GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 1, 1998, 1-24

CONDITIONS OF THE EXISTENCE AND UNIQUENESS OF SOLUTIONS OF THE MULTIPOINT BOUNDARY

VALUE PROBLEM FOR A SYSTEM OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

M. ASHORDIA

Abstract. Effective sufficient conditions are established for the solvability and unique solvability of the boundary value problem

dx(t) = dA(t) · f(t, x(t)),
x_{i}(t_{i}) = φ_{i}(x) (i = 1, …, n),

where x = (x_{i})_{i=1}^{n}, A : [a, b] → R^{n×n} is a matrix-function with bounded variation components, f : [a, b] × R^{n} → R^{n} is a vectorfunction belonging to the Carathéodory class corresponding to A; t_{1}, …, t_{n} ∈ [a, b] and φ_{1}, …, φ_{n} are the continuous functionals (in general nonlinear) defined on the set of all vector-functions of bounded variation.

1. Statement of the Problem and Formulation of the Main Results

Let t_{1}, …, t_{n} ∈ [a, b]; a_{mik} : [a, b] → R be a nondecreasing function on the intervals [a, t_{i}[ and ]t_{i}, b] for m ∈ {1, 2} and i, k ∈ {1, …, n}; a_{ik}(t) ≡ a_{1ik}(t) − a_{2ik}(t), A = (a_{ik})_{i,k=1}^{n}; f = (f_{k})_{k=1}^{n} : [a, b] × R^{n} → R^{n} be a vectorfunction belonging to the Carathéodory class corresponding to the matrixfunction A, and φ_{i} : BV_{s}([a, b], R^{n}) → 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)),  (1.1)

where x = (x_{i})_{i=1}^{n}, consider the multipoint boundary value problem

x_{i}(t_{i}) = φ_{i}(x) (i = 1, …, n).  (1.2)

1

1991 Mathematics Subject Classification. 34B10.

Key words and phrases. Generalized ordinary differential equations, multipoint boundary value problem, solvability, unique solvability.

1072-947X/98/0100-0001$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

In this paper sufficient conditions are given for the existence and uniqueness of solutions of the boundary value problem (1.1), (1.2). Analogous results are contained in [1–4] for the multipoint boundary value problems for systems of ordinary differential equations.

The theory of generalized ordinary differential equations enables one to investigate ordinary differential and difference equations from the commonly accepted 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 appropriate boundary value problems for systems of generalized ordinary differential equations [5–15].

Throughout the paper the following notation and definitions will be used. R = ] − ∞, +∞[ , R_{+} = [0, +∞[ ; [a, b] (a, b ∈ R) is a closed segment. R^{n×m} is the 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×m} = {(x_{ij})_{i,j=1}^{n,m} : x_{ij} ≥ 0 (i = 1, …, n; j = 1, …, m)}.

If X = (x_{ij})_{i,j=1}^{n,m}, then |X| = (|x_{ij}|)_{i,j=1}^{n,m}, [X]_{±} = (|X| ± X)/2.

R^{n} = R^{n×1} is the space of all real column n-vectors x = (x_{i})_{i=1}^{n}; R_{+}^{n} = R_{+}^{n×1}.

If X ∈ R^{n×n}, then X^{−1} and det(X) are, respectively, the matrix inverse to X and the determinant of X; I_{n} is the identity n × n-matrix; δ_{ij} is Kronecker symbol, i.e., δ_{ij} = 1 if i = j and δ_{ij} = 0 if i ≠ j (i, j = 1, …, n).

∨^{b}_{a}(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+) are the left and the right limit of the matrix-function X : [a, b] → R^{n×m} at the point t^{1};

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], R^{n×m}) is the set of all matrix-functions of bounded variation

X : [a, b] → R^{n×m} (i.e., such that ∨^{b}_{a}(X) < +∞); BV_{v}([a, b], R^{n}) is the Banach space (BV ([a, b], R^{n}), ∥ · ∥_{v}) with the norm

∥x∥_{v} = ∥x(a)∥ + ∨^{b}_{a}(x);

BV_{s}([a, b], R^{n}) is the normed space (BV ([a, b], R^{n}), ∥ · ∥_{s});

BV_{s}([a, b], R_{+}^{n}) = {x ∈ BV_{s}([a, b], R^{n}) : x(t) ∈ R_{+}^{n} for t ∈ [a, b]}.

1We shall assume X(t) = X(a) for t ≤ a and X(t) = X(b) for t ≥ b, if necessary.

If D ⊂ R, then C(D, R^{n}) is the set of all continuous vector-functions x : D → R^{n}; C(D, R_{+}^{n}) = {x ∈ C(D, R^{n}) : x(t) ∈ R_{+}^{n} for t ∈ [a, b]}.

If α ∈ BV ([a, b], R) has not more than a finite number of discontinuity points and m ∈ {1, 2}, then D_{αm} = {t_{αm1}, …, t_{αmn_{αm}}} (t_{αm1} < ⋯ < t_{αmn_{αm}}) is the set of all points t ∈ [a, b] for which d_{m}α(t) ≠ 0;

μ_{αm} = max {d_{m}α(t) : t ∈ D_{αm}} (m = 1, 2);
ν_{αmβj} = max {d_{j}β(t_{αml}) + ∑_{t_{αm,l+1−m}<τ<t_{αm,l+2−m}}d_{j}β(τ) : l = 1, …, n_{αm}}

for β ∈ BV ([a, b], R) (j, m = 1, 2), here t_{α1n_{α1}+1} = b + 1, t_{α20} = a − 1. If β ∈ BV ([a, b], R), then

μ_{βji}(t) ≡ (−1)^{j}[β(t) − β(t_{i})] − d_{j}β(t_{i}) (j = 1, 2; i = 1, …, n).

If g : [a, b] → R is a nondecreasing function, x : [a, b] → R and a ≤ s < t ≤ b, then

∫_{s}^{t} x(τ)g(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 interval ]s, t[ with respect to the measure μ_{g} corresponding of the function g (if s = t, then ∫_{s}^{t} x(τ)dg(τ) = 0);

L^{p}([a, b], R; g) (1 ≤ p < +∞) is the space of all μ_{g}-measurable functions x : [a, b] → R such that ∫_{a}^{b} |x(t)|^{p}dg(t) < +∞ with the norm

∥x∥_{p,g} = ( ∫_{a}^{b} |x(t)|^{p}dg(t))^{\frac{1}{p}};

L^{+∞}([a, b], R; g) is the space of all μ_{g}-measurable essentially bounded functions x : [a, b] → R with the norm

∥x∥_{+∞,g} = ess sup {|x(t)| : t ∈ [a, b]}.

s_{k} : BV ([a, b], R) → BV ([a, b], R) (k = 0, 1, 2) are the operators defined by

s_{1}(x)(a) = s_{2}(x)(a) = 0,

s_{1}(x)(t) = ∑_{a<τ≤t} d_{1}x(τ), s_{2}(x)(t) = ∑_{a≤τ<t} d_{2}x(τ) for t ∈ (a, b],

s_{0}(x)(t) ≡ x(t) − s_{1}(x)(t) − s_{2}(x)(t).

A matrix-function is said to be nondecreasing if each of its components is such.

If G = (g_{ik})_{i,k=1}^{l,n} : [a, b] → R^{l×n} is a nondecreasing matrix-function and D ⊂ R^{n×m}, then L([a, b], D; G) is the set of all matrix-functions X = (x_{kj})_{k,j=1}^{n,m} : [a, b] → D such that x_{kj} ∈ L^{1}([a, b], R; g_{ik}) (i = 1, …, l; k = 1, …, n; j = 1, …, m);

∫_{s}^{t} dG(τ) · X(τ) = ( ∑_{k=1}^{n} ∫_{s}^{t} x_{kj}(τ)dg_{ik}(τ))_{i,j=1}^{l,m} for a ≤ s ≤ t ≤ b,

S_{j}(G)(t) ≡ (s_{j}(g_{ik})(t))_{i,k=1}^{l,n} (j = 0, 1, 2).

