Mem. Differential Equations Math. Phys. 10(1997), 122–124

M. Ashordia

ON SYSTEMS OF LINEAR GENERALIZED ORDINARY DIFFERENTIAL AND INTEGRAL INEQUALITIES

(Reported on April 15–22, 1996)

In the present note, we consider the questions of estimates of the solutions of the system of differential inequlities

dx(t) · sign(t − t_{0}) ≤ dC(t) · x(t) + dq(t) for t ∈ [a, b] \ {t_{0}},  (1)

satisfying the condition

x(t_{0}) + (−1)^{j}d_{j}x(t_{0}) ≤ c_{0} + d_{j}C(t_{0}) · c_{0} + d_{j}q(t_{0}) (j = 1, 2),  (2)

and of the solutions of the system of integral inequalities

x(t) ≤ c_{0} + ( ∫_{t_{0}}^{t} dC(τ) · x(τ) + q(t) − q(t_{0})) sign(t − t_{0}) for t ∈ [a, b],  (3)

satisfying the condition (2), where t_{0} ∈ [a, b], c_{0} ∈ R^{n}, q ∈ BV([a, b], R^{n}) and C = (c_{ik})_{i,k=1}^{n} ∈ BV([a, b], R^{n×n}).

The following notation and definitions will be used: R =]−∞, +∞[, [a, b] (a, b ∈ R) is a closed segment, R^{n×m} is the set of all real n×m-matrices X = (x_{ik})_{i,k=1}^{n,m}; if X ∈ R^{n×n}, then det(X) is the determinant of X, I_{n} is the identity n × n-matrix; R^{n} = R^{n×1} is the set of all real column n-vectors x = (x_{i})_{i=1}^{n}.

BV([a, b], R^{n×m}) is the set of all matrix-functions X = (x_{ik})_{i,k=1}^{n,m} : [a, b] → R^{n×m} such that every its component x_{ik} has bounded total variation on [a, b]. If I ⊂ R is an interval, then BV(I, R^{n×m}) is the set of all matrix-functions X : I → R^{n×m} such that X ∈ BV([c, d], R^{n×m}) for every c, d ∈ I. X(t−) = (x_{ik}(t−))_{i,k=1}^{n,m} and X(t+) = (x_{ik}(t+))_{i,k=1}^{n,m} are the left and the right limits of X at the point t ∈ [