GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 1996, 501-524

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

ORDINARY DIFFERENTIAL EQUATIONS

M. ASHORDIA

Abstract. The concept of a strongly isolated solution of the nonlinear boundary value problem

dx(t) = dA(t) · f(t, x(t)), h(x) = 0,

is introduced, where A : [a, b] → R^{n×n} is a matrix-function of bounded variation, f : [a, b] × R^{n} → R^{n} is a vector-function belonging to a Carathéodory class, and h is a continuous operator from the space of n-dimensional vector-functions of bounded variation into R^{n}.

It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given.

1. Statement of the Problem and Formulation of the Results

Let A = (a_{ij})_{i,j=1}^{n} : [a, b] → R^{n×n} be a matrix-function of bounded variation, f = (f_{i})_{i=1}^{n} : [a, b] × R^{n} → R^{n} be a vector-function belonging to the Carathéodory class K([a, b] × R^{n}, R^{n}; A), and let h : BV_{s}([a, b], R^{n}) → R^{n} be a continuous operator such that the nonlinear boundary value problem

dx(t) = dA(t) · f(t, x(t)),  (1.1)
h(x) = 0  (1.2)

has a solution x^{0}.

Consider a sequence of matrix-functions of bounded variation A_{k} : [a, b] → R^{n×n} (k = 1, 2, …), a sequence of vector-functions f_{k} : [a, b] × R^{n} → R^{n}, f_{k} ∈ K([a, b] × R^{n}, R^{n}; A_{k}) (k = 1, 2, …) and a sequence of continuous operators h_{k} : BV_{s}([a, b], R^{n}) → R^{n} (k = 1, 2, …).

501

1991 Mathematics Subject Classification. 34B15.

Key words and phrases. Strongly isolated solution, correct problem, Carathéodory class, Opial condition.

1072-947X/96/1100-0501$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

In this paper sufficient conditions are given guaranteeing both the solvability of the problem

dx(t) = dA_{k}(t) · f_{k}(t, x(t)),  (1.1_{k})
h_{k}(x) = 0  (1.2_{k})

for any sufficiently large k and the convergence of its solutions as k → +∞ to the solution of problem (1.1), (1.2).

An analogous question was studied in [1–4] for initial and boundary value problems for nonlinear systems of ordinary differential equations.

The theory of generalized ordinary differential equations enables one to investigate ordinary differential and difference equations from a unified standpoint. Moreover, the convergence conditions for difference schemes corresponding to the boundary value problems for systems of ordinary differential equations can be obtained from the results concerning the correctness of the boundary value problems for systems of generalized ordinary differential equations [5–8].

Throughout the paper the following notation and definitions will be used: R =]−∞, +∞[, R_{+} = [0, +∞[; R^{n×m} is a space of all real n×m matrices

X = (x_{ij})_{i,j=1}^{n,m} with the norm ∥X∥ = max_{j=1,...,m} ∑_{i=1}^{n} |x_{ij}|; R^{n} = R^{n×1}.

|X| = (|x_{ij}|)_{i,j=1}^{n,m}, [X]_{+} = (\frac{|x_{ij}| + x_{ij}}{2} )_{i,j=1}^{n,m}.

R_{+}^{n×m} = {(x_{ij})_{i,j=1}^{n,m} ∈ R^{n×m} : x_{ij} ≥ 0 (i = 1, …, n; j = 1, …, m)},

R_{+}^{n} = R_{+}^{n×1}.

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

V_{a}^{b}X is the total variation of the matrix-function X : [a, b] → R^{n×m}, i.e., the sum of total variations of the latter’s components; X(t−) and X(t+) (X(a−) = X(a), X(b+) = X(b)) are the left and the right limit of the matrix-function X at the point t ∈ [a, b],

d_{1}X(t) = X(t) − X(t−), d_{2}X(t) = X(t+) − x(t);
∥X∥_{s} = sup {∥X(t)∥ : t ∈ [a, b]}.

BV([a, b], D) is a set of all X : [a, b] → D ⊂ R^{n×m} such that V_{a}^{b}X < +∞. BV_{s}([a, b], R^{n}) is a normed space (BV([a, b], R^{n}); ∥ · ∥_{s}). If y ∈ BV_{s}([a, b], R^{n}), then U(y; r) = {x ∈ BV_{s}([a, b], R^{n}) : ∥x − y∥_{s} <

r}; D(y; r) is a set of all x ∈ R^{n} such that inf{∥x − y(τ)∥ : τ ∈ [a, b]} < r.

If J ⊂ R and D_{1} ⊂ R^{n}, then C(J, D_{1}) is a set of all continuous vectorfunctions x : J → D_{1}.

CORRECTNESS OF NONLINEAR BOUNDARY VALUE PROBLEMS 503

If g ∈ BV([a, b], R), x : [a, b] → R and a ≤ s < t ≤ b, then v(g) : [a, b] → R is defined by v(g)(a) = 0 and v(g)(t) = V_{a}^{t}g for t ∈ (a, b];

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

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

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

If G = (g_{ij})_{i,j=1}^{l,n} ∈ BV([a, b], R^{l×n}), X = (x_{jk})_{j,k=1}^{n,m} : [a, b] → D_{2} ⊂ R^{n×m}, and a ≤ s ≤ t ≤ b, then

V (X) = (v(x_{jk}))_{j,k=1}^{n,m},

∫_{s}^{t} dG(τ) · X(τ) = ( ∑_{j=1}^{n} ∫_{s}^{t} x_{jk}(τ)dg_{ij}(τ))_{i,k=1}^{l,m} .

L([a, b], D_{2}; G) is a set of all matrix-functions (x_{jk})_{j,k=1}^{n,m} : [a, b] → D_{2} such that x_{jk} is integrable with respect to g_{ij} (i = 1, …, l).

K([a, b] × D_{1}, D_{2}; G) is a Carathéodory class, i.e., the set of all mappings F = (f_{jk})_{j,k=1}^{n,m} : [a, b] × D_{1} → D_{2} such that (a) f_{ik}(·, x) is measurable with respect to the measures V (g_{ij}) and V (g_{ij}) − g_{ij} for x ∈ D_{1} (i = 1, …, l),

(b) F (t, ·) : D_{1} → D_{2} is continuous for t ∈ [a, b] and

sup {|F (·, x)| : x ∈ D_{0}} ∈ L([a, b], R^{n×m}; G)

for every compact D_{0} ⊂ D_{1}.

K^{0}([a, b] × D_{1}, D_{2}; G) is a set of all mappings F = (f_{jk})_{j,k=1}^{n,m} : [a, b] × D_{1} → D_{2} such that for every vector-function of bounded variation x : [a, b] → D_{1} the function f_{jk}(·, x(·)) is measurable with respect to the measures V (g_{ij}) and V (g_{ij}) − g_{ij} (i = 1, …, l).

If B ∈ BV([a, b], R^{n}), then M([a, b] × R_{+}, R_{+}^{n}; B) is a set of all vectorfunctions ω ∈ K([a, b] × R_{+}, R_{+}^{n}; B) such that ω(t, ·) is nondecreasing and ω(t, ·) = 0 for every t ∈ [a, b].

The inequalities between both the vectors and the matrices are understood to be componentwise.

If B_{1} and B_{2} are normed spaces, then an operator g : B_{1} → B_{2} is called positive homogeneous if g(λx) = λg(x) for every λ ∈ R_{+} and x ∈ B_{1}.