If D_{1} ⊂ R^{n} and D_{2} ⊂ R^{n×m}, then K([a, b] × D_{1}, D_{2}; G) is the Carathéodory class, i.e., the set of all mappings F = (f_{kj})_{k,j=1}^{n,m} : [a, b] × D_{1} → D_{2} such that for each i ∈ {1, …, l}, j ∈ {1, …, m} and k ∈ {1, …, n}:

a) the function f_{kj}(·, x) : [a, b] → D_{2} is μ_{g_{ik}}-measurable for every x ∈ D_{1};

b) the function f_{kj}(t, ·) : D_{1} → D_{2} is continuous for μ_{g_{ik}}-almost every t ∈ [a, b], and sup {|f_{kj}(·, x)| : x ∈ D_{0}} ∈ L([a, b], R; g_{ik}) for every compact D_{0} ⊂ D_{1}.

If G_{j} : [a, b] → R^{l×n} (j = 1, 2) are nondecreasing matrix-functions, G = G_{1} − G_{2} and X : [a, b] → R^{n×m}, then

∫_{s}^{t} dG(τ) · X(τ) = ∫_{s}^{t} dG_{1}(τ) · X(τ) − ∫_{s}^{t} dG_{2}(τ) · X(τ) for s ≤ t,

S_{k}(G) = S_{k}(G_{1}) − S_{k}(G_{2}) (k = 0, 1, 2),

K([a, b] × D_{1}, D_{2}; G) = ∩^{2}_{j=1}K([a, b] × D_{1}, D_{2}; G_{j}).

The inequalities between the vectors and between the matrices are understood componentwise.

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

An operator φ : BV_{s}([a, b], R^{n}) → R^{n} is called nondecreasing if for every x, y ∈ BV_{s}([a, b], R^{n}) such that x(t) ≤ y(t) for t ∈ [a, b], the inequality φ(x)(t) ≤ φ(y)(t) is fulfilled for t ∈ [a, b].

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 differential inequalities

dx(t) − dA(t) · f(t, x(t)) ≤ 0 (≥ 0)

we understand a vector-function x ∈ BV ([a, b], R^{n}) such that

x(t) − x(s) − ∫_{s}^{t} dA(τ) · f(τ, x(τ)) ≤ 0 (≥ 0) for a ≤ s ≤ t ≤ b.

Definition 1.1. We shall say the pair ((c_{il})_{i,l=1}^{n}; (φ_{0i})_{i=1}^{n}), consisting of a matrix-function (c_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) and a positive homogeneous nondecreasing operator (φ_{0i})_{i=1}^{n} : BV_{s}([a, b], R_{+}^{n}) → R_{+}^{n}, belongs to the set U(t_{1}, …, t_{n}) if the functions c_{il} (i ≠ l; i, l = 1, …, n) are nondecreasing on [a, b] and continuous at the point t_{i},

d_{j}c_{ii}(t) ≥ 0 for t ∈ [a, b] (j = 1, 2; i = 1, …, n)  (1.3)

and the problem

[dx_{i}(t) − sign(t − t_{i}) ∑_{l=1}^{n} x_{l}(t)dc_{il}(t)] sign(t − t_{i}) ≤ 0 (i = 1, …, n),

(−1)^{j}d_{j}x_{i}(t_{i}) ≤ x_{i}(t_{i})d_{j}c_{ii}(t_{i}) (j = 1, 2; i = 1, …, n);

(1.4)

x_{i}(t_{i}) ≤ φ_{0i}(|x_{1}|, …, |x_{n}|) (i = 1, …, n)  (1.5)

has no nontrivial non-negative solution.

Let the nondecreasing matrix-functions A_{m} = (a_{ik}^{(m)})_{i,k=1}^{n} : [a, b] → R^{n×n} (m = 1, 2) be defined by

a_{ik}^{(m)}(t_{i}) = a_{mik}(t_{i}), a_{ik}^{(m)}(t) = a_{mik}(t) +

+(−1)^{j}([d_{j}a_{1ik}(t_{i})]_{−} + [d_{j}a_{2ik}(t_{i})]_{−}) for (−1)^{j}(t − t_{i}) > 0 (j = 1, 2).

Theorem 1.1. Let the conditions

(−1)^{m+1}f_{k}(t, x_{1}, …, x_{n}) sign[(t − t_{i})x_{i}] ≤

≤ ∑_{l=1}^{n} p_{mikl}(t)|x_{l}| + q_{k}(t, ∑_{l=1}^{n} |x_{l}|)

for μ_{a_{mik}}- almost every t ∈ [a, b]\{t_{i}} (i, k = 1, …, n)  (1.6)

and

[(−1)^{m+j+1}f_{k}(t_{i}, x_{1}, …, x_{n}) sign x_{i} − ∑_{l=1}^{n} α_{mikjl}|x_{l}| −

−q_{k}(t_{i}, ∑_{l=1}^{n} |x_{l}|)]d_{j}a_{mik}(t_{i}) ≤ 0 (j = 1, 2; i, k = 1, …, n) (1.7)

be fulfilled on R^{n} for m ∈ 1, 2, and let the inequalities

|φ_{i}(x_{1}, …, x_{n})| ≤
≤ φ_{0i}(|x_{1}|, …, |x_{n}|) + γ( ∑_{l=1}^{n} ∥x_{l}∥_{s}) (i = 1, …, n)  (1.8)

be fulfilled on BV ([a, b], R^{n}), where α_{mikjl} ∈ R (j, m = 1, 2; i, k, l = 1, …, n), (p_{mikl})_{k,l=1}^{n} ∈ L([a, b], R^{n×n}; A_{m}) (m = 1, 2; i = 1, …, n), q = (q_{k})_{k=1}^{n} ∈ ∩_{m=1}^{2}K([a, b] × R_{+}, R_{+}^{n}; A_{m}) is a vector-function nondecreasing with respect to the second variable, γ ∈ C(R_{+}, R_{+}) and

lim_{ρ→+∞} \frac{1}{ρ} ∫_{a}^{b} d(A_{1}(t) + A_{2}(t)) · q(t, ρ) = 0, lim_{ρ→+∞} \frac{γ(ρ)}{ρ} = 0 . (1.9)

Let, moreover, there exists a matrix-function (c_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) such that

((c_{il})_{i,l=1}^{n}; (φ_{0i})_{i=1}^{n}) ∈ U(t_{1}, …, t_{n}),  (1.10)

∑_{m=1}^{2} ∑_{k=1}^{n} ∫_{s}^{t} p_{mikl}(τ)da_{mik}(τ) ≤ c_{il}(t) − c_{il}(s)

for a ≤ s < t < t_{i} and t_{i} < s < t ≤ b (i, l = 1, …, n) (1.11)

and

∑_{m=1}^{2} ∑_{k=1}^{n} α_{mikjl}d_{j}a_{mik}(t_{i}) ≤ δ_{il}d_{j}c_{ii}(t_{i}) (j = 1, 2; i, l = 1, …, n). (1.12)

Then problem (1.1), (1.2) is solvable.

