Georgian Mathematical Journal 1(1994), No. 4, 343-351

ON THE CORRECTNESS OF LINEAR BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF GENERALIZED

ORDINARY DIFFERENTIAL EQUATIONS

M. ASHORDIA

Abstract. The sufficient conditions are established for the correctness of the linear boundary value problem

dx(t) = dA(t) · x(t) + df(t), l(x) = c_{0}, where A : [a, b] → ℝ^{n×n} and f : [a, b] → ℝ^{n} are matrix- and vectorfunctions of bounded variation, c_{0} ∈ ℝ^{n}, and l is a linear continuous operator from the space of n-dimentional vector-functions of bounded variation into ℝ^{n}.

Let the matrix- and vector-functions, A : [a, b] → ℝ^{n×n} and f : [a, b] → ℝ^{n}, respectively, be of bounded variation, c_{0} ∈ ℝ^{n}, and let l : BV_{n}(a, b) → ℝ^{n} be a linear continuous operator such that the boundary value problem

dx(t) = dA(t) · x(t) + df(t),  (1)
l(x) = c_{0}  (2)

has the unique solution x_{0}.

Consider the sequences of matrix- and vector-functions of bounded variation A_{k} : [a, b] → ℝ^{n×n} (k = 1, 2, …) and f_{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} : BV_{n}(a, b) → ℝ^{n} (k = 1, 2, …).

In this paper the sufficient conditions are given for the problem

dx(t) = dA_{k}(t) · x(t) + df_{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)

343

1991 Mathematics Subject Classification. 34B05.

1072-947X/1994/0700-0353$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

An analogous question is studied in [2–4] for the boundary value problem for a system of ordinary differential equations.

The theory of generalized ordinary differential equations enables one to investigate ordinary differential and difference equations from the common standpoint. Moreover, the convergence conditions for difference schemes corresponding to boundary value problems for systems of ordinary differential equations can be deduced from the correctness results for aprropriate boundary value problems for systems of generalized ordinary differential equations [1, 5, 6].

The following notations and definitions will be used throughout the paper:

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∥ = max_{j=1,...,n} ∑_{i=1}^{n} |x_{ij}|;

If X ∈ ℝ^{n×n}, then X^{−1} and det(X) are the matrix inverse to X and the determinant of X, respectively; E is the identity n × n matrix;

∨_{a}^{b}x and ∨_{a}^{b}X are the sums of total variations of components of vector- and

matrix-functions, x : [a, b] → ℝ^{n} and X : [a, b] → ℝ^{n×n}, respectively; BV_{n}(a, b) is a space of all vector-functions of bounded variation x :

[a, b] → ℝ^{n} (i.e., such that ∨_{a}^{b}x < +∞) with the norm

∥x∥_{sup} = sup{∥x(t)∥ : t ∈ [a, b]}^{1};

x(t−) and x(t+) (x(a−) = x(a), x(b+) = x(b)) are the left and the right limits of the vector-function x : [a, b] → ℝ^{n} at the point t;

d_{1}x(t) = x(t) − x(t−), d_{2}x(t) = x(t+) − x(t); BV_{n×n}(a, b) is a set of all matrix-functions of bounded variation X :

[a, b] → ℝ^{n×n}, i.e., such that ∨_{a}^{b}X < +∞;

d_{1}X(t) = X(t) − X(t−), d_{2}X(t) = X(t+) − X(t); If X = (x_{ij})_{i,j=1}^{n} ∈ BV_{n×n}(a, b), then V (X) : [a, b] → ℝ^{n×n} is defined by

V (X)(a) = 0, V (X)(t) = ( ∨^{t}_{a} x_{ij})_{i,j=1}^{n} (a < t ≤ b);

1BV_{n}(a, b) is not the Banach space with respect to this norm.