A vector-function x ∈ BV([a, b], R^{n}) is said to be a solution of system (1.1) if

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

By a solution of the system of generalized ordinary differential inequalities dx(t) ≤ dA(t)·f(t, x(t)) (≥) we mean a vector-function x ∈ BV([a, b], R^{n}) such that

x(t) ≤ x(s) + ∫_{s}^{t} dA(τ) · f(τ, x(τ)) (≥) a ≤ s < t ≤ b.

Definition 1.1. Let l : BV_{s}([a, b], R^{n}) → R^{n} be a linear continuous operator and let l_{0} : BV_{s}([a, b], R^{n}) → R_{+}^{n} be a positive homogeneous continuous operator. We shall say that a matrix-function P : [a, b]×R^{n} → R^{n×m} satisfies the Opial condition with respect to the triplet (l, l_{0}; A) if:

(a) P ∈ K([a, b] × R^{n}, R^{n×n}; A) and there exists a matrix-function Φ ∈ L([a, b], R_{+}^{n×n}; A) such that

|P (t, x)| ≤ Φ(t) on [a, b] × R^{n};

(b) for every B ∈ BV([a, b], R^{n×n})

det (I + (−1)^{j}d_{j}B(t)) ≠ 0 for t ∈ [a, b] (j = 1, 2)  (1.3)

and the problem

dx(t) = dB(t) · x(t), |l(x)| ≤ l_{0}(x)  (1.4)

has only the trivial solution provided there exists a sequence

y_{k} ∈ BV([a, b], R^{n}) (k = 1, 2, …)

such that

lim_{k→+∞} ∫_{a}^{t} dA(τ) · P (τ, y_{k}(τ)) = B(t) uniformly on [a, b].

Let r be a positive number.

Definition 1.2. x^{0} is said to be strongly isolated in the radius r if there exist P ∈ K([a, b]×R^{n}, R^{n×n}; A), q ∈ K([a, b]×R^{n}, R^{n}; A), a linear continuous operator l : BV_{s}([a, b], R^{n}) → R^{n}, and a positive homogeneous operator l̃: BV_{s}([a, b], R^{n}) → R^{n} such that:

(a) f(t, x) = P (t, x)x + q(t, x) for t ∈ [a, b], ∥x − x^{0}(t)∥ < r and the equality h(x) = l(x) + l̃(x) is fulfilled on U(x^{0}; r);

(b) the vector-functions α(t, ρ) = max{|q(t, x)| : ∥x∥ ≤ ρ} and β(ρ) = sup{[|l̃(x)| − l_{0}(x)]_{+} : ∥x∥_{s} ≤ ρ} satisfy the conditions

lim_{ρ→+∞} \frac{1}{ρ} ∫_{a}^{b} dV (A)(t) · α(t, ρ) = 0, lim_{ρ→+∞} \frac{β(ρ)}{ρ} = 0 ;

(c) the problem

dx(t) = dA(t) · [P (t, x(t))x(t) + q(t, x(t))],  (1.5)
l(x) + l̃(x) = 0  (1.6)

has no solution different from x^{0};

(d) the matrix-function P satisfies the Opial condition with respect to the triplet (l, l_{0}; A).

The notation

((A_{k}, f_{k}, h_{k}))_{k=1}^{+∞} ∈ W_{r}(A, f, h; x^{0})

means that:

(a) for every x ∈ D(x^{0}; r)

lim_{k→+∞} ∫_{a}^{t} dA_{k}(τ) · f_{k}(τ, x) = ∫_{a}^{t} dA(τ) · f(τ, x)  (1.7)

uniformly on [a, b]; (b)

lim_{k→+∞} h_{k}(x) = h(x) uniformly on U(x^{0}; r);  (1.8)

(c) there exists a sequence ω_{k} ∈ M([a, b] × R_{+}, R_{+}^{n}; A_{k}) (k = 1, 2, …) such that

sup {∥ ∫_{a}^{b} dV (A_{k})(τ) · ω_{k}(τ, r)∥ : k = 1, 2, … } < +∞,  (1.9)
lim_{s→0+} sup {∥ ∫_{a}^{b} dV (A_{k})(τ) · ω_{k}(τ, s)∥ : k = 1, 2, … } = 0, (1.10)
|f_{k}(t, x) − f_{k}(t, y)| ≤ ω_{k}(t, ∥x − y∥)  (1.11)

on [a, b] × D(x^{0}; r) (k = 1, 2, …).

Remark 1.1. If for every natural m there exists a positive number μ_{m} such that

ω_{k}(t, mσ) ≤ μ_{m}ω_{k}(t, σ) for σ > 0, t ∈ [a, b] (k = 1, 2, …),

then (1.9) follows from (1.10).

In particular, the sequence of functions

ω_{k}(t, σ) = max {|f_{k}(t, x) − f_{k}(t, y)| : ∥x∥ ≤ ∥x^{0}∥_{s} + r,
∥y∥ ≤ ∥x^{0}∥_{s} + r, ∥x − y∥ ≤ σ}

(k = 1, 2, …) has the latter property.

Definition 1.3. Problem (1.1), (1.2) is said to be (x^{0}; r)-correct if for every ε ∈]0, r[ and ((A_{k}, f_{k}, h_{k}))_{k=1}^{+∞} ∈ W_{r}(A, f, h; x^{0}) there exists a natural number k_{0} such that problem (1.1_{k}), (1.2_{k}) has at least one solution contained in U(x^{0}; r), and any such solution belongs to the ball U(x^{0}; r) for any k ≥ k_{0}.

Definition 1.4. Problem (1.1), (1.2) is said to be correct if it has the unique solution x^{0} and for every r > 0 it is (x^{0}; r)-correct.

Theorem 1.1. If problem (1.1), (1.2) has a solution x^{0} which is strongly isolated in the radius r, then it is (x^{0}; r)-correct.

Corollary 1.1. Let the inequality

|f(t, x) − P (t, x)x| ≤ α(t, ∥x∥)  (1.12)

be fulfilled on [a, b] × R^{n} and

|h(x) − l(x)| ≤ l_{0}(x) + l_{1}(∥x∥_{s}) for x ∈ BV([a, b], R^{n}), (1.13)

where l : BV_{s}([a, b], R^{n}) → R^{n} and l_{0} : BV_{s}([a, b], R^{n}) → R_{+}^{n} are respectively a linear continuous and a positive homogeneous continuous operator, the matrix-function P satisfies the Opial condition with respect to the triplet (l, l_{0}; A), α ∈ K([a, b] × R_{+}, R_{+}^{n}; A) is nondecreasing in the second variable, l_{1} ∈ C(R_{+}, R_{+}^{n}), and

lim_{ρ→+∞} \frac{1}{ρ} ∫_{a}^{b} dV (A)(t) · α(t, ρ) = lim_{ρ→+∞} \frac{l_{1}(ρ)}{ρ} = 0.  (1.14)

If problem (1.1), (1.2) has at most one solution, then it is correct.

