has the unique solution z_{k}.

It can easily be shown that the vector-function z_{k} satisfies the system

\frac{dz}{dt} = P_{k}^{∗}(t)z + r_{k}^{∗}(t),  (28)

where r_{k}^{∗}(t) = [Φ_{k}(t) · z_{k}(t)]′ + q_{k}^{∗}(t).

Show that

lim_{k→+∞} sup ∥z_{k}∥_{c} < +∞.  (29)

Let this proposal be invalid. Assume without loss of generality that

lim_{k→+∞} ∥z_{k}∥_{c} = +∞.  (30)

Put

u_{k}(t) = ∥z_{k}∥_{c}^{−1} · z_{k}(t) for t ∈ [a, b] (k = 1, 2, …).

Then in view of (25) and (28), for every natural k the vector-function u_{k}(t) will be the solution of the boundary value problem

\frac{du}{dt} = P_{k}^{∗}(t)u + s_{k}(t),

l_{k}(u) = ∥z_{k}∥_{c}^{−1} · c_{k1},

where s_{k}(t) = ∥z_{k}∥_{c}^{−1} · r_{k}^{∗}(t). Equations (11),(14),(23), and (30) imply

lim_{k→+∞}[∥z_{k}∥_{c}^{−1} · c_{k1}] = 0

and

lim_{k→+∞} ∫_{a}^{t} s_{k}(τ) dτ = 0

uniformly on [a, b]. Hence, according to (10) and (13), the conditions of Theorem 2_{0} are fulfilled for the sequence of the last boundary value problems. Therefore

lim_{k→+∞} ∥u_{k}∥_{c} = 0.

This equality contradicts the conditions ∥u_{k}∥_{c} = 1 (k = 1, 2, …). The inequality (29) is proved.

In view of (11),(14), and (29)

lim_{k→+∞} ∫_{a}^{t} r_{k}^{∗}(τ) dτ = ∫_{a}^{t} q_{0}(τ) dτ

uniformly on [a, b].

Applying Theorem 2_{0} to the sequence of the problems (28),(25), we again show that

lim_{k→+∞} ∥z_{k} − x_{0}∥_{c} = 0.

Corollary 1. Let (6)–(8),

lim_{k→+∞} sup ∫_{a}^{b} ∥P_{k}(τ) − Φ_{k}(τ)P_{k}(τ) − Φ′_{k}(τ)∥ dτ < +∞

hold and let the conditions (11),(21),(22),

lim_{k→+∞} ∫_{a}^{t} Φ_{k}(τ)P_{k}(τ) dτ = ∫_{a}^{t} P^{∗}(τ) dτ

and

lim_{k→+∞} ∫_{a}^{t} Φ_{k}(τ)q_{k}(τ) dτ = ∫_{a}^{t} q^{∗}(τ) dτ

be fulfilled uniformly on [a, b], where Φ_{k} ∈ C̃([a, b]; ℝ^{n×n}) (k = 1, 2, …), P^{∗} ∈ L([a, b]; ℝ^{n×n}), q^{∗} ∈ L([a, b]; ℝ^{n}). Let, moreover, the system

\frac{dx}{dt} = P_{0}^{∗}(t)x + q_{0}^{∗}(t),

where P_{0}^{∗}(t) ≡ P_{0}(t) − P^{∗}(t), q_{0}^{∗}(t) ≡ q_{0}(t) − q^{∗}(t), have a unique solution satisfying the condition (2). Then

((P_{k}, q_{k}, l_{k}))_{k=1}^{+∞} ∈ S(P_{0}^{∗}, q_{0}^{∗}, l_{0}).

Proof. It suffices to assume in Theorem 2 that φ_{k}(t) ≡ 0 and to notice that

lim_{k→+∞} ∫_{a}^{t}[E − Φ_{k}(τ)] · P_{k}(τ) dτ = ∫_{a}^{t} P_{0}^{∗}(τ) dτ

and

lim_{k→+∞} ∫_{a}^{t}[E − Φ_{k}(τ)] · q_{k}(τ) dτ = ∫_{a}^{t} q_{0}^{∗}(τ) dτ

uniformly on [a, b].