Corollary 1.1. Let conditions (1.3), (1.9), (1.11), (1.12) and

|c_{il}(t) − c_{il}(s)| ≤
≤ ∫_{s}^{t} h_{il}(τ)dα_{l}(τ) for a ≤ s < t ≤ b (i, l = 1, …, n)  (1.13)

hold and let conditions (1.6) and (1.7) be fulfilled on R^{n} for every m ∈ {1, 2}, where α_{mikjl} ∈ R (j, m = 1, 2; i, k, l = 1, …, n), (p_{mikl})_{k,l=1}^{n} ∈ L([a, b], R^{n×n}; A_{m}) (m = 1, 2; i = 1, …, n); q = (q_{k})_{k=1}^{n} ∈ ∩_{m=1}^{2}K([a, b] × R_{+}, R_{+}^{n}; A_{m}) is a vector-function, nondecreasing with respect to the second variable, γ ∈ C(R_{+}, R_{+}); c_{il} (i ≠ l; i, l = 1, …, n) are the functions nondecreasing on [a, b] and continuous at the point t_{i}, c_{ii} ∈ BV ([a, b], R) (i = 1, …, n); α_{l} (l = 1, …, n) are the functions nondecreasing on [a, b] and having not more than a finite number of discontinuity points; h_{il} ∈ L^{μ}([a, b], R_{+}; α_{l}) (i, l = 1, …, n), 1 ≤ μ ≤ ∞. Let, moreover, the inequalities

|φ_{i}(x_{1}, …, x_{n})| ≤

≤ ∑_{m=0}^{2} ∑_{k=1}^{n} l_{mik}∥x∥_{ν,s_{m}(α_{k})} + γ( ∑_{l=1}^{n} ∥x_{l}∥_{s}) (i = 1, …, n)

be fulfilled for x = (x_{l})_{l=1}^{n} ∈ BV ([a, b], R^{n}) and the module of every characteristic value of the 3n × 3n-matrix H = (H_{j+1,m+1})_{j,m=0}^{2} be less than 1, where l_{mik} ∈ R_{+} (m = 0, 1, 2; i, k = 1, …, n), \frac{1}{μ} + \frac{2}{ν} = 1;

H_{j+1,m+1} = (ξ_{ij}l_{mik} + λ_{kmij}∥h_{ik}∥_{μ,s_{m}(α_{i})})_{i,k=1}^{n} (j, m = 0, 1, 2);
ξ_{ij} = [s_{j}(α_{i})(b) − s_{j}(α_{i})(a)]^{\frac{1}{ν}} (j = 0, 1, 2; i = 1, …, n);
λ_{k0k0} = [(2π^{−1})^{\frac{1}{ν}} ξ_{k0}]^{2} (k = 1, …, n),
λ_{kmij} = ξ_{km}ξ_{ij} for m^{2} + j^{2} + (i − k)^{2} > 0, mj = 0
(j, m = 0, 1, 2; i, k = 1, …, n);
λ_{kmij} = (\frac{1}{4} μ_{α_{k}m}ν_{α_{k}mα_{i}j} sin^{−2} \frac{π}{4n_{α_{k}m} + 2})^{\frac{1}{ν}}
(j, m = 1, 2; i, k = 1, …, n).

Then problem (1.1), (1.2) is solvable.

Corollary 1.2. Let inequalities (1.12) hold for i ≠ l (i, l = 1, …, n), let there exist m, m_{1} ∈ {1, 2} such that m+m_{1} = 3 and conditions (1.6), (1.7),

(−1)^{m_{1}+1}f_{k}(t, x_{1}, …, x_{n}) sign[(t − t_{i})x_{i}] ≤ ∑_{l=1}^{n} η_{il}|x_{l}| + q_{k}(t, ∑_{l=1}^{n} |x_{l}|)

for μ_{a_{m_{1}ik}}- almost every t ∈ [a, b]\{t_{i}} (i, k = 1, …, n),

[(−1)^{m_{1}+j+1}f_{k}(t_{i}, x_{1}, …, x_{n}) sign x_{i} − ∑_{l=1}^{n} α_{m_{1}ikjl}|x_{l}| −

−q_{k}(t_{i}, ∑_{l=1}^{n} |x_{l}|)]d_{j}a_{m_{1}ik}(t_{i}) ≤ 0 (j = 1, 2; i, k = 1, …, n)

be fulfilled on R^{n}, and let the inequalities

|φ_{i}(x_{1}, …, x_{n})| ≤ c_{i}|x_{i}(τ_{i})| + γ( ∑_{l=1}^{n} ∥x_{l}∥_{s}) (i = 1, …, n)

be fulfilled on BV ([a, b], R^{n}), where α_{mikjl} ∈ R and α_{m_{1}ikjl} ∈ R (j = 1, 2; i, k = 1, …, n), (p_{mikl})_{k,l=1}^{n} ∈ L([a, b], R_{+}^{n×n}; A_{m}) (i = 1, …, n), η_{il} ∈ R_{+} (i ≠ l; i, l = 1, …, n), η_{ii} < 0 (i = 1, …, n), a_{m_{1}ik} (i, k = 1, …, n) are the functions, nondecreasing and continuous on every interval [a, t_{i}[ and ]t_{i}, b], satisfying the condition

∑_{k=1}^{n} d_{j}a_{m_{1}ik}(t_{i}) ≤ 0 (j = 1, 2; i = 1, …, n);  (1.14)

q = (q_{k})_{k=1}^{n} ∈ ∩_{m=1}^{2}K([a, b] × R_{+}, R_{+}^{n}; A_{m}) is the vector-function nondecreasing with respect to the second variable, γ ∈ C(R_{+}, R_{+}) and conditions (1.9) hold; c_{i} ∈ R_{+} and τ_{i} ∈ [a, b], τ_{i} ≠ t_{i} (i = 1, …, n) are such that

c_{i}γ_{ij}ζ_{ij} < 1 if (−1)^{j}(τ_{i} − t_{i}) > 0 (j = 1, 2; i = 1, …, n); (1.15)

γ_{ij} = 1 + η_{ii}d_{j}α_{i}(t_{i}) + ∑_{k=1}^{n} p_{miki}(t_{i})d_{j}a_{mik}(t_{i}),

ζ_{ij} = exp (η_{ii}μ_{α_{i}ji}(τ_{i})), α_{i}(t) ≡ ∑_{k=1}^{n} a_{m_{1}ik}(t) (i, j = 1, …, n).

Let, moreover,

g_{iij} < 1 if (−1)^{j}(τ_{i} − t_{i}) > 0 (j = 1, 2; i = 1, …, n) (1.16)

and the real part of every characteristic value of the matrix (ξ_{il})_{i,l=1}^{n} be negative, where

ξ_{il} = η_{il}[δ_{il} + (1 − δ_{il})h_{ij}] − η_{ii}g_{ilj} for (−1)^{j}(τ_{i} − t_{i}) > 0
(j = 1, 2; i, l = 1, …, n),
g_{ilj} = c_{i}γ_{ij}(1 − c_{i}γ_{ij}ζ_{ij})^{−1}μ_{β_{il}ji}(τ_{i}) + max {μ_{β_{il}1i}(a), μ_{β_{il}2i}(b)},
β_{il}(t) = ∑_{k=1}^{n} ∫_{a}^{t} p_{mikl}(τ)da_{mik}(τ),

h_{ij} = 1 for c_{i}γ_{ij} ≤ 1 and h_{ij} = 1 + (c_{i}γ_{ij} − 1)(1 − c_{i}γ_{ij}ζ_{ij})^{−1} for c_{i}γ_{ij} > 1. Then problem (1.1), (1.2) is solvable.

Theorem 1.2. Let the conditions

(−1)^{m+1}[f_{k}(t, x_{1}, …, x_{n}) − f_{k}(t, y_{1}, …, y_{n})] ×

× sign [(t − t_{i})(x_{i} − y_{i})] ≤ ∑_{l=1}^{n} p_{mikl}(t)|x_{l} − y_{l}|

for μ_{a_{mik}}- almost every t ∈ [a, b]\{t_{i}} (i, k = 1, …, n),  (1.17)

{(−1)^{m+j+1}[f_{k}(t_{i}, x_{1}, …, x_{n}) − f_{k}(t_{i}, y_{1}, …, y_{n})] sign(x_{i} − y_{i}) −

− ∑_{l=1}^{n} α_{mikjl}|x_{l} − y_{l}|}d_{j}a_{mik}(t_{i}) ≤ 0 (j = 1, 2; i, k = 1, …, n) (1.18)

be fulfilled on R^{n} for every m ∈ {1, 2}, and let the inequalities

|φ_{i}(x_{1}, …, x_{n}) − φ_{i}(y_{1}, …, y_{n})| ≤
≤ φ_{0i}(|x_{1} − y_{1}|, …, |x_{n} − y_{n}|) (i = 1, …, n)  (1.19)

be fulfilled on BV ([a, b], R^{n}), where α_{mikjl} ∈ R (j, m = 1, 2; i, k, l = 1, …, n), (p_{mikl})_{k,l=1}^{n} ∈ L([a, b], R^{n×n}; A_{m}) (m = 1, 2; i = 1, …, n). Let, moreover, there exist a matrix-function (c_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) such that conditios (1.10)–(1.12) hold. Then problem (1.1), (1.2) has only one solution.