Corollary 1.2. Let inequalities (1.12) and

P_{1}(t) ≤ P_{1}(t, x) ≤ P_{2}(t)  (1.15)

be fulfilled on [a, b] × R^{n} and let (1.13) hold, where l : BV_{s}([a, b], R^{n}) → R^{n} and l_{0} : BV_{s}([a, b], R^{n}) → R^{n} are respectively a linear continuous and a positive homogeneous continuous operator, P ∈ K^{0}([a, b] × R^{n}, R^{n×n}; A), P_{k} ∈ L([a, b], R^{n×n}; A) (k = 1, 2), α ∈ K([a, b] × R_{+}, R_{+}^{n}; A) is nondecreasing in the second variable, l_{1} ∈ C(R_{+}, R_{+}^{n}) and (1.14) holds. Let, moreover, (1.3) hold and let problem (1.4) have only a trivial solution for every B ∈ BV([a, b], R^{n×n}) satisfying the inequality

|B(t) − B(s) − \frac{1}{2} ∫_{s}^{t} dA(τ) · [P_{1}(τ) + P_{2}(τ)]| ≤

≤ \frac{1}{2} ∫_{s}^{t} dV (A)(τ) · [P_{2}(τ) − P_{1}(τ)]  (1.16)

for a ≤ s ≤ t ≤ b. If problem (1.1), (1.2) has at most one solution, then it is correct.

Remark 1.2. Corollary 1.2 is of interest only in the case of P ∉ K([a, b]× R^{n}, R^{n×n}; A), since for P ∈ K([a, b] × R^{n}, R^{n×n}; A) it follows immediately from Corollary 1.1.

Corollary 1.3. Let the conditions

|f(t, x) − P_{0}(t)x| ≤ Q(t)|x| + q(t, ∥x∥),
det (I + (−1)^{j}d_{j}A(t) · P_{0}(t)) ≠ 0 (j = 1, 2)  (1.17)

and

∥|d_{j}A(t)| · Q(t)∥∥(I + (−1)^{j}d_{j}A(t) · P_{0}(t))^{−1}∥ < 1 (j = 1, 2) (1.18)

be fulfilled on [a, b] × R^{n} and let (1.13) hold, where l : BV_{s}([a, b], R^{n}) → R^{n} and l_{0} : BV_{s}([a, b], R^{n}) → R_{+}^{n} are respectively a linear continuous and a positive homogeneous continuous operator, P_{0} ∈ L([a, b], R^{n×n}; A), Q ∈ L([a, b], R_{+}^{n×n}; A), q ∈ K([a, b] × R_{+}, R_{+}^{n}; A) is nondecreasing in the second variable, l_{1} ∈ C(R_{+}, R_{+}^{n}), and

lim_{ρ→+∞} \frac{1}{ρ} ∫_{a}^{b} dV (A)(t) · q(t, ρ) = lim_{ρ→+∞} \frac{l_{1}(ρ)}{ρ} = 0.

Moreover, let the problem

|dx(t) − dA(t) · P_{0}(t)x(t)| ≤ dV (A)(t) · Q(t)|x(t)|, |l(x)| ≤ l_{0}(x) (1.19)

have only the trivial solution. If problem (1.1), (1.2) has at most one solution, then it is correct.

Corollary 1.4. Let the conditions

|f(t, x) − P (t)x| ≤ α(t, ∥x∥)

and

det (I + (−1)^{j}d_{j}A(t) · P (t)) ≠ 0 (j = 1, 2)  (1.20)

be fulfilled on [a, b] × R^{n} and

|h(x) − l(x)| ≤ l_{1}(∥x∥_{s}) for x ∈ BV([a, b], R^{n}),

where l : BV_{s}([a, b], R^{n}) → R^{n} is a linear continuous operator, P ∈ L([a, b], R^{n×n}; A), α ∈ K([a, b]×R_{+}, R_{+}^{n}; A) is nondecreasing in the second variable, l_{1} ∈ C(R_{+}, R_{+}^{n}), and (1.14) holds. Moreover, let the problem

dx(t) = dA(t) · P (t)x(t), l(x) = 0

have only the trivial solution. If problem (1.1), (1.2) has at most one solution, then it is correct.

Corollary 1.5. Let conditions (1.17), (1.18) and

|f(t, x) − f(t, y) − P_{0}(t)(x − y)| ≤ Q(t)|x − y|

be fulfilled on [a, b] × R^{n} and

|h(x) − h(y) − l(x − y)| ≤ l_{0}(x − y) on BV([a, b], R^{n}),

where l : BV_{s}([a, b], R^{n}) → R^{n} and l_{0} : BV_{s}([a, b], R^{n}) → R_{+}^{n} are respectively a linear continuous and a positive homogeneous continuous operator, P_{0} ∈ L([a, b], R^{n×n}; A), Q ∈ L([a, b], R_{+}^{n×n}; A), and problem (1.19) has only the trivial solution. Then problem (1.1), (1.2) is correct.

Let t_{0} ∈ [a, b]. For every j ∈ {1, 2}, (−1)^{j}(t − t_{0}) < 0, c ∈ R^{n} and ρ ≥ ∥c∥ by X_{j}(t, c, ρ) we denote a set of all solutions z of system (1.1) such that z(t_{0}) = c and ∥z(s)∥ < ρ for (t − s)(s − t_{0}) > 0.

Corollary 1.6. Let h(x) = x(t_{0}) − g(x), t_{0} ∈ [a, b], then the set

G = {g(x) : x ∈ BV([a, b], R^{n})}

is bounded and for any c ∈ G every local solution of system (1.1) satisfying the condition

x(t_{0}) = c  (1.21)

can be continued to the whole segment [a, b]. Let, moreover, the matrixfunction A be continuous at the point t_{0}, and for every j ∈ {1, 2}

lim_{ρ→+∞} sup ( inf {∥x + (−1)^{j}d_{j}A(t) · f(t, y)∥ − σ_{3−j}(t, z) : ∥x∥ ≥ ρ,
∥y∥ = ρ, z ∈ X_{j}(t, c, ρ), c ∈ G}) > 0  (1.22)

uniformly on {t ∈ [a, b] : (−1)^{j}(t − t_{0}) < 0}, where σ_{j}(t, z) = max{∥z(t)∥, ∥z(t) + (−1)^{j}d_{j}z(t)∥}. Then the unique solvability of problem (1.1), (1.2) guarantees its correctness.

Corollary 1.7. Let t_{0} ∈ [a, b], c ∈ R^{n}, and let system (1.1) have the unique solution x_{0} defined on the whole segment [a, b], satisfying the initial condition (1.21). Moreover, let the matrix-function A be continuous at the point t_{0} and for every j ∈ {1, 2}

lim_{ρ→+∞} sup inf {∥x + (−1)^{j}d_{j}A(t) · f(t, y)∥ : |x∥ ≥ ρ,

∥y∥ = ρ} > ∥x^{0}∥_{s}  (1.23)

uniformly on {t ∈ [a, b] : (−1)^{j}(t − t_{0}) < 0}. Then problem (1.1), (1.2) is correct.

Remark 1.3. If the matrix-function A is continuous on [a, b], then condition (1.23) is fulfilled. In the case of discontinuous A in (1.23) the strict inequality cannot be replaced by the nonstrict one. Below we shall give the corresponding example.