Corollary 2. Let (6)–(8) hold, and let there exist a natural number m and matrix-functions P_{oj} ∈ L([a, b], ℝ^{n×n}) (j = 1, …, m) such that

lim_{k→+∞}[P_{km}(t) − P_{k}(t)] = 0,

lim_{k→+∞} ∫_{a}^{t}[E + P_{km}(τ) − P_{k}(τ)] · P_{k}(τ) dτ = ∫_{a}^{t} P_{0}(τ) dτ,

lim_{k→+∞} ∫_{a}^{t}[E + P_{km}(τ) − P_{k}(τ)] · q_{k}(τ) dτ = ∫_{a}^{t} q_{0}(τ) dτ

uniformly on [a, b], where

P_{k1}(t) ≡ P_{k}(t), P_{k j+1}(t) ≡ P_{kj}(t) − ∫_{a}^{t}[P_{kj}(τ) − P_{0j}(τ)] dτ
(j = 1, …, m).

Let, moreover,

lim_{k→+∞} sup ∫_{a}^{b} ∥[E + P_{km}(τ) − P_{k}(τ)] · P_{k}(τ) +

+[P_{km}(τ) − P_{k}(t)]′∥dτ < +∞.

Then (9) holds.

Theorem 2′. Let there exist sequences of matrix- and vector-functions, Φ_{k} ∈ C̃([a, b]; ℝ^{n×n}) (k = 1, 2, …) and φ_{k} ∈ C̃([a, b]; ℝ^{n}) (k = 1, 2, …), respectively, such that the conditions (17),(23) hold, and let the conditions (11),(13), and (18) be fulfilled uniformly on [a, b]. Here P_{k}^{∗}(t) and q_{k}^{∗}(t) are the matrix- and vector-functions defined by (19) and (16), respectively. Then the conclusion of Theorem 2 is true.

Proof. In view of (14), we may assume without loss of generality that for every natural k the matrix E − Φ_{k}(t) is invertible for t ∈ [a, b].

For every k ∈ {0, 1, … } and t ∈ [a, b] assume

P_{0}^{∗}(t) = P_{0}(t), q_{0}^{∗}(t) = q_{0}(t), Φ_{0}(t) = 0, φ_{0}(t) = 0,

c_{k1} = c_{k} − l_{k}(φ_{k})_{,} Q_{k}(t) = H_{k}(t) · [P_{k}^{∗}(t) − diag P_{k}^{∗}(t)] · H_{k}^{−1}(t),

r_{k}(t) = H_{k}(t) · q_{k}^{∗}(t),

where

H_{k}(t) = exp ( − ∫_{a}^{t} diag P_{k}^{∗}(τ) dτ).

Moreover, assume

l_{k}^{∗}(z) = l_{k}(x) for z ∈ C([a, b]; ℝ^{n}),

where x(t) = [E − Φ_{k}(t)]^{−1} · H_{k}^{−1}(t) · z(t).

From (13) it follows that l_{k}^{∗} : C([a, b]; ℝ^{n}) → ℝ^{n} (k = 0, 1, …) is a sequence of linear continuous operators for which conditions (7) and (8) are satisfied.

For every k ∈ {0, 1, … } the transformation

z(t) = H_{k}(t) · [E − Φ_{k}(t)] · [x(t) − φ_{k}(t)] for t ∈ [a, b]  (31)

reduces the problem (3),(4) to

\frac{dz}{dt} = Q_{k}(t)z + r_{k}(t),  (32)
l_{k}^{∗}(z) = c_{k1}  (33)

and the problem (1),(2) to

\frac{dz}{dt} = Q_{0}(t)z + r_{0}(t),  (34)
l_{0}^{∗}(z) = c_{0}.  (35)

In view of (13) and (17) from Lemma 1.1 ([1], p.9) it follows that

lim_{k→+∞} ∫_{a}^{t} Q_{k}(τ) dτ = ∫_{a}^{t} Q_{0}(τ) dτ

uniformly on [a, b]. According to Theorem 2_{0} from the above and from (7),(8),(17),(18),(23) it follows that the problem (32),(33) has the unique solution z_{k} for any sufficiently large k, and

lim_{k→+∞} ∥z_{k} − z_{0}∥_{c} = 0,

where z_{0} is the unique solution of the problem (34),(35). Therefore (11),(13) and (31) show that the statement of the theorem is true.

Corollary 3. Let the conditions (6)–(8),

lim_{k→+∞} sup ∫_{a}^{b} ∥P_{k}(τ) − diag P_{k}(τ)∥ dτ < +∞

hold and let (21) and

lim_{k→+∞} ∫_{a}^{t} exp ( − ∫_{a}^{τ} diag P_{k}(s)ds) · q_{k}(τ) dτ =

= ∫_{a}^{t} exp ( − ∫_{a}^{τ} diag P_{0}(s)ds) · q_{0}(τ) dτ

be fulfilled uniformly on [a, b]. Then (9) holds.