If α ∈ BV_{1}(a, b), x : [a, b] → ℝ and a ≤ s < t ≤ b, then

∫_{s}^{t} x(τ) dα(τ) = x(t)d_{1}α(t) + x(s)d_{2}α(s) + ∫_{]s,t[} x(τ) dα(τ),

where ∫_{]s,t[} x(τ) dα(τ) is the Lebesgue–Stieltjes integral over the open inter-

val ]s, t[ (if s = t, then ∫_{s}^{t} x(τ) dα(τ) = 0);

If A = (a_{ij})_{i,j=1}^{n} ∈ BV_{n×n}(a, b), X = (x_{ij})_{i,j=1}^{n} : [a, b] → ℝ^{n×n}, x = (x_{i})_{i=1}^{n} : [a, b] → ℝ^{n} and a ≤ s ≤ t ≤ b, then

∫_{s}^{t} dA(τ) · X(τ) = ( ∑_{k=1}^{n} ∫_{s}^{t} x_{kj}(τ) da_{ik}(τ))_{i,j=1}^{n},

∫_{s}^{t} dA(τ) · x(τ) = ( ∑_{k=1}^{n} ∫_{s}^{t} x_{k}(τ) da_{ik}(τ))_{i=1}^{n};

∥l∥ is the norm of the linear continuous operator l : BV_{n}(a, b) → ℝ^{n}; If X ∈ BV_{n×n}(a, b) is the matrix-function with columns x_{1}, …, x_{n}, then

l(X) is the matrix with columns l(x_{1}), …, l(x_{n}).

A function x ∈ BV_{n}(a, b) is said to be a solution of problem (1), (2) if it satisfies condition (2) and

x(t) = x(s) + ∫_{s}^{t} dA(τ) · x(τ) + f(t) − f(s) for a ≤ s ≤ t ≤ b.

Alongside with (1) and (3), we shall consider the corresponding homogeneous systems

dx(t) = dA(t) · x(t)  (1_{0})

and

dx(t) = dA_{k}(t) · x(t),  (3_{0})

respectively.

A matrix-function Y ∈ BV_{n×n}(a, b) is said to be a fundamental matrix of the homogeneous system (1_{0}) if

Y (t) = Y (s) + ∫_{s}^{t} dA(τ) · Y (τ) for a ≤ s ≤ t ≤ b

and det (Y (t)) ≠ 0 for t ∈ [a, b].

Theorem 1. Let the conditions

det (E + (−1)^{j}d_{j}A(t)) ≠ 0 for t ∈ [a, b] (j = 1, 2),  (6)
lim_{k→+∞} l_{k}(y) = l(y) for y ∈ BV_{n}(a, b),  (7)
lim_{k→+∞} c_{k} = c_{0},  (8)
lim_{k→+∞} sup ∥l_{k}∥ < +∞,  (9)
lim_{k→+∞} sup ∨_{a}^{b}A_{k} < +∞  (10)

be satisfied and let the conditions

lim_{k→+∞} [A_{k}(t) − A_{k}(a)] = A(t) − A(a),  (11)
lim_{k→+∞} [f_{k}(t) − f_{k}(a)] = f(t) − f(a)  (12)

be fulfilled uniformly on [a, b]. Then for any sufficiently large k problem (3),

(4) has the unique solution x_{k} and (5) is valid.

To prove the theorem we shall use the following lemmas.

Lemma 1. Let α_{k}, β_{k} ∈ BV_{1}(a, b) (k = 0, 1 …),

lim_{k→+∞} ∥β_{k} − β_{0}∥_{sup} = 0,  (13)
r = sup { ∨_{a}^{b} α_{k} : k = 0, 1, … } < +∞  (14)

and the condition

lim_{k→+∞} [α_{k}(t) − α_{k}(a)] = α_{0}(t) − α_{0}(a)  (15)

be fulfilled uniformly on [a, b]. Then

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

uniformly on [a, b].

Proof. Let ε be an arbitrary positive number. We denote

D_{j}(a, b, ε; g) = {t ∈ [a, b] : d_{j}g(t) ≥ ε} (j = 1, 2)

where

g(t) = V (β_{0})(t) for t ∈ [a, b].