Example 1.1. Let n = 1 and let m > 2 be a fixed natural number, τ_{i} = a + \frac{i}{m}(b − a) (i = 0, …, m); h(x) = x(t_{0}) − c_{0}, h_{k}(x) = x(t_{0}) − c_{k}, where c_{0} = 0, c_{k} = \frac{1}{k} (k = 1, 2, …); A(t) = i for t ∈ [τ_{i}, τ_{i+1}[ (i = 0, …, m − 1), A(b) = m, A_{k}(t) ≡ A(t) (k = 1, 2, …); f(t, x) = f_{k}(t, x) = 0 for t ∈ [a, τ_{1}[ ∪]τ_{1}, b], x ∈ R^{n} (k = 1, 2, …);

f(τ_{1}, x) = \begin{cases}0 & for x ∈] − ∞, 0[ , \\ (1 + c_{j+1} − c_{j})(x − j) + j + c_{j} & for x ∈ [j, j + 1[ \\ & (j = 0, 1, …); \end{cases}

f_{k}(τ_{1}, x) = \begin{cases}f(τ_{1}, x) & for x ∈] − ∞, k − 1[∪]k + 1, +∞[ , \\ (1 − c_{k−1} − c_{k})(x − k)+ \\ +k − c_{k} & for x ∈ [k − 1, k[ , \\ (1 + c_{k+1} + c_{k})(x − k)+ \\ +k − c_{k} & for x ∈ [k, k + 1[ \end{cases}

(k = 1, 2, …).

Then x^{0}(t) ≡ 0, ((A_{k}, f_{k}, h_{k}))_{k=1}^{+∞} ∈ W_{r}(A, f, h; x^{0}) for every positive number r and for every natural k problem (1.1_{k}), (1.2_{k}) has the unique solution x_{k}(t) ≡ k. As for condition (1.23), it is transformed into the equality only for t = τ_{1}.

Corollary 1.8. Let there exist a solution x^{0} of problem (1.1), (1.2) and a positive number r such that

|f(t, x) − f(t, x^{0}(t)) − P (t)(x − x^{0}(t))| ≤ Q(t)|x − x^{0}(t)|
for t ∈ [a, b], ∥x − x^{0}(t)∥ ≤ r (1.24)

and

|h(x) − l(x − x^{0})| ≤ l^{∗}(|x − x^{0}|) for x ∈ U(x^{0}; r),  (1.25)

where l : BV_{s}([a, b], R^{n}) → R^{n} and l^{∗} : BV_{s}([a, b]; R_{+}^{n}) → R_{+}^{n} are respectively a linear continuous and a positive homogeneous nondecreasing continuous operator, P ∈ L([a, b], R^{n×n}; A), Q ∈ L([a, b], R_{+}^{n×n}; A). Let, moreover, (1.20) be fulfilled on [a, b] and let the problem

\begin{matrix}|dx(t) − dA(t) · P (t)x(t)| ≤ dV (A)(t) · Q(t)|x(t)|, \\ |l(x)| ≤ l^{∗}(|x|) \end{matrix}  (1.26)

have only the trivial solution. Then problem (1.1), (1.2) is (x^{0}; r)-correct.

Corollary 1.9. Let every component f_{j} (j = 1, …, n) of the vectorfunction f have partial derivatives in the last n variables belonging to K([a, b] ×R^{n}, R; a_{ij}) (i = 1, …, n) and let there exist a solution x^{0} of problem (1.1), (1.2) such that the operator h has the Frechet derivative l in x^{0}. Let, moreover,

det (I + (−1)^{j}d_{j}A(t) · F (t, x^{0}(t))) ≠ 0 for t ∈ [a, b] (j = 1, 2)

and the problem

dx(t) = dA(t) · F (t, x^{0}(t))x(t), l(x) = 0,  (1.27)

where F (t, x) = \frac{∂f(t,x)}{∂x} have only the trivial solution. Then problem (1.1), (1.2) is (x^{0}; r)-correct for any sufficiently small r.

2. Auxiliary Propositions

For every positive number ξ and a nondecreasing vector-function g : [a, b] → R^{n} we put

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

Let R(a, b, ξ; g) be a set of all subdivisions {α_{0}, τ_{1}, α_{1}, …, τ_{m}, α_{m}} of the segment [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, ξ; g), then ∥g(τ_{i})−g(α_{i−1})∥ < ξ, and if τ_{i} ∈D_{1}(a, b, ξ; g), then α_{i−1} < τ_{i} and ∥g(τ_{i}−) − g(α_{i−1})∥ < ξ (i = 1, …, n);

(c) if τ_{i} ∉ D_{2}(a, b, ξ; g), then ∥g(α_{i})−g(τ_{i})∥ < ξ, and if τ_{i} ∈ D_{2}(a, b, ξ; g), then τ_{i} < α_{i} and ∥g(α_{i}) − g(τ_{i}+)∥ < ξ (i = 1, …, n).

Lemma 2.1. The set R(a, b, ξ; g) is not empty.

We omit the proof of this lemma for it is analogous to that of Lemma 1.1.1 from [5].

Lemma 2.2. Let D ⊂ R^{n} and A_{k} ∈ BV([a, b], R^{n×n}) (k = 1, 2, …), f_{k} ∈ K([a, b] × D, R^{n}; A_{k}) (k = 1, 2, …), ω_{k} ∈ M([a, b] × R_{+}, R_{+}^{n}; A_{k}) (k = 0, 1, …), A_{0}(t) ≡ A(t), and y_{k} ∈ BV([a, b], D) (k = 1, 2, …) be sequences such that (1.7) is fulfilled uniformly on [a, b] × D and (1.11) is fulfilled on [a, b] × D for k ∈ {0, 1, … }, where f_{0}(t, x) ≡ f(t, x). Let, moreover, conditions (1.10) and

∥y_{k}(t) − y_{k}(s)∥ ≤ l_{k} + ∥g(t) − g(s)∥
for a ≤ s < t ≤ b (k = 1, 2, …) (2.1)

hold, where l_{k} ≥ 0, l_{k} → 0 as k → +∞ and g : [a, b] → R^{n} is a nondecreasing vector-function. Then

lim_{k→+∞} [ ∫_{a}^{t} dA_{k}(τ) · f_{k}(τ, y_{k}(τ)) − ∫_{a}^{t} dA(τ) · f(τ, y_{k}(τ))] = 0

uniformly on [a, b].  (2.2)

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

x_{k}(t) = ∫_{a}^{t} dA_{k}(τ) · f_{k}(τ, y_{k}(τ)) − ∫_{a}^{t} dA(τ) · f(τ, y_{k}(τ)).

We are to show that

lim_{k→+∞} ∥x_{k}∥_{s} = 0.  (2.3)

Let ε be an arbitrary positive number. In view of (1.10) there exists a positive number δ such that

∥ ∫_{a}^{b} dV (A_{k})(τ) · ω_{k}(τ, 2δ)∥ < ε (k = 1, 2, …).  (2.4)

According to Lemma 2.1 the set R(a, b, δ; g) is not empty.

Let {α_{0}, τ_{1}, α_{1}, …, τ_{m}, α_{m}} ∈ R(a, b, δ; g) be fixed. For every natural k we assume that

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

It is not difficult to see that

∥y_{k}(t) − ỹ_{k}(t)∥ < l_{k} + δ for t ∈ [a, b] (k = 1, 2, …).

Hence there exists a natural k_{0} such that

∥y_{k}(t) − ỹ_{k}(t)∥ < 2δ for t ∈ [a, b] (k ≥ k_{0}).