Corollary 1.3. Let conditions (1.3), (1.11)–(1.13) hold and let conditions (1.17) and (1.18) be fulfilled on R^{n} for every m ∈ {1, 2}, where α_{mikjl} ∈ R (j, m = 1, 2; i, k, l = 1, …, n), (p_{mikl})_{k,l=1}^{n} ∈ L([a, b], R^{n×n}; A_{m}) (m = 1, 2; i = 1, …, n), c_{il} (i ≠ l; i, l = 1, …, n) are the functions nondecreasing on [a, b] and continuous at the point t_{i}, c_{ii} ∈ BV ([a, b], R) (i = 1, …, n); α_{l} (l = 1, …, n) are the functions nondecreasing on [a, b] and having not more than a finite number of discontinuity points; h_{il} ∈ L^{μ}([a, b], R_{+}; α_{l}) (i, l = 1, …, n), 1 ≤ μ ≤ +∞. Let, moreover, the inequalities

|φ_{i}(x_{1}, …, x_{n}) − φ_{i}(y_{1}, …, y_{n})| ≤ ∑_{m=0}^{2} ∑_{k=1}^{n} l_{mik}∥x_{k} − y_{k}∥_{ν,s_{m}(α_{k})}

(i, k = 1, …, n)

be fulfilled on BV ([a, b], R^{n}), where l_{mik} ∈ R_{+} (m = 0, 1, 2; i, k = 1, …, n); \frac{1}{μ} + \frac{2}{ν} = 1 and the module of every characteristic value of the 3n × 3nmatrix H = (H_{j+1,m+1})_{j,m=0}^{2} appearing in Corollary 1.1 is less than 1. Then problem (1.1), (1.2) has only one solution.

Corollary 1.4. Let inequalities (1.12) hold for i ≠ l (i, l = 1, …, n), let there exist m, m_{1} ∈ {1, 2} such that m + m_{1} = 3 and conditions (1.17), (1.18),

(−1)^{m_{1}+1}[f_{k}(t, x_{1}, …, x_{n}) − f_{k}(t, y_{1}, …, y_{n})] sign [(t − t_{i})(x_{i} − y_{i})] ≤

≤ ∑_{l=1}^{n} η_{il}|x_{l} − y_{l}| for μ_{a_{m_{1}ik}}- almost every t ∈ [a, b]\{t_{i}}

(i, k = 1, …, n),

{(−1)^{m_{1}+j+1}[f_{k}(t_{i}, x_{1}, …, x_{n}) − f_{k}(t_{i}, y_{1}, …, y_{n})] sign(x_{i} − y_{i}) −

− ∑_{l=1}^{n} α_{m_{1}ikjl}|x_{l} − y_{l}|}d_{j}a_{m_{1}ik}(t_{i}) ≤ 0 (j = 1, 2; i, k = 1, …, n)

be fulfilled on R^{n}, where α_{mikjl} ∈ R and α_{m_{1}ikjl} ∈ R (j = 1, 2; i, k, l = 1, …, n), (p_{mikl})_{k,l=1}^{n} ∈ L([a, b], R_{+}^{n×n}; A_{m}) (i = 1, …, n), η_{il} ∈ R_{+} (i ≠ l; i, l = 1, …, n), η_{ii} < 0 (i = 1, …, n); a_{m_{1}ik} (i, k = 1, …, n) are the functions, nondecreasing and continuous on the intervals [a, t_{i}[ and ]t_{i}, b], satisfying condition (1.14). Let, moreover, c_{i} ∈ R_{+} and τ_{i} ∈ [a, b], τ_{i} ≠ t_{i} (i = 1, …, n) be such that conditions (1.15) and (1.16) hold and the real part of every characteristic value of the matrix (ξ_{il})_{i,l=1}^{n} be negative, where γ_{ij}, ζ_{ij}, ξ_{il}, h_{ij}, g_{ilj} (j = 1, 2; i, l = 1, …, n) and α_{i}(t) (i = 1, …, n) are the numbers and the functions, respectively, appearing in Corollary 1.2. Then for every λ_{i} ∈ [−c_{i}, c_{i}] and γ_{i} ∈ R_{+} (i = 1, …, n) system (1.1) has only one solution satisfying the boundary conditions

x_{i}(t_{i}) = λ_{i}x_{i}(τ_{i}) + γ_{i} (i = 1, …, n).  (1.20)

Remark 1.1. The 3n × 3n-matrix H appearing in Corollaries 1.1 and 1.3 can be replaced by the n × n-matrix

( max { ∑_{j=0}^{2} (ξ_{ij}l_{mik} + λ_{kmij}∥h_{ik}∥_{μ,s_{m}(α_{k})}) : m = 0, 1, 2})_{i,k=1}^{n}.

Theorem 1.3. Let φ_{0i}: BV_{s}([a, b], R_{+}^{n}) → R_{+} (i = 1, …, n) be the linear continuous functionals, a matrix-function C = (c_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) be such that the functions c_{il} (i ≠ l; i, l = 1, …, n) are nondecreasing on [a, b] and continuous at the point t_{i}, c_{ii} ∈ BV ([a, b], R) (i = 1, …, n), and condition (1.3) holds, but condition (1.10) is violated. Then there exist nondecreasing matrix-functions A_{m} = (a_{mkl})_{k,l=1}^{n} : [a, b] → R^{n×n} (m = 1, 2), matrix-functions (p_{mikl})_{k,l=1}^{n} ∈ L([a, b], R^{n×n}; A_{m}) (m = 1, 2; i = 1, …, n), numbers α_{mikjl} ∈ R (j, m = 1, 2; i, k, l = 1, …, n), a vector-function f = (f_{k})_{k=1}^{n} ∈ ∩_{m=1}^{2}K([a, b] × R^{n}; A_{m}) and functionals φ_{i} : BV_{s}([a, b], R^{n}) → R (i = 1, …, n) for which (1.11), (1.12) hold and conditions (1.17), (1.18) and (1.19) are fulfilled respectively on R^{n} for m ∈ {1, 2} and on BV ([a, b], R^{n}), but problem (1.1), (1.2) has no solution.

2. Auxiliary Propositions

Lemma 2.1. Let g ∈ BV ([a, b], R). Then

∫_{a}^{b} sign g(t)dg(t) = |g(b)| − |g(a)| + ∑_{a<t≤b} [|g(t−)| − g(t−) sign g(t)] −

− ∑_{a≤t<b} [|g(t+)| − g(t+) sign g(t)].  (2.1)

Proof. Let (α_{k}, β_{k}) ⊂ [a, b] (k = 1, 2, …) be a system of all maximal nonoverlapping intervals on which the function sign x(t) is constant. By G = {γ_{1}, γ_{2}, … } we denote a closure of the set A∪B, where A = {α_{1}, α_{2}, … } and B = {β_{1}, β_{2}, … }. Taking into account the relations [a, b] = ∪_{k=1}^{+∞}(α_{k}, β_{k})∪G, g(γ_{k}−) = 0 for γ_{k} ∉ {a} ∪ B and g(γ_{k}+) = 0 for γ_{k} ∈ {b} ∪ A, we have

∫_{a}^{b} sign g(t)dg(t) =

= ∑_{k=1}^{+∞} ( lim_{ε→0+} ∫_{α_{k}+ε}^{β_{k}−ε}sign g(t)dg(t) + [g(γ_{k}+) − g(γ_{k}−)] sign g(γ_{k})) =

= |g(b)| + |g(b−)| − g(b−) sign g(b) − |g(a)| − |g(a+)| + g(a+) sign g(a) +

+ ∑_{γ_{k}∈A∪B\{a,b}} {|g(γ_{k}−)| − |g(γ_{k}+)| + [g(γ_{k}+) − g(γ_{k}−)] sign g(γ_{k})}.