By Lemma 1.1.1 from [5] there exists a finite subdivision {α_{0}, τ_{1}, α_{1}, …, τ_{m}, α_{m}} of [a, b] such that

a) a = α_{0} < α_{1} < ⋯ < α_{m} = b, α_{0} ≤ τ_{1} ≤ α_{1} ≤ ⋯ ≤ τ_{m} ≤ α_{m}; b) If τ_{i} ∉ D_{1}(a, b, ε; g), then g(τ_{i}) − g(α_{i−1}) < ε;

If τ_{i} ∈ D_{1}(a, b, ε; g), then α_{i−1} < τ_{i} and g(τ_{i}−) − g(α_{i−1}) < ε;

c) If τ_{i} ∉ D_{2}(a, b, ε; g), then g(α_{i}) − g(τ_{i}) < ε; If τ_{i} ∈ D_{2}(a, b, ε; g), then τ_{i} < α_{i} and g(α_{i}) − g(τ_{i}+) < ε.

We set

η(t) = \begin{cases}β_{0}(t) & for t ∈ \{α_{0}, τ_{1}, α_{1}, …, τ_{m}, α_{m}\}; \\ β_{0}(τ_{i}−) & for t ∈]α_{i−1}, τ_{i}[, τ_{i} ∈ D_{1}(a, b, ε; g); \\ β_{0}(τ_{i}) & for t ∈]α_{i−1}, τ_{i}[, τ_{i} ∉ D_{1}(a, b, ε; g) or \\ & for t ∈]τ_{i}, α_{i}[, τ_{i} ∉ D_{2}(a, b, ε; g); \\ β_{0}(τ_{i}+) & for t ∈]τ_{i}, α_{i}[, τ_{i} ∈ D_{2}(a, b, ε; g); \\ & (i = 1, …, m). \end{cases}

It can be easily shown that η ∈ BV_{1}(a, b) and

|β_{0}(t) − η(t)| < 2ε for t ∈ [a, b].  (16)

For every natural k and t ∈ [a, b] we assume

γ_{k}(t) = ∫_{a}^{t} β_{k}(τ) dα_{k}(t) − ∫_{a}^{t} β_{0}(τ) dα_{0}(τ)

and

δ_{k}(t) = ∫_{a}^{t} η(t) d[α_{k}(τ) − α_{0}(τ)].

It follows from (15) that

lim_{k→+∞} ∥δ_{k}∥_{sup} = 0.  (17)

On the other hand, by (14) and (16) we have

∥γ_{k}∥_{sup} ≤ 4rε + r∥β_{k} − β_{0}∥_{sup} + ∥δ∥_{sup} (k = 1, 2, …).

Hence in view of (13) and (17) lim_{k→+∞} ∥γ_{k}∥_{sup} = 0 since ε is arbitrary.

Lemma 2. Let condition (6) be fulfilled and

lim_{k→+∞} Y_{k}(t) = Y (t) uniformly on [a, b],  (18)

where Y and Y_{k} are the fundamental matrices of the homogeneous systems (1_{0}) and (3_{0}), respectively. Then

inf {| det(Y (t))| : t ∈ [a, b]} > 0,  (19)
inf {| det(Y^{−1}(t))| : t ∈ [a, b]} > 0  (20)

and

lim_{k→+∞} Y_{k}^{−1}(t) = Y^{−1}(t) uniformly on [a, b].  (21)

Proof. It is known ([6], Theorem III.2.10) that d_{j}Y (t) = d_{j}A(t) · Y (t) for t ∈ [a, b] (j = 1, 2). Therefore (6) implies

det (Y (t−) · Y (t+)) = [ det(Y (t))^{2}] · ∏_{j=1}^{2} det (E + (−1)^{j}d_{j}A(t)) ≠ 0

for t ∈ [a, b]. (22)

Let us show that (19) is valid. Assume the contrary. Then it can be easily shown that there exists a point t_{0} ∈ [a, b] such that det (Y (t_{0}−) · Y (t_{0}+)) = 0. But this equality contradicts (22). Inequality (19) is proved.