From this and (1.11) we have

∥x_{k}∥_{s} ≤ ∥ ∫_{a}^{b} dV (A_{k})(τ) · ω_{k}(τ, 2δ)∥ + ∥ ∫_{a}^{b} dV (A)(τ) · ω_{0}(τ, 2δ)∥ +

+ μ_{0k} + ∑_{i=1}^{m}(λ_{ik} + 2μ_{ik} + 2ν_{ik}) for k ≥ k_{0},  (2.5)

where

λ_{ik} = sup {∥ ∫_{]s,t[} dA_{k}(τ) · f_{k}(τ, x) − ∫_{]s,t[} dA(τ) · f(τ, x)∥ : s, t ∈ [a, b],
x ∈ {y_{k}(τ_{i}−), y_{k}(τ_{i}), y_{k}(τ_{i}+)}} (i = 1, …, m),
μ_{ik} = max {∥d_{j}A_{k}(α_{i}) · f_{k}(α_{i}, y_{k}(α_{i})) −
− d_{j}A(α_{i}) · f(α_{i}, y_{k}(α_{i}))∥ : j ∈ {1, 2}} (i = 0, …, m),
ν_{ik} = max {∥d_{j}A_{k}(τ_{i}) · f_{k}(τ_{i}, y_{k}(τ_{i})) −
− d_{j}A(τ_{i}) · f(τ_{i}, y_{k}(τ_{i}))∥ : j ∈ {1, 2}} (i = 1, …, m).

It follows from the conditions of the lemma and (1.7) that

lim_{k→+∞} d_{j}A_{k}(t) · f_{k}(t, x) = d_{j}A(t) · f(t, x) (j = 1, 2)

and

lim_{k→+∞} ∫_{]s,t[} dA_{k}(τ) · f_{k}(τ, x) = ∫_{]s,t[} dA(τ) · f(τ, x)

uniformly on [a, b] × D. Therefore

lim_{k→+∞} μ_{0k} = 0 and

lim_{k→+∞}(λ_{ik} + 2μ_{ik} + 2ν_{ik}) = 0 (i = 1, …, m). (2.6)

On the other hand, we may assume without loss of generality that

∥ ∫_{a}^{b} dV (A)(τ) · ω_{0}(τ, 2δ)∥ < ε.

Taking into account this fact, (2.4) and (2.6), it follows from (2.5) that (2.3) is valid.

Remark 2.1. If the set D is bounded and continuous, (1.10) and (1.11) hold, then condition (1.7) is fulfilled uniformly on [a, b] × D if and only if for every x ∈ D it is fulfilled uniformly on [a, b].

Lemma 2.3. Let y, y_{k} ∈ BV([a, b], R^{n}) (k = 1, 2, …) be vector-functions such that

lim_{k→+∞} y_{k}(t) = y(t) for t ∈ [a, b].

Let condition (2.1) hold, where l_{k} ≥ 0,l_{k} → 0 as k → +∞, and let g : [a, b] → R^{n} be a nondecreasing vector-function. Then

lim_{k→+∞} ∥y_{k} − y∥_{s} = 0.

Proof. Let ε be an arbitrary positive number, {α_{0}, τ_{1}, α_{1}, …, τ_{m}, α_{m}} ∈ R(a, b, \frac{ξ}{5}; g), and let n_{ε} be a natural number such that

l_{i} < \frac{ε}{5} and ∥y_{i}(τ) − y_{k}(τ)∥ < \frac{ε}{5}

for τ ∈ {α_{0}, τ_{1}, α_{1}, …, τ_{m}, α_{m}} (i, k ≥ n_{ε}).

Assume that α_{j−1} < t < τ_{j} (j = 1, …, m). Then in view of (2.1) we have

∥y_{i}(t) − y_{k}(t)∥ ≤
≤ ∥y_{i}(t) − y_{i}(τ_{j})∥ + ∥y_{i}(τ_{j}) − y_{k}(τ_{j})∥ + ∥y_{k}(τ_{j}) − y_{k}(t)∥ ≤
≤ l_{i} + l_{k} + 2∥g(t) − g(τ_{j})∥ + ∥y_{i}(τ_{j}) − y_{k}(τ_{j})∥ <

< \frac{3ε}{5} + 2∥g(τ_{j}) − g(α_{j−1})∥ < ε

for τ_{j} ∉ D_{1}(a, b, \frac{ε}{5}; g) (i, k ≥ n_{ε}) and

∥y_{i}(t) − y_{k}(t)∥ ≤

≤ ∥y_{i}(t) − y_{i}(α_{j−1})∥ + ∥y_{i}(α_{j−1}) − y_{k}(α_{j−1})∥ + ∥y_{k}(α_{j−1}) − y_{k}(t)∥ ≤ ≤ l_{i} + l_{k} + 2∥g(t) − g(α_{j−1})∥ + ∥y_{i}(α_{j−1}) − y_{k}(α_{j−1})∥ <

< \frac{3ε}{5} + 2∥g(τ_{j}−) − g(α_{j−1})∥ < ε

for τ_{j} ∈ D_{1}(a, b, \frac{ε}{5}; g) (i, k ≥ n_{ε}).

The case τ_{j} < t < α_{j} (j = 1, …, m) is considered analogously.

Lemma 2.4. Let condition (1.13) hold and inequality (1.12) be fulfilled on [a, b] × R^{n}, where l : BV_{s}([a, b], R^{n}) → R^{n} and l_{0} : BV_{s}([a, b], R^{n}) → R_{+}^{n} are respectively a linear continuous and a positive homogeneous continuous operator, let the matrix-function P satisfy the Opial condition with respect to the triplet (l, l_{0}; A), α ∈ K([a, b] × R_{+}, R_{+}^{n}; A) be nondecreasing in the second variable, l_{1} ∈ C(R_{+}, R_{+}^{n}), and let (1.14) hold. Moreover, let g ∈ K([a, b] × R^{n}, R^{n}; G) be a vector-function such that

lim_{ρ→+∞} \frac{1}{ρ} ∫_{a}^{b} dV (G)(t) · γ(t, ρ) = 0,

where γ(t, ρ) = max{|g(t, x)| : ∥x∥ ≤ ρ}. Then the boundary value problem

dx(t) = dA(t) · f(t, x(t)) + dG(t) · g(t, x(t)),^{1}, h(x) = 0 (2.7)

is solvable.

Proof. Let problem (2.7) have a solution x. Put

z(t) = ∫_{a}^{t} dG(τ) · g(τ, x(τ)) and y(t) = x(t) − z(t) for t ∈ [a, b].

Then the vector-function x̃ = \binom{y}{z} will be a solution of the problem

dx(t) = dÃ(t) · f̃(t, x(t)), h̃(x) = 0,  (2.8)

where

Ã(t) = (\begin{matrix}A(t), & 0 \\ 0, & G(t) \end{matrix}) ,
f̃(t, x) = (\begin{matrix}f(t, x_{1} + x_{n+1}, …, x_{n} + x_{2n}) \\ g(t, x_{1} + x_{n+1}, …, x_{n} + x_{2n}) \end{matrix}) on [a, b] × R^{2n},

and

h̃(x) = (\begin{matrix}h(x_{1} + x_{n+1}, …, x_{n} + x_{2n}) \\ (x_{n+i}(a))_{i=1}^{n} \end{matrix}) for x = (x_{i})_{i=1}^{2n} ∈ BV([a, b], R^{2n}).

Conversely, if x̃ = (x_{i})_{i=1}^{2n} is a solution of problem (2.8), then x = (x_{i} + x_{n+i})_{i=1}^{n} will be a solution of problem (2.7). It is not difficult to show that the conditions of the existence theorem (see [9, Theorem 1]) are fulfilled for problem (2.8). Hence it is solvable as problem (2.7).

1A vector-function x ∈ BV([a, b], R^{n}) is said to be a solution of this system if it satisfies the corresponding integral equality.