Remark. As compared with Theorem 2_{0} and the results of [2], it is not assumed in Theorems 2 and 2′ that the equalities (21) and (22) hold uniformly on [a, b]. Below we will give an example of a sequence of boundary value problems for linear systems for which (9) holds but (21) is not fulfilled uniformly on [a, b].

Example. Let a = 0, b = 2π, n = 2, and for every natural k and t ∈ [0, 2π], let

P_{k}(t) = ( \begin{matrix}0 & p_{k1}(t) \\ 0 & p_{k2}(t) \end{matrix} ) , P_{0}(t) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) ,

φ_{k}(t) = q_{k}(t) = q_{0}(t) = ( \begin{matrix}0 \\ 0 \end{matrix} );

p_{k1}(t) = \begin{cases}(\sqrt{k} + \sqrt[4]{k}) sin kt & for t ∈ I_{k}, \\ \sqrt{k} sin kt & for t ∈ [0, 2π]\I_{k}; \end{cases}

p_{k2}(t) = \begin{cases}−α′_{k}(t) · [1 − α_{k}(t)]^{−1} & for t ∈ I_{k}, \\ 0 & for t ∈ [0, 2π]\I_{k}; \end{cases}

β_{k}(t) = ∫_{0}^{t}[1 − α_{k}(τ)] · p_{k1}(τ) dτ;

α_{k}(t) = \begin{cases}4π^{−1}(\sqrt[4]{k} + 1)^{−1} sin kt & for t ∈ I_{k}, \\ 0 & for t ∈ [0, 2π]\I_{k}, \end{cases}

where I_{k} = ∪_{m=0}^{k−1}]2mk^{−1}π, (2m + 1)k^{−1}π[. Let, moreover, for every k ∈ {0, 1, … }, Y_{k}(t) be the fundamental matrix of the system (3_{0}) satisfying

Y_{k}(a) = E.

It can easily be shown that for every natural k we have

Y_{0}(t) = E, Y_{k}(t) = ( \begin{matrix}1 & β_{k}(t) \\ 0 & 1 − α_{k}(t) \end{matrix} ) for t ∈ [0, 2π]

and

lim_{k→+∞} Y_{k}(t) = Y_{0}(t)

uniformly on [0, 2π], since

lim_{k→+∞} ∥α_{k}∥_{c} = lim_{k→+∞} ∥β_{k}∥_{c} = 0.

Note that

lim_{k→+∞} ∫_{0}^{2π} p_{k1}(t) dt = 2 lim_{k→+∞} \sqrt[4]{k} = +∞.

Therefore neither the conditions of Theorem 2_{0} nor the results of [2] are fulfilled.

On the other hand, if we assume that

Φ_{k}(t) = E − Y_{k}^{−1}(t) for t ∈ [0, 2π] (k = 1, 2, …),

then the conditions of Theorems 2 and 2′ will be fulfilled, and if we put

Φ_{k}(t) = ( \begin{matrix}α_{k}(t) & β_{k}(t) \\ 0 & 0 \end{matrix} ) for t ∈ [0, 2π] (k = 1, 2, …),

then in this case only the conditions of Theorem 2 will be fulfilled, since

lim_{k→+∞} sup ∫_{0}^{2π} |p_{k2}(t)| dt = +∞.

Proof of Theorem 1. The sufficiency follows from Theorem 2, since in view of (6),(8), and (12), condition (23) holds.

Let us show the necessity. Let c_{k} ∈ ℝ^{n} (k = 0, 1, …) be an arbitrary sequence satisfying (6) and let e_{j} = (δ_{ij})_{i=1}^{n} (j = 1, …, n), where δ_{ij} = 1 if i = j and δ_{ij} = 0 if i ≠ j.

In view of (9), we may assume without loss of generality that for every natural k the problem (3),(4) has the unique solution x_{k}.

For any k ∈ {0, 1, … } and j ∈ {1, …, m} assume

y_{kj}(t) = x_{k}(t) − x_{kj}(t) (t ∈ [a, b]),

where x_{oj} and x_{kj} (k = 1, 2, …) are the unique solutions of (1) and (3) satisfying

l_{0}(x) = c_{0} − e_{j} and l_{k}(x) = c_{k} − e_{j},

respectively. Moreover, for every k ∈ {0, 1, … } denote by Y_{k}(t) the matrixfunction whose columns are y_{k1}(t), …, y_{kn}(t).