The proof of inequality (20) is analogous.

In view of (18) and (19) there exists a positive number q such that inf {| det(Y_{k}(t))| : t ∈ [a, b]} > q > 0 for any sufficiently large k. From this and (18) we obtain (21).

Proof of the Theorem. Let us show that

det (E + (−1)^{j}d_{j}A_{k}(t)) ≠ 0 for t ∈ [a, b] (j = 1, 2)  (23)

for any sufficiently large k.

By (11)

lim_{k→+∞} d_{j}A_{k}(t) = d_{j}A(t) (j = 1, 2)  (24)

uniformly on [a, b]. Since ∨_{a}^{b}A < +∞, the series ∑_{t∈[a,b]} ∥d_{j}A(t)∥ (j = 1, 2) converges. Thus for any j ∈ {1, 2} the inequality

∥d_{j}A(t)∥ ≥ \frac{1}{2}

may hold only for some finite number of points t_{j1}, …, t_{j m_{j}} in [a, b]. Therefore

∥d_{j}A(t)∥ < \frac{1}{2} for t ∈ [a, b], t ≠ t_{ji} (i = 1, …, m_{j}).  (25)

It follows from (6), (24) and (25) that for any sufficiently large k and for j ∈ {1, 2}

det (E + (−1)^{j}d_{j}A_{k}(t_{ji})) ≠ 0 (i = 1, …, m_{j})  (26)

and

∥d_{j}A_{k}(t)∥ < \frac{1}{2} for t ∈ [a, b], t ≠ t_{ji} (i = 1, …, m_{j}).  (27)

The latter inequality implies that the matrices E + (−1)^{j}d_{j}A_{k}(t) (j = 1, 2) are invertible for t ∈ [a, b], t ≠ t_{ji} (i = 1, …, m_{j}) too. Therefore (23) is proved.

Besides, by (26) and (27) there exists a positive number r_{0} such that for any sufficiently large k

∥[E + (−1)^{j}d_{j}A_{k}(t)]^{−1}∥ ≤ r_{0} for t ∈ [a, b] (j = 1, 2).  (28)

Let k be a sufficiently large natural number. In view of (6) and (23) there exist ([6], Theorem III.2.10) fundamental matrices Y and Y_{k} of systems (1_{0}) and (3_{0}), respectively, satisfying Y (a) = Y_{k}(a) = E. Moreover, Y_{k}^{−1} ∈ BV_{n×n}(a, b).

Let us prove (18). We set Z_{k}(t) = Y_{k}(t) − Y (t) for t ∈ [a, b] and B_{k}(t) = A_{k}(t−) for t ∈ [a, b]. Obviously, for every t ∈ [a, b]

d_{1}[B_{k}(t) − A_{k}(t)] = −d_{2}[B_{k}(t) − A_{k}(t)] = −d_{1}A_{k}(t)

and

∫_{a}^{t} d[B_{k}(τ) − A_{k}(τ)] · Z_{k}(τ) = −d_{1}A_{k}(t) · Z_{k}(t).

Consequently,

Z_{k}(t) ≡ [E − d_{1}A_{k}(t)]^{−1}[ ∫_{a}^{t} d[A_{k}(τ) − A(τ)] · Y (τ) + ∫_{a}^{t} dB_{k}(τ) · Z_{k}(τ)].

From this and (28) we get

∥Z_{k}(t)∥ ≤ r_{0}(ε_{k} + ∫_{a}^{t} d∥V (B_{k})(τ)∥ · ∥Z_{k}(τ)∥) for t ∈ [a, b],

where

ε_{k} = sup {∥ ∫_{a}^{t} d[A_{k}(τ) − A(τ)] · Y (τ)∥ : t ∈ [a, b]}.