3. Proof of the Main Results

Proof of Theorem 1.1. Let us assume that the theorem does not hold. Then there exist ε ∈]0, r[ and ((A_{k}, f_{k}, h_{k}))_{k=1}^{+∞} ∈ W_{r}(A, f, h; x^{0}) such that for every natural number k problem (1.1_{k}), (1.2_{k}) has either no solution belonging to the ball U(x^{0}; r) or at least one solution belonging to U(x^{0}; r)\U(x^{0}; ε).

Put

χ(t, x) = \begin{cases}x & for ∥x − x^{0}(t)∥ ≤ r, \\ x^{0}(t) + \frac{r}{∥x−x^{0}(t)∥}(x − x^{0}(t)) & for ∥x − x^{0}(t)∥ > r; \end{cases} (3.1)
χ̃(x)(t) = χ(t, x(t)),
f̃_{k}(t, x) = f_{k}(t, χ(t, x)) (k = 1, 2, …)

and

h̃_{k}(x) = h_{k}(χ̃(x)) (k = 1, 2 …).

By virtue of Lemma 2.4 the problem

dx(t) = dA(t) · [P (t, x(t))x(t) + q(t, x(t))] + dη_{k}(χ̃(x))(t),
l(x) + l̃(x) + γ_{k}(χ̃(x)) = 0,

where

γ_{k}(y) = h_{k}(y) − h(y) for y ∈ BV([a, b], R^{n})

and

η_{k}(y)(t) = ∫_{a}^{t} dA_{k}(τ) · f_{k}(τ, y(τ)) − ∫_{a}^{t} dA(τ) · f(τ, y(τ))
for y ∈ BV([a, b], R^{n}),

is solvable for every natural k. From the above it is evident that it has a solution x_{k} satisfying the inequality

∥x_{k} − x^{0}∥_{s} ≥ ε.  (3.2)

It is clear that

x_{k}(t) = x_{k}(a) + z_{k}(t) + η_{k}(x^{0})(t) for t ∈ [a, b] (k = 1, 2, …), (3.3)

where

z_{k}(t) = ∫_{a}^{t} dA(τ) · [P (τ, x_{k}(τ))x_{k}(τ) + q(τ, x_{k}(τ))] +
+ ∫_{a}^{t} dA_{k}(τ) · [f_{k}(τ, y_{k}(τ)) − f_{k}(τ, x^{0}(τ))] +
+ ∫_{a}^{t} dA(τ) · [f(τ, x^{0}(τ)) − f(τ, y_{k}(τ))]

and

y_{k}(t) = χ(t, x_{k}(t)).

In view of (1.11), (3.1) and Lemma 2.2

|η_{k}(y_{k})(t) − η_{k}(y_{k})(s)| ≤ | ∫_{s}^{t} dV (A_{k})(τ) · ω_{k}(τ, r)| +
+| ∫_{s}^{t} dV (A)(τ) · ω_{0}(τ, r)| + δ_{k} for a ≤ s ≤ t ≤ b (k = 1, 2, …) (3.4)

and

lim_{k→+∞} δ_{k} = 0,  (3.5)

where

ω_{0}(τ, σ) =

= max {|f(τ, x) − f(τ, y)| : ∥x∥≤∥x^{0}∥_{s} + r, ∥y∥≤∥x^{0}∥_{s} + r, ∥x − y∥≤σ}

and

δ_{k} = sup {|η_{k}(x^{0})(t) − η_{k}(x^{0})(s)| : a ≤ s ≤ t ≤ b}.

On the other hand, according to Lemma 1 from [9], there exists a positive number ρ_{0} such that

∥x_{k}∥_{s} < ρ_{0}[∥β(∥x_{k}∥_{s})∥ + ∥ ∫_{a}^{b} dV (A)(t) · α(t, ∥x_{k}∥_{s})∥ + ζ_{k}]
(k = 1, 2, …),

where

ζ_{k} = sup {∥η_{k}(y_{k})(t)∥ : t ∈ [a, b]} + ∥γ_{k}(y_{k})∥.

From this in view of (1.9), (3.4), (3.5) and the condition (b) of Definition 1.2 we have

ρ_{1} = sup {∥x_{k}∥_{s} : k = 1, 2, … } < +∞  (3.6)

and

|z_{k}(t) − z_{k}(s)| ≤ g(t) − g(s) for a ≤ s ≤ t ≤ b (k = 1, 2, …), (3.7)

where

g(t) =∫_{a}^{t} dV (A)(τ) · ψ(τ)+sup {∫_{a}^{t}dV (A_{k})(τ) · ω_{k}(τ, r) : k = 1, 2, … },
ψ(τ) = ρ_{1} max {|P (τ, x)| : ∥x∥ ≤ ρ_{1}} + α_{1}(τ, ρ_{1}) + ω_{0}(τ, r).

Hence the sequence (z_{k})_{k=1}^{+∞} satisfies the conditions of Helly’s choice theorem and condition (2.1), where l_{k} = 0 (k = 1, 2, …). Therefore with regard to Lemma 2.3 and (3.6) we may assume without loss of generality that

lim_{k→+∞} ∥z_{k} − z^{∗}∥_{s} = 0 and lim_{k→+∞} x_{k}(a) = c^{∗},

where z^{∗} ∈ BV([a, b], R^{n}), c^{∗} ∈ R^{n}. By this, (3.5) and (3.7) we have from (3.3) that

|x_{k}(t) − x_{k}(s)| ≤ δ_{k} + g(t) − g(s)
for a ≤ s ≤ t ≤ b (k = 1, 2, …)  (3.8)

and

lim_{k→+∞} ∥x_{k} − x^{∗}∥_{s} = 0,  (3.9)

where x^{∗}(t) ≡ c^{∗} + z^{∗}(t). Using (3.8), it is not difficult to show that

∥y_{k}(t) − y_{k}(s)∥ ≤ ρ_{2}(∥δ_{k}∥ + ∥V (x^{0})(t) + g(t) − V (x^{0})(s) − g(s)∥)

for a ≤ s ≤ t ≤ b (k = 1, 2, …), where ρ_{2} = 2 + r^{−1}(ρ_{1} + ∥x^{0}∥_{s}).

Consequently, by virtue of Lemma 2.2 condition (2.2) is fulfilled. On the other hand, in view of (1.8)

lim_{k→+∞} γ_{k}(y_{k}) = 0.  (3.10)

By (2.2), (3.9) and (3.10), passing to the limit as k → +∞ in equalities (3.3) and

l(x_{k}) + l̃(x_{k}) + γ_{k}(y_{k}) = 0,

we find that x^{∗} is the solution of the problem (1.5), (1.6).

Further, from (3.2) we have

∥x^{∗} − x^{0}∥ ≥ ε.

But this is impossible, since according to the condition (c) of Definition 1.2 the latter problem has no solution differing from x^{0}.

Proof of Corollary 1.1. According to Theorem 1 from [9] problem (1.1), (1.2) has the unique solution x^{0} which, in view of conditions (1.12)–(1.14), is strongly isolated in every radius r > 0. Hence, the corollary follows from Theorem 1.1.

Proof of Corollary 1.2. According to Theorem 2 from [9], problem (1.1), (1.2) has the unique solution x^{0}. Let r be an arbitrary positive number. It suffices to show that x^{0} is strongly isolated in the radius r.