From this immediately follows equality (2.1), since

|x(t−)| = x(t−) sign x(t) and |x(t+)| = x(t+) sign x(t)

for t ∈ (α_{k}, β_{k}) (k = 1, 2, …).

Lemma 2.2. Let q = (q_{i})_{i=1}^{n} ∈ BV ([a, b], R^{n}) and (c_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) be such that the functions c_{il} (i ≠ l; i, l = 1, …, n) are nondecreasing and

d_{j}c_{ii}(t_{i}) ≥ 0, d_{j}c_{ii}(t) ≥ −1
for (−1)^{j}(t − t_{i}) > 0 (j = 1, 2; i = 1, …, n).  (2.2)

Let, moreover, B = (b_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) be a matrix-function satisfying the conditions

s_{0}(b_{ii})(t) − s_{0}(b_{ii})(s) ≤ [s_{0}(c_{ii})(t) − s_{0}(c_{ii})(s)] sign(t − s)
for (t − s)(s − t_{i}) > 0 (i = 1, …, n),  (2.3)
(−1)^{j+m}(|1 + (−1)^{m}d_{m}b_{ii}(t)| − 1) ≤ d_{m}c_{ii}(t)

12 M. ASHORDIA

for (−1)^{j}(t − t_{i}) ≥ 0 (j, m = 1, 2; i = 1, …, n),  (2.4)
|s_{0}(b_{il})(t) − s_{0}(b_{il})(s)| ≤ s_{0}(c_{il})(t) − s_{0}(c_{il})(s)
for a ≤ s < t ≤ b (i ≠ l; i, l = 1, …, n)  (2.5)

and

|d_{j}b_{il}(t)| ≤ d_{j}c_{il}(t) for t ∈ [a, b] (i ≠ l; i, l = 1, …, n). (2.6)

Then every solution x = (x_{i})_{i=1}^{n} of the system

dx(t) = dB(t) · x(t) + dq(t)  (2.7)

will be a solution of the system

[d|x_{i}(t)| − sign(t − t_{i}) ∑_{l=1}^{n} |x_{l}(t)|dc_{il}(t) −
− sign x_{i}(t)dq_{i}(t)] sign(t − t_{i}) ≤ 0 (i = 1, …, n),
(−1)^{j}d_{j}|x_{i}(t_{i})| ≤ ∑_{l=1}^{n} |x_{l}(t_{i})|d_{j}c_{il}(t_{i}) + (−1)^{j} sign x_{i}(t_{i})d_{j}q_{i}(t_{i})
(j = 1, 2; i = 1, …, n).

(2.8)

Proof. Taking into account (2.1) and the definition of a solution of system (2.7), it can be easily shown that

|x_{i}(t)| − |x_{i}(s)| = ∫_{s}^{t} |x_{i}(τ)|ds_{0}(b_{ii})(τ) +

+ ∑_{l≠i, l=1}^{n} ∫_{s}^{t} x_{l}(τ) sign x_{i}(τ)ds_{0}(b_{il})(τ) +

+ ∑_{s<τ≤t} [|x_{i}(τ)| − |x_{i}(τ−)|] + ∑_{s≤τ<t} [|x_{i}(τ+)| − |x_{i}(τ)|] +

+ ∫_{s}^{t} sign x_{i}(τ)dq_{i}(τ) for a ≤ s ≤ t ≤ b (i = 1, …, n).

By (2.2)–(2.6) from this we have

|x_{i}(t)| − |x_{i}(s)| ≤ ∑_{l=1}^{n} ∫_{s}^{t} |x_{l}(τ)|ds_{0}(c_{il})(τ) + ∑_{s<τ≤t} [|x_{i}(τ)| − |x_{i}(τ−)|] +

+ ∑_{s≤τ<t} [|x_{i}(τ+)| − |x_{i}(τ)|] + ∫_{s}^{t} sign x_{i}(τ)dq_{i}(τ) ≤

≤ ∑_{l=1}^{n} ∫_{s}^{t} |x_{l}(τ)|dc_{il}(τ) + ∑_{s<τ≤t} {|x_{i}(τ)|(1 − d_{1}c_{ii}(τ)) +

+ | ∑_{l≠i, l=1}^{n} x_{l}(τ)d_{1}b_{il}(τ)| − |x_{i}(τ)| |1 − d_{1}b_{ii}(τ)|} −

− ∑_{s≤τ<t} {|x_{i}(τ)|(1 + d_{2}c_{ii}(τ)) −

− | ∑_{l≠i, l=1}^{n} x_{l}(τ)d_{2}b_{il}(τ)| − |x_{i}(τ)| |1 + d_{2}b_{ii}(τ)|} −

− ∑_{s<τ≤t} ∑_{l≠i, l=1}^{n} |x_{l}(τ)|d_{1}c_{il}(τ) −

− ∑_{s≤τ<t} ∑_{l≠i, l=1}^{n} |x_{l}(τ)|d_{2}c_{il}(τ) + ∫_{s}^{t} sign x_{i}(τ)dq_{i}(τ) ≤

≤ ∑_{l=1}^{n} ∫_{s}^{t} |x_{l}(τ)|dc_{il}(τ) + ∫_{s}^{t} sign x_{i}(τ)dq_{i}(τ) +

+ ∑_{s<τ≤t} {|x_{i}(τ)| [1 − d_{1}c_{ii}(τ) − |1 − d_{1}b_{ii}(τ)|] +

+ ∑_{l≠i, l=1}^{n} |x_{l}(τ)| [|d_{1}b_{il}(τ)| − d_{1}c_{il}(τ)]} −

− ∑_{s≤τ<t} {|x_{i}(τ)| [1 + d_{2}c_{ii}(τ) − |1 + d_{2}b_{ii}(τ)|] −

− ∑_{l≠i, l=1}^{n} |x_{l}(τ)| [|d_{2}b_{il}(τ)| − d_{2}c_{il}(τ)]} ≤

≤ ∑_{l=1}^{n} ∫_{s}^{t} |x_{l}(τ)|dc_{il}(τ) + ∫_{s}^{t} sign x_{i}(τ)dq_{i}(τ)

for t_{i} < s ≤ t ≤ b (i = 1, …, n).

Therefore inequalities (2.8) are fulfilled for t > t_{i} and j = 2.

Analogously, we shall show that

|x_{i}(t)| − |x_{i}(s)| ≥ − ∑_{l=1}^{n} ∫_{s}^{t} |x_{l}(τ)|dc_{il}(τ) + ∫_{s}^{t} sign x_{i}(τ)dq_{i}(τ)

for a ≤ s ≤ t < t_{i} (i = 1, …, n).

The above inequality implies (2.8) for t < t_{i} and j = 1.

Lemma 2.3. Let a matrix-function (c_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) be such that the functions c_{il} (i ≠ l; i, l = 1, …, n) are nondecreasing, the condition (2.2) holds and system (1.4) has no nontrivial non-negative solution satisfying the condition

x_{i}(t_{i}) = 0 (i = 1, …, n).  (2.9)

Let, moreover, a matrix-function B = (b_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n}) be such that the functions b_{il} (i ≠ l; i, l = 1, …, n) are continuous at the point t_{i},

1 + (−1)^{j}d_{j}b_{ii}(t_{i}) ≠ 0 (j = 1, 2; i = 1, …, n)  (2.10)

and conditions (2.3)–(2.6) and

|1 + (−1)^{j}d_{j}b_{ii}(t)| ≥ 1 − d_{3−j}c_{ii}(t)
for (−1)^{j}(t − t_{i}) > 0 (j = 1, 2; i = 1, …, n)  (2.11)

hold. Then

det (I_{n} + (−1)^{j}d_{j}B(t)) ≠ 0 for t ∈ [a, b] (j = 1, 2).  (2.12)

Proof. Assume the contrary. Then there exist j ∈ {1, 2}, τ_{0} ∈ [a, b] and ξ = (ξ_{l})_{l=1}^{n} ∈ R^{n} such that

det (I_{n} + (−1)^{j}d_{j}B(τ_{0})) = 0

and the following system of linear algebraic equations

(I_{n} + (−1)^{j}d_{j}B(τ_{0}))y = ξ  (2.13)

is not solvable.