Hence, according to the Gronwall inequality ([6], Theorem I.4.30),

∥Z_{k}(t)∥ ≤ r_{0}ε_{k} exp (r_{0} ∨_{a}^{b} B_{k}) ≤ r_{0}ε_{k} exp (r_{0} ∨_{a}^{b} A_{k}) for t ∈ [a, b].

By (10), (11) and Lemma 1 this inequality implies (18).

It is known ([6], Theorem III.2.13) that if x_{k} is the solution of (3), then

x_{k}(t) ≡ Y_{k}(t)x_{k}(a) + f_{k}(t) − f_{k}(a) − Y_{k}(t) ∫_{a}^{t} dY_{k}^{−1}(τ) · [f_{k}(τ) − f_{k}(a)].

Thus problem (3), (4) has the unique solution if and only if

det (l_{k}(Y_{k})) ≠ 0.  (29)

Since problem (1), (2) has the unique solution x_{0}, we have

det (l(Y)) ≠ 0.  (30)

Besides, by (7), (9) and (18)

lim_{k→+∞} l_{k}(Y_{k}) = l(Y).

Therefore, in view of (30), there exists a natural number k_{0} such that condition (29) is fulfilled for every k ≥ k_{0}. Thus problem (3), (4) has the unique solution x_{k} for k ≥ k_{0} and

x_{k}(t) ≡ Y_{k}(t)[l_{k}(Y_{k})]^{−1}[c_{k} − l_{k}(F_{k}(f_{k}))] + F_{k}(f_{k})(t),  (31)

where

F_{k}(f_{k})(t) = f_{k}(t) − f_{k}(a) − Y_{k}(t) ∫_{a}^{t} dY_{k}^{−1}(τ) · [f_{k}(τ) − f_{k}(a)].

According to Lemma 2 condition (21) is fulfilled and

ρ = sup {∥Y_{k}^{−1}(t)∥ + ∥Y_{k}(t)∥ : t ∈ [a, b], k ≥ k_{0}} < +∞.  (32)

The equality

Y_{k}^{−1}(t) − Y_{k}^{−1}(s) = Y_{k}^{−1}(s) ∫_{t}^{s} dA_{k}(τ) · Y_{k}(τ) Y_{k}^{−1}(t)

implies

∥Y_{k}^{−1}(t) − Y_{k}^{−1}(s)∥ ≤ ρ^{3} ∨_{s}^{t} A_{k} for a ≤ s ≤ t ≤ b (k ≥ k_{0}).

This inequality, together with (10) and (32), yields

lim_{k→+∞} sup ∨_{a}^{b}Y_{k}^{−1} < +∞.

By this, (12) and (21) it follows from Lemma 1 that

lim_{k→+∞} ∫_{a}^{t} dY_{k}^{−1}(τ) · [f_{k}(τ) − f_{k}(a)] =

= ∫_{a}^{t} dY^{−1}(τ) · [f(τ) − f(a)]  (33)

uniformly on [a, b].

Using (7)-(9), (12), (18), (29), (30) and (33), from (31) we get lim_{k→+∞} x_{k}(t)

= z(t) uniformly on [a, b], where

z(t) = Y (t)[l(Y)]^{−1}[c_{0} − l(F (f))] + F (f)(t),

F (f)(t) = f(t) − f(a) − Y (t) ∫_{a}^{t} dY^{−1}(τ) · [f(τ) − f(a)].

It is easy to verify that the vector-function z : [a, b] → ℝ^{n} is the solution of problem (1), (2). Therefore

x_{0}(t) = z(t) for t ∈ [a, b].

References

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6. S. Schwabik, M. Tvrdy and O. Vejvoda, Differential and integral equations: Boundary value problems and adjoints. Academia, Praha, 1979.

(Received 2.07.1993)

Author’s address:

I. Vekua Institute of Applied Mathematics

of Tbilisi State University

2 University St., 380043 Tbilisi

Republic of Georgia