Let S be a set of all matrix-functions B ∈ BV([a, b], R^{n×n}), B(a) = 0, satisfying (1.16) for a ≤ s ≤ t ≤ b. Then by Lemma 1 from [9] there exists a positive number ρ_{0} such that

∥x∥_{s} ≤ ρ_{0}[∥[|l(x)| − l_{0}(x)]_{+}∥ +
+ sup {∥x(t) − x(a) − ∫_{a}^{t} dB(τ) · x(τ)∥ : t ∈ [a, b]}]
for x ∈ BV([a, b], R^{n}), B ∈ S.  (3.11)

Moreover, in view of (1.14) there exists a number ρ_{1} > ∥x^{0}∥_{s} + r such that

ρ_{0}[∥l_{1}(ρ)∥ + ∥ ∫_{a}^{b} dV (A)(t) · α(t, ρ)∥] < ρ for ρ ≥ ρ_{1}. (3.12)

It is evident that

f(t, x) = P̃(t)x + q̃(t, x) for t ∈ [a, b], ∥x − x^{0}(t)∥ < r,

where

P̃(t) = \frac{1}{2}[P_{1}(t) + P_{2}(t)],
q̃(t, x) = [f(t, x) − P̃(t)x]χ(∥x∥) on [a, b] × R^{n}

and

χ(s) = \begin{cases}1 & for 0 ≤ s < ρ_{1}, \\ 2 − \frac{τ}{ρ_{1}} & for ρ_{1} ≤ s < 2ρ_{1}, \\ 0 & for s ≥ 2ρ_{1}. \end{cases}  (3.13)

It follows from (1.13) and (1.14) that the conditions (a), (b), and (d) of Definition 1.2 are fulfilled for P = P̃, q = q̃, l, l_{0}, l̃= h − l and A.

Let us show that the problem

dx(t) = dA(t) · [P̃(t)x + q̃(t, x)]  (3.14)

has no solution differing from x^{0} and satisfying (1.6). Let y^{0} be an arbitrary solution of problem (3.14), (1.6). Then y^{0} will be the solution of the system

dx(t) = dB(t) · x(t) + dA(t) · [f(t, y^{0}(t)) − P (t, y^{0}(t))y^{0}(t)]χ(∥y^{0}(t)∥),

where

B(t) = ∫_{a}^{b} dA(τ) · [P̃(τ) + (P (τ, y^{0}(τ)) − P̃(τ))χ(∥y^{0}(τ)∥)].

On the other hand, it follows from (1.15) and (3.13) that B ∈ S. Hence, using (1.12), (1.13), (3.11) and (3.12), we obtain

∥y^{0}∥_{s} < ρ_{1}.

But by (3.13) it is clear that every solution of system (3.14), admitting such an estimate, is the solution of problem (1.1), (1.2). Therefore x^{0}(t) ≡ y^{0}(t), and the condition (c) of Definition 1.2 is fulfilled, i.e., x^{0} is strongly isolated in every radius r.

Proof of Corollary 1.3. Let

f(t, x) = (f_{i}(t, x))_{i=1}^{n}, q(t, ρ) = (q_{i}(t, ρ))_{i=1}^{n},

P_{0}(t) = (p_{0ij}(t))_{i,j=1}^{n}, Q(t) = (q_{ij}(t))_{i,j=1}^{n},

η_{i}(t, x) = [ ∑_{j=1}^{n} q_{ij}(t)|x_{j}| + q_{i}(t, ∥x∥) + 1]^{−1} ×

×[f_{i}(t, x) − ∑_{j=1}^{n} p_{0ij}(t)x_{j}] (i = 1, …, n)

and

p_{ij}(t, x) = p_{0ij}(t) + q_{ij}(t)η_{i}(t, x) sign x_{j} (i, j = 1, …, n)

for t ∈ [a, b], x = (x_{i})_{i=1}^{n} ∈ R^{n}, ρ ∈ R_{+}. It is not difficult to verify that the matrix-functions

P (t, x) = (p_{ij}(t, x))_{i,j=1}^{n}, P_{1}(t) = P_{0}(t) − Q(t), P_{2}(t) = P_{0}(t) + Q(t)

and the vector-function

α(t, ρ) = (q_{i}(t, ρ) + 1)_{i=1}^{n}

satisfy the condition of Corollary 1.2. Hence Corollary 1.3 follows from Corollary 1.2.

Corollary 1.4 is a special case of Corollary 1.3, when Q(t) ≡ 0 and l_{0}(x) ≡ 0.

Corollary 1.5 follows from Corollary 1.3 and Theorem 4 from [9].

Proof of Corollary 1.6. By (1.22) there exist sequences of positive numbers (ρ_{k})_{k=0}^{+∞} and (μ_{k})_{k=1}^{+∞} such that ρ_{k} → +∞ as k → +∞,

sup {∥g(x)∥ : x ∈ BV([a, b], R^{n})} < ρ_{k} (k = 1, 2, …) (3.15)

and

∥x + (−1)^{j}d_{j}A(t) · f(t, y)∥ > μ_{k} > σ_{3−j}(t, z) + ρ_{0}
for ∥x∥ ≥ ρ_{k}, ∥y∥ = ρ_{k}, (−1)^{j}(t − t_{0}) < 0,

z ∈ X_{j}(t, c, ρ_{k}), c ∈ G (j = 1, 2; k = 1, 2, …).  (3.16)

Let x^{0} be the unique solution of problem (1.1), (1.2). According to Theorem 1.1 it is sufficient to establish that x^{0} is strongly isolated in the radius r_{k} = ρ_{k} − ∥x^{0}∥_{s} for every natural k.

Put

P (t, x) = 0, q(t, x) = f(t, χ_{k}(x)) on [a, b] × R^{n}

and

l(x) = x(t_{0}), l_{0}(x) = 0, l̃(x) = −g(x) for x ∈ BV([a, b], R^{n}),

where

χ_{k}(x) = \begin{cases}x & for ∥x∥ ≤ ρ_{k}, \\ \frac{r}{∥x∥}x & for ∥x∥ > ρ_{k}. \end{cases}

It is obvious that the conditions (a), (b), and (d) of Definition 1.2 are fulfilled. Let now y be an arbitrary solution of problem (1.5), (1.6), and let c = g(y). By (3.15) and the fact that the matrix-function A is continuous at the point t_{0}, we have

t^{∗} > t_{0},

where

t^{∗} = sup {t : ∥y(s)∥ < ρ_{k} for t_{0} ≤ s ≤ t}.

Obviously, y ∈ X_{1}(t^{∗}, c, ρ_{k}), there exists y(t^{∗}−), and

∥y(t^{∗}−)∥ ≤ ρ_{k}.

From the above argument, due to (3.16) and the equality

y(t^{∗}−) = y(t^{∗}) − d_{1}A(t^{∗}) · f(t^{∗}, χ_{k}(y(t^{∗})))

we obtain

∥y(t^{∗})∥ < ρ_{k}.

Assume t^{∗} < b. If there exists a sequence τ_{m} > t^{∗} (m = 1, 2, …) such that τ_{m} → t^{∗} as m → +∞ and

∥y(τ_{m})∥ ≥ ρ_{k} (m = 1, 2, …),

then by the first inequality of (3.16)

∥y(τ_{m}−)∥ = ∥y(τ_{m}) − d_{1}A(τ_{m}) · f(τ_{m}, χ_{k}(y(τ_{m})))∥ > μ_{k} (m = 1, 2, …).

Hence,

∥y(t^{∗}+)∥ = lim_{m→+∞} ∥y(τ_{m}−)∥ ≥ μ_{k}.