Let N_{0} = {i_{1}, …, i_{n_{0}}} be a set of all indices i ∈ {1, …, n} for which t_{i} = τ_{0}. Put η = ξ if N_{0} = ∅ and η = (η_{l})_{l=1}^{n}, η = ξ − ∑_{k=1}^{n_{0}} ξ^{(k)} if N_{0} ≠ ∅, where ξ^{(k)} = (δ_{li_{k}}ξ_{i_{k}})_{l=1}^{n} (k = 1, …, n_{0}). Then the system

(I_{n} + (−1)^{j}d_{j}B(τ_{0}))y = η  (2.14)

is not solvable, either, since in view of (2.1) the system

(I_{n} + (−1)^{j}d_{j}B(τ_{0}))y = ξ^{(k)}

has a solution (δ_{li_{k}}ξ_{i_{k}}(1 + (−1)^{j}d_{j}b_{i_{k}i_{k}}(t_{i_{k}}))^{−1})_{l=1}^{n} for every k ∈{1, …, n_{0}}.

Let B^{∗}(t) = (b_{il}^{∗}(t))_{i,l=1}^{n} be a matrix-function whose elements are defined by the equalities b_{il}^{∗}(τ_{0}) = 0, b_{il}^{∗}(t) = (−1)^{j}d_{j}b_{il}(τ_{0}) for t ≠ τ_{0} (i ≠ l); b_{ii}^{∗}(t) ≡ b_{ii}(t) if t_{i} = τ_{0}; b_{ii}^{∗}(t) = s_{0}(b_{ii})(t) for (τ_{0} − t)(τ_{0} − t_{i}) ≤ 0 (t_{i} ≠ τ_{0}) and b_{ii}^{∗}(t) = s_{0}(b_{ii})(t) + (−1)^{j}d_{j}b_{ii}(τ_{0}) for (τ_{0} − t)(τ_{0} − t_{i}) > 0 (t_{i} ≠ τ_{0}).

By (2.3)–(2.6) and (2.11) it is easy to show that the matrix-function B^{∗} satisfies conditions (2.3)–(2.6). Consequently, according to Lemma 2.2 it is not difficult to ascertain that the system

dx(t) = dB^{∗}(t) · x(t)

has only a trivial solution satisfying condition (2.9). This, in turn, is equivalent to the condition

Ker(I_{n} − F) = ∅,

where F : BV_{v}([a, b], R^{n}) → BV_{v}([a, b], R^{n}) is a completely continuous operator given by

(F x)(t) ≡ ( ∑_{l=1}^{n} ∫_{t_{i}}^{t} x_{l}(τ)db_{il}^{∗}(τ))_{i=1}^{n} for x = (x_{l})_{l=1}^{n} ∈ BV_{v}([a, b], R^{n}).

Therefore the operator equation

(I_{n} − F)x = φ

has a unique solution y = (y_{i})_{i=1}^{n}, where φ(t) = 0 for t ≠ τ_{0} and φ(τ_{0}) = η (see [7], p. 28). It is clear that

y_{i}(t) = ∑_{l=1}^{n} ∫_{t_{i}}^{t} y_{l}(τ)db_{il}^{∗}(τ) for t ≠ τ_{0} (i = 1, …, n)  (2.15)

and

y_{i}(τ_{0}) = ∑_{l=1}^{n} ∫_{t_{i}}^{τ_{0}}y_{l}(τ)db_{il}^{∗}(τ) + η_{i} (i = 1, …, n).  (2.16)

Assume that an index i ∈ {1, …, n} is such that τ_{0} ≠ t_{i}. Then by (2.15) and the definition of B^{∗} the function y_{i} is a solution of the problem

dy_{i}(t) = y_{i}(t)ds_{0}(b_{ii})(t), y_{i}(t_{i}) = 0

on [τ_{0} + ε, t_{i}] for t_{i} > τ_{0} and on [t_{i}, τ_{0} − ε] for t_{i} < τ_{0}, where ε > 0 is an arbitrary sufficiently small number. But the latter problem has only a trivial solution ([7], p. 106). Therefore

y_{i}(τ_{0}) + (−1)^{m}d_{m}y_{i}(τ_{0}) = 0 for (−1)^{m}(t_{i} − τ_{0}) > 0 (m = 1, 2).

On the other hand, it follows from (2.15) and (2.16) that

d_{m}y_{i}(τ_{0}) = ∑_{l=1}^{n} y_{l}(τ_{0})d_{m}b_{il}^{∗}(τ_{0}) − (−1)^{m}η_{i} (m = 1, 2).

Hence, with regard for the equalities

d_{m}b_{il}^{∗}(τ_{0}) = (−1)^{m+j}d_{j}b_{il}(τ_{0})
for (−1)^{m}(t_{i} − τ_{0}) > 0 (m = 1, 2; l = 1, …, n),

we have

y_{i}(τ_{0}) + (−1)^{j} ∑_{l=1}^{n} y_{l}(τ_{0})d_{j}b_{il}(τ_{0}) = η_{i} for t_{i} ≠ τ_{0}.

The latter equality is obviously true for t_{i} = τ_{0} as well because η_{i} = 0 and functions b_{il} (i ≠ l; l = 1, …, n) are continuous at the point t_{i}.

According to what has been said, it is clear that

(I_{n} + (−1)^{j}d_{j}B(τ_{0}))y(τ_{0}) = η,

i.e., system (2.14) is solvable. The contradiction abtained proves the lemma.

Lemma 2.4. Let condition (1.10) hold. Then there exists a positive number ρ_{∗} such that every solution of the problem

[d|x_{i}(t)| − sign(t − t_{i})( ∑_{l=1}^{n} |x_{l}(t)|dc_{il}(t) +

+du_{i}(t))] sign(t − t_{i}) ≤ 0 (i = 1, …, n),

(−1)^{j}d_{j}|x_{i}(t_{i})| ≤ |x_{i}(t_{i})|d_{j}c_{ii}(t_{i}) + d_{j}u_{i}(t_{i}) (j = 1, 2; i = 1, …, n);

(2.17)

|x_{i}(t_{i})| ≤ φ_{0i}(|x_{1}|, …, |x_{n}|) + γ (i = 1, …, n)  (2.18)

admits an estimate

∑_{i=1}^{n} ∥x_{i}∥_{s} ≤ ρ_{∗}[γ + \frac{1}{n} ∥u(·) − u(a)∥_{s}],  (2.19)

where γ ∈ R_{+} and u = (u_{i})_{i=1}^{n} ∈ BV ([a, b], R_{+}^{n}) are an arbitrary number and vector-function, respectively.

Proof. Let

g(x) = (|x_{i}(t_{i})| − φ_{0i}(|x_{1}|, …, |x_{n}|))_{i=1}^{n}

and let S be a set of all matrix-functions B = (b_{il})_{i,l=1}^{n} ∈ BV ([a, b], R^{n×n}) for which conditions (2.3)–(2.6), (2.10) and (2.11) are fulfilled and the function b_{il} is continuous at the point t_{i} for every i ≠ l (i, l = 1, …, n). According to Lemma 2.2 and (1.10) the problem

dx(t) = dB(t) · x(t), g(x) ≤ 0

has no nontrivial solution if only B ∈ S. Moreover, by Lemma 2.3 the condition (2.12) holds. By Lemma 1 from [13] there exists a positive number ρ_{0} such that

∥y∥_{s} ≤ ρ_{0}[∥[g(y)]_{+}∥ +

+ sup {∥y(t) − y(a) − ∫_{a}^{t} dB(τ) · y(τ)∥ : t ∈ [a, b]}]  (2.20)

for every y ∈ BV ([a, b], R^{n}) and B ∈ S.