It can easily be shown that y_{oj} and y_{kj} satisfy (1_{0}) and (3_{0}), respectively, and

l_{k}(y_{kj}) = e_{j} (j = 1, …, n; k = 0, 1, …).  (36)

If for some k and α_{j} ∈ ℝ (j = 1, …, n)

∑_{j=1}^{n} α_{j}y_{kj}(t) = 0 (t ∈ [a, b]),

then, using (36), we have

∑_{j=1}^{n} α_{j}e_{j} = 0,

and therefore

α_{1} = ⋯ = α_{n} = 0,

i.e., Y_{0} and Y_{k} are the fundamental matrices of the systems (1_{0}) and (3_{0}), respectively. Hence, (5) implies

lim_{k→+∞} Y_{k}^{−1}(t) = Y_{0}^{−1}(t) uniformly on [a, b].  (37)

Let, for every natural k and t ∈ [a, b],

Φ_{k}(t) = E − Y_{0}(t)Y_{k}^{−1}(t),  (38)
φ_{k}(t) = x_{k}(t) − x_{0}(t).  (39)

Let us show (10)–(14). Equations (11) and (12) are evident. Moreover, using the equality

[Y_{k}^{−1}(t)]′ = −Y_{k}^{−1}(t)P_{k}(t) for t ∈ [a, b] (k = 1, 2, …),

it can be easily shown that

P_{k}^{∗}(t) = P_{0}(t)Y_{0}(t)Y_{k}^{−1}(t) for t ∈ [a, b] (k = 1, 2, …)

and

∫_{a}^{t} q_{k}^{∗}(τ) dτ = Y_{0}(t)Y_{k}^{−1}(t)x_{0}(t) − Y_{0}(a)Y_{k}^{−1}(a)x_{0}(a) −
− ∫_{a}^{t} P_{0}(τ)Y_{0}(τ)Y_{k}^{−1}(τ)x_{0}(τ) dτ for t ∈ [a, b] (k = 1, 2, …).

Therefore, according to (37) the conditions (10),(13), and (14) are fulfilled uniformly on [a, b]. This completes the proof.

The proof of Theorem 1′ is analogous. We note only that Φ_{k} and φ_{k} are defined as above.

The behavior at k → +∞ of the solution of the Cauchy problem (l_{k}(x) = x(t_{0}), t_{0} ∈ [a, b]) and of the Cauchy-Nicoletti problem (l_{k}(x) = (x_{i}(t_{i}))_{i=1}^{n}, t_{i} ∈ [a, b]) is considered in [3-5]. Moreover, in [6] the necessary conditions for the stability of the Cauchy problem are investigated.

References

1. I.T.Kiguradze, Boundary value problems for systems of ordinary differential equations. (Russian) Modern problems in mathematics. The latest achievements (Itogi nauki i tekhniki. VINITI Acad. Sci. USSR). Moscow, 1987, V.30, 3-103.

2. M.T.Ashordia and D.G.Bitsadze, On the correctness of linear boundary value problems for systems of ordinary differential equations. (Russian) Bull. Acad. Sci. Georgian SSR 142(1991), No. 3, 473-476.

3. D.G.Bitsadze, On the problem of dependence on the parameter of the solution of multipoint nonlinear boundary value problems. (Russian) Proc. I.N.Vekua Inst. Appl. Math. Tbilis. State Univ. 22(1987), 42-55.

4. J.Kurzweil and J.Jarnik, Iterated Lie brackets in limit processes in ordinary differential equations. Results in Mathematics, Birkhäuser Verlag. Basel, 1988, V.14, 125-137.

5. A.M.Samoilenko, Investigation of differential equations with ”non-re-gular” right part. (Russian) Abhandl. der Deutsch. Akad. Wiss. Berlin. Kl. Math., Phys. und Tech. 1965, No. 1, 106-113.

6. N.N.Petrov, Necessary conditions of continuity with respect to the parameter for some classes of equations. (Russian) Vestnik Leningrad. Univ. Mat. Mech. Astronom. (1965), No. 1, 47-53.

(Received 22.09.1992)

Author’s address:

I.Vekua Institute of Applied Mathematics

of Tbilisi State University

2 University St., 380043 Tbilisi

Republic of Georgia