On the other hand, by the second inequality of (3.16) and the definition of σ_{2}(t, z) we have

μ_{k} > ∥y(t^{∗}+)∥.

The obtained contradiction shows that there exists a positive number δ such that

∥y(t)∥ < ρ_{k} for t ∈ [t^{∗}, t^{∗} + δ].

But this contradicts the definition of the point t^{∗}. Therefore t^{∗} = b, and y is the solution of system (1.1) on [a, b].

Analogously, we can show that y is the solution of system (1.1) on [a, t_{0}]. It is now clear that y is the solution of problem (1.1), (1.2), and y(t) ≡ x^{0}(t). Thus the condition (c) of Definition 1.2 is likewise fulfilled.

Corollary 1.7 follows from Corollary 1.6 for g(x) = const.

Proof of Corollary 1.8. Put

P (t, x) = P (t), q(t, x) = f(t, χ(t, x)) − P (t)χ(t, x),
l̃(x) = h(χ̃(x)) − l(χ̃(x)), l_{0}(x) = 0,

where χ is the function defined by (3.1), and χ̃(x)(t) = χ(t, x(t)). Then the conditions (a), (b), and (d) of Definition 1.2 are fulfilled. According to Theorem 1.1 it remains to show that the condition (c) is fulfilled. Let x̃ be an arbitrary solution of problem (1.5), (1.6). Assume

x(t) = x̃(t) − x^{0}(t).

As |χ(t, x̃(t)) − x^{0}(t)| ≤ |x(t)| for t ∈ [a, b] and the operator l^{∗} is nondecreasing, it follows from (1.24) and (1.25) that x is the solution of the problem (1.26). But the latter problem has only the trivial solution. Thus problem (1.5), (1.6) has no solution differing from x^{0}.

Proof of Corollary 1.9. Let

P (t) = F (t, x^{0}(t)).

By Theorem 1 from [7] the unique solvability of problem (1.27) guarantees the existence of a positive number r such that problem (1.26) has no nontrivial solution if

l^{∗}(|x|) = α∥x∥_{s},  (3.17)
α ∈ R_{+}^{n}, Q ∈ L([a, b], R_{+}^{n×n}; A), ∥α∥ ≤ δ,
∥ ∫_{a}^{b} dV (A)(t) · Q(t)∥ ≤ δ.  (3.18)

Choosing a number r > 0 such that

∥h(x) − l(x − x^{0})∥ ≤ \frac{δ}{n}∥x − x^{0}∥_{s} for x ∈ U(x^{0}; r)  (3.19)

inequality (3.18) will be fulfilled, where

Q(t) = max {|F (t, x) − P (t)| : ∥x − x^{0}(t)∥ ≤ r}.

From the representation

f(t, x) − f(t, x^{0}(t)) = ∫_{0}^{1} F (t, sx + (1 − s)x^{0}(t))ds · (x − x^{0}(t))

and condition (3.19) follow inequalities (1.24) and (1.25), where l^{∗} is the operator defined by (3.17), and α is the n-dimensional vector all of whose components are equal to \frac{δ}{n}. On the other hand, by our choice of Q and l^{∗} problem (1.26) has only the trivial solution. Thus, all conditions of Corollary 1.8 that guarantee the (x^{0}; r)-correctness of the problem (1.1), (1.2) are fulfilled.

4. Application

Let E_{k}^{n} and Ẽ_{k}^{n} be the spaces of all vector-function x : N_{k} → R^{n} and

x : Ñ_{k} → R^{n} with the norms ∥x∥_{k} = max{∥x(i)∥ : i ∈ N_{k}} and ∥x∥_{k} =

max{∥x(i)∥ : i ∈ Ñ_{k}}, respectively, where N_{k} = {1, …, k}, Ñ_{k} = {0, …, k} (k = 1, 2, …); let Δ be the first order difference operator, i.e., Δx(i − 1) = x(i) − x(i − 1) for x ∈ Ẽ_{k}^{n} and i ∈ N_{k} (k = 1, 2, …).

By M(N_{n}×R_{+}, R_{+}^{n}) we denote a set of all vector-functions ω : N_{n}×R_{+} → R_{+}^{n} such that ω(i, ·) is continuous and nondecreasing, and ω(i, 0) = 0 for i ∈ N_{n}.

For the system of ordinary differential equations

\frac{dx(t)}{dt} = f(t, x(t))  (4.1)

consider the boundary value problem

h(x) = 0,  (4.2)

where f ∈ K([a, b] × R^{n}; R^{n}), and h : BV_{s}([a, b], R^{n}) → R^{n} is a continuous operator.

Along with problem (4.1), (4.2) let us consider its difference analogue

Δy(i − 1) = \frac{1}{k} f_{k}(i, y(i)) (i = 1, …, k),  (4.1_{k})
h_{k}(y) = 0  (4.2_{k})

(k = 2, 3, …), where f_{k}(i, ·) : R^{n} → R^{n} is a continuous function for every i ∈ N_{k}, and h_{k} : Ẽ_{k}^{n} → R^{n} is a continuous operator.

Let ν_{k} : [a, b] → R (k = 1, 2, …) be the functions defined by equations ν_{k}(t) = i for t ∈ I_{ik} ∩ [a, b] (i = 0, …, k), where I_{ik} = [a + \frac{2i−1}{2k}(b − a), a + \frac{2i+1}{2k}(b − a)[ (i = 0, …, k).

Theorem 4.1. Let x^{0} be a solution of problem (4.1), (4.2), strongly isolated in the radius r > 0, and let

lim_{k→+∞} ∥\frac{1}{k} ∑_{i=1}^{ν_{k}(t)} f_{k}(i, x) − ∫_{0}^{t} f(τ, x)dτ∥ = 0

uniformly on [a, b] for every x ∈ D(x^{0}; r). Moreover, let there exist a sequence ω_{rk} ∈ M(N_{n} × R_{+}, R_{+}^{n}) (k = 1, 2, …) such that

sup \{∥\frac{1}{k} ∑_{i=1}^{k} ω_{rk}(i, r)∥ : k = 1, 2, … \} < +∞,

lim_{s→0+} sup \{∥\frac{1}{k} ∑_{i=1}^{k} ω_{rk}(i, s)∥ : k = 1, 2, … \} = 0

and

|f_{k}(i, x) − f_{k}(i, y)| ≤ ω_{rk}(i, ∥x − y∥)

on N_{k} × D(x^{0}; r) (k = 1, 2, …). Then for every ε > 0 there exists a natural number k_{0} such that a set Y_{rk} of solutions of problem (4.1_{k}), (4.2_{k}), satisfying the inequality

max \{∥y(i) − x^{0}(a + \frac{i}{k} (b − a))∥ : i ∈ Ñ_{k}\} < r

is not empty, and

max \{∥y(i) − x^{0}(a + \frac{i}{k} (b − a))∥ : i ∈ Ñ_{k}\} < ε for y ∈ Y_{rk} (k > k_{0}).

This theorem follows from Theorem 1.1, since problem (4.1_{k}), (4.2_{k}) can be written in the form of problem (1.1_{k}), (1.2_{k}) for every k ∈ {2, 3, … }.

References

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9. M. Ashordia, The conditions for existence and uniqueness of solutions of nonlinear boundary value problems for systems of generalized ordinary differential equations. (Russian) Differentsial’nye Uravneniya (in print).

(Received 22.12.1994)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Alexidze St., Tbilisi 380093

Republic of Georgia