Let (x_{i})_{i=1}^{n} be a solution of problem (2.17), (2.18). For every i∈{1, …, n} we assume

b_{i}(t) ≡ [s_{0}(c_{ii})(t) − s_{0}(c_{ii})(t_{i})] sign(t − t_{i}),  (2.21)
ψ_{i}(t) ≡ [ ∑_{l=1}^{n} ∫_{t_{i}}^{t} |x_{l}(τ)|dc_{il}(τ) −

− ∫_{t_{i}}^{t} |x_{i}(τ)|ds_{0}(c_{ii})(τ) + u_{i}(t) − u_{i}(t_{i})] sign(t − t_{i}).

Since the function b_{i} is continuous, the Cauchy problem

dy_{i}(t) = y_{i}(t)db_{i}(t) + dψ_{i}(t), y_{i}(t_{i}) = |x_{i}(t_{i})|

has a unique solution y_{i} for every i ∈ {1, …, n} (see [7], p. 106).

Taking (2.17) into account it is not difficult to verify that for every i ∈ {1, …, n} the function

z_{i}(t) ≡ |x_{i}(t)| − y_{i}(t)

satisfies the conditions of Lemma 2.4 from [15], where t_{0} = t_{i}, c_{0} = 0, α(t) ≡ b_{i}(t) and φ(t, x) ≡ x. In addition, the problem

du(t) = u(t)db_{i}(t), u(t_{i}) = 0  (2.22)

has only a trivial solution. Hence, by this lemma

|x_{i}(t)| ≤ y_{i}(t) for t ∈ [a, b] (i = 1, …, n)  (2.23)

and

|x_{i}(t)| ≡ η_{i}(t)y_{i}(t) (i = 1, …, n),

where η_{i} : [a, b] → [0, 1] (i = 1, …, n) are measurable functions. Moreover, it is easy to see that

y_{i}(t) ≡ y_{i}(t_{i}) + ∑_{l=1}^{n} ∫_{t_{i}}^{t} y_{l}(τ)db_{il}(τ) +
+ [u_{i}(t) − u_{i}(t_{i})] sign(t − t_{i}) (i = 1, …, n)  (2.24)

and

∥[g(y)]_{+}∥ ≤ nγ,  (2.25)

where

b_{ii}(t) = b_{i}(t) + sign(t − t_{i}) ∫_{t_{i}}^{t} η_{i}(τ)d(c_{ii}(τ) − s_{0}(c_{ii})(τ)), (2.26)

b_{il}(t) = sign(t − t_{i}) ∫_{t_{i}}^{t} η_{l}(τ)dc_{il}(τ) (i ≠ l; i, l = 1, …, n). (2.27)

On the other hand, according to Lemma 2.3 and condition (1.3) it can be easily shown that B = (b_{il})_{i,l=1}^{n} ∈ S. Therefore, inequality (2.20) holds. By (2.23)–(2.25) inequality (2.20) inplies estimate (2.19), where ρ_{∗} = n^{2}ρ_{0}.

3. Proof of the Main Results

Proof of Theorem 1.1. Let ρ_{∗} be a positive number such that the conclusion of Lemma 2.4 is true.

By (1.9) there exists a positive number ρ such that

ρ_{∗}[γ(ρ) + \frac{1}{n} ∥ ∫_{a}^{b} d(A_{1}(t) + A_{2}(t)) · q(t, ρ)∥] < ρ for ρ ≥ ρ_{0}. (3.1)

Let b_{i}(t) (i = 1, …, n) be defined by (2.21),

χ(s) = \begin{cases}1 & for |s| ≤ ρ_{0}, \\ 2 − \frac{|s|}{ρ_{0}} & for ρ_{0} < |s| < 2ρ_{0}, \\ 0 & for |s| ≥ 2ρ_{0}; \end{cases}  (3.2)

φ̃_{i}(x_{1}, …, x_{n}) = χ( ∑_{m=1}^{n} ∥x_{m}∥_{s})φ_{i}(x_{1}, …, x_{n}) (i = 1, …, n), (3.3)

and let B(t) be a diagonal matrix with diagonal elements b_{1}(t), …, b_{n}(t). Then

sup {χ( ∑_{m=1}^{n} |x_{m}|) · |f(t, x_{1}, …, x_{n})| : (x_{i})_{i=1}^{n} ∈ R^{n}} ∈

∈ L([a, b], R_{+}^{n}; A_{1} + A_{2})  (3.4)

and

sup { ∑_{m=1}^{n} |φ̃_{m}(x_{1}, …, x_{n})| : (x_{i})_{i=1}^{n} ∈ BV_{s}([a, b], R^{n})} < +∞. (3.5)

It is clear that the problem

dx(t) = dB(t) · x(t), x_{i}(t_{i}) = 0 (i = 1, …, n)

has only a trivial solution. With regard to this, conditions (3.4) and (3.5), by Lemma 2.4 from [14] the following problem

dx(t) = dB(t) · x(t)[1 − χ( ∑_{m=1}^{n} |x_{m}(t)|)] +

+ dA(t) · χ( ∑_{m=1}^{n} |x_{m}(t)|)f(t, x(t))^{2},  (3.6)
x_{i}(t_{i}) = φ̃_{i}(x_{1}, …, x_{n}) (i = 1, …, n)  (3.7)

is solvable. Let x = (x_{i})_{i=1}^{n} be an arbitrary solution of problem (3.6), (3.7). According to Lemma 2.2, (1.6), (1.7), (1.11) and (1.12) we have

(−1)^{j}[|x_{i}(t)| − |x_{i}(s)|] ≤

≤ ∫_{s}^{t} |x_{i}(τ)| [1 − (−1)^{j}χ( ∑_{m=1}^{n} |x_{m}(τ)|)]db_{i}(τ) +

+(−1)^{j} ∑_{k=1}^{n} ∫_{s}^{t} χ( ∑_{m=1}^{n} |x_{m}(τ)|) · f_{k}(τ, x(τ)) sign x_{i}(τ)da_{ik}(τ) ≤

≤ ∫_{s}^{t} |x_{i}(τ)|ds_{0}(c_{ii})(τ)+∫_{s}^{t} χ( ∑_{m=1}^{n} |x_{m}(τ)|) · |x_{i}(τ)|d(c_{ii}(τ) − s_{0}(c_{ii})(τ)) +

+ ∑_{l≠i, l=1}^{n} ∫_{s}^{t} χ( ∑_{m=1}^{n} |x_{m}(τ)|) · |x_{l}(τ)|dc_{il}(τ) +

2A vector-function from BV ([a, b], R^{n}) is said to be a solution of this system if it satisfies the corresponding integral equality.

+ ∑_{k=1}^{n} ∫_{s}^{t} q_{k}(τ, ∥x(τ)∥)d(a_{1ik}(τ) + a_{2ik}(τ)) ≤

≤ ∑_{l=1}^{n} ∫_{s}^{t} |x_{l}(τ)|dc_{il}(τ) + ∑_{k=1}^{n} ∫_{s}^{t} q_{k}(τ, ∥x(τ)∥)d(a_{1ik}(τ) + a_{2ik}(τ))

for s ≤ t, (−1)^{j}(t − t_{i}) ≥ 0, (−1)^{j}(s − t_{i}) ≥ 0 (j = 1, 2; i = 1, …, n),

since by conditions (1.3) and (1.10) the functions c_{ii} − s_{0}(c_{ii}) and c_{il} (i ≠ l) are nondecreasing. Consequently, condition (2.17) holds, where

u_{i}(t) ≡ ∑_{k=1}^{n} ∫_{a}^{t} q_{k}(τ, ∥x(τ)∥)d(a_{1ik}(τ) + a_{2ik}(τ)) (i = 1, …, n).

On the other hand, by (1.8), (3.2), (3.3) and (3.7) condition (2.18) holds. Therefore, according to the conclusion of Lemma 2.4

∑_{i=1}^{n} ∥x_{i}∥_{s} ≤ ρ_{∗}[γ( ∑_{i=1}^{n} ∥x_{i}∥_{s}) +

+ \frac{1}{n}∥ ∫_{a}^{b} d(A_{1}(t) + A_{2}(t)) · q(t, ∑_{i=1}^{n} ∥x_{i}∥_{s})∥].

From here with regard for (3.1) we conclude that

∑_{i=1}^{n} ∥x_{i}∥_{s} < ρ_{0}.

According to this inequality from (3.2), (3.3) (3.6) and (3.7) it follows that x = (x_{i})_{i=1}^{n} is a solution of problem (1.1), (1.2), too.

Proof of Theorem 1.2. (1.17)–(1.19) imply inequalities (1.6)–(1.8), where

q_{k}(t, ∥x∥) ≡ |f_{k}(t, 0, …, 0)| (k = 1, …, n), γ(r) ≡ ∑_{i=1}^{n} |φ_{i}(0, …, 0)|.

Hence all conditions of Theorem 1.1 are fulfilled so that the solvability of the problem (1.1), (1.2) is guaranteed. It remains to prove that problem (1.1), (1.2) has not more than one solution.

Let (x_{i})_{i=1}^{n} and (y_{i})_{i=1}^{n} be arbitrary solutions of problem (1.1), (1.2) and

z_{i}(t) ≡ x_{i}(t) − y_{i}(t) (i = 1, …, n).

According to Lemma 2.2

[d|z_{i}(t)| − sign z_{i}(t) ∑_{k=1}^{n} φ_{k}(t)da_{ik}(t)] sign(t − t_{i}) ≤ 0

for t ∈ [a, b] (i = 1, …, n)

and

(−1)^{j}d_{j}|z_{i}(t_{i})| ≤ (−1)^{j} sign z_{i}(t_{i}) ∑_{k=1}^{n} φ_{k}(t_{i})da_{ik}(t_{i})

(j = 1, 2; i = 1, …, n),

where

φ_{k}(t) ≡ f_{k}(t, x_{1}(t), …, x_{n}(t)) − f_{k}(t, y_{1}(t), …, y_{n}(t)) (k = 1, …, n).

By virtue of (1.11), (1.12) and (1.17)–(1.19) this implies that (|z_{i}(t)|)_{i=1}^{n} is a non-negative solution of problem (1.4), (1.5). Therefore, in view of (1.10) z_{i}(t) ≡ 0 (i = 1, …, n).

Corollaries 1.1–1.4 are immediately obtained from the proved theorems and Lemmas 2.6 and 2.7 from [15]. In addition, c_{il} (i, l = 1, …, n) are defined in just the same way as in [15] (see p. 36) for Corollaries 1.2 and 1.4.

Remark 1.1 follows from the proof of Lemma 2.6 from [15].

Proof of Theorem 1.3. As condition (1.10) is violated, problem (1.4), (1.5) has a nontrivial non-negative solution (x_{i})_{i=1}^{n}.

By y_{i} we shall denote unique solution of the following Cauchy problem

dy(t) = y(t)db_{i}(t) + dψ_{i}(t), y(t_{i}) = x_{i}(t_{i}),  (3.8)

where b_{i}(t) is defined by (2.21), and

ψ_{i}(t) ≡ [ ∑_{l=1}^{n} ∫_{t_{i}}^{t} x_{l}(τ)dc_{il}(τ) − ∫_{t_{i}}^{t} x_{i}(τ)ds_{0}(c_{ii})(τ)] sign(t − t_{i})

for every i ∈ {1, …, n}. Moreover, it is easy to verify that the function

z_{i}(t) ≡ x_{i}(t) − y_{i}(t)

satisfies the conditions of Lemma 2.4 from [15], where t_{0} = t_{i}, c_{0} = 0, α(t) ≡ b_{i}(t), φ(t, x) ≡ x and problem (2.22) has only a trivial solution for every i ∈ {1, …, n}. Consequently, according to this lemma

x_{i}(t) ≤ y_{i}(t) for t ∈ [a, b] (i = 1, …, n)

and

x_{i}(t) = η_{i}(t)y_{i}(t) for t ∈ [a, b] (i = 1, …, n),

where η_{i} : [a, b] → [0, 1] (i = 1, …, n) are some measurable functions. By this, (1.5) and (3.8) the vector-function y = (y_{i})_{i=1}^{n} is a nontrivial nonnegative solution of the linear homogeneous problem

dy(t) = dA(t) · y(t), y_{i}(t_{i}) = δ_{i}φ_{0i}(y_{1}, …, y_{n}) (i = 1, …, n),

where δ_{i} ∈ [0, 1] (i = 1, …, n), A(t) = (a_{il}(t))_{i,l=1}^{n} for t ∈ [a, b], and a_{ii}(t) and a_{il}(t) (i ≠ l) are equal to the right-hand side of (2.26) and (2.27), respectively.

Assume that system (1.4) has no nontrivial non-negative solution satisfying condition (2.9). With regard to (1.3) it is easy to verify that the matrix-function A = (a_{il})_{i,l=1}^{n} satisfies the conditions of Lemma 2.3. Therefore, there exist numbers c_{i} (i = 1, …, n) such that problem (1.1), (1.2) has no solution if for every j, m ∈ {1, 2} and i, k, l ∈ {1, …, n}

f_{k}(t, x_{1}, …, x_{n}) ≡ x_{k}, φ_{i}(x_{1}, …, x_{n}) ≡ δ_{i}φ_{0i}(x_{1}, …, x_{n}) + c_{i};
p_{mikl}(t) ≡ 0 and α_{mikjl} = 0 for l ≠ k,
p_{mikk}(t) ≡ 1 and α_{mikjk} = 1 for i ≠ k,
p_{mkkk}(t) = (−1)^{m+1} sign(t − t_{k}), α_{mkkjk} = (−1)^{m+j+1};
a_{mil}(t) = 0 for (−1)^{m}(t − t_{i}) ≥ 0 (i ≠ l),
a_{mil}(t) = a_{il}(t) sign(t − t_{i}) for (−1)^{m}(t − t_{i}) < 0 (i ≠ l),

and a_{1ii} and a_{2ii} are arbitrary nondecreasing functions for which

a_{1ii}(t) − a_{2ii}(t) ≡ a_{ii}(t).

Moreover, f_{k}, φ_{i}, a_{mkl}, p_{mikl} and α_{mikjl} (j, m = 1, 2; i, k, l = 1, …, n) satisfy conditions (1.11), (1.12), (1.17)–(1.19).

Assume now that system (1.4) has a nontrivial non-negative solution (x_{i})_{i=1}^{n} satisfying condition (2.9). Then it is clear that the conditions of Theorem 1.3 are fulfilled for functionals φ_{0i}(x_{1}, …, x_{n}) ≡ 0 (i = 1, …, n). Reasoning as above, it is not difficult to ascertain that the problem

dy(t) = dA(t) · y(t), y_{i}(t_{i}) = 0 (i = 1, …, n)

has a nontrivial solution. Therefore

Ker(I_{n} − F) ≠ ∅,

where F is a completely continuous operator of the type appearing in the proof of Lemma 2.3. By Fredholm’s alternative (see [7], p. 28) there exists a vector-function q = (q_{i})_{i=1}^{n} ∈ BV ([a, b], R^{n}) such that the problem

dy(t) = dA(t) · y(t) + dq(t), y_{i}(t_{i}) = 0 (i = 1, …, n)

is not solvable. Assuming z(t) = y(t) − q(t) we find that the problem

dz(t) = dA(t) · (z(t) + q(t)), z_{i}(t_{i}) = c_{i} (i = 1, …, n),

where c_{i} = q_{i}(t_{i}) (i = 1, …, n) is not solvable either.

Let a_{mil}, p_{mikl} and α_{mikjl} (j, m = 1, 2; i, k, l = 1, …, n) be defined as above and

f_{k}(t, x_{1}, …, x_{n}) ≡ x_{k} + q_{k}(t) and φ_{i}(x_{1}, …, x_{n}) ≡ q_{i}(t_{i}).

Then conditions (1.11), (1.12) and (1.17)–(1.19) are fulfilled and problem (1.1), (1.2) is not solvable either.

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(Received 12.10.1995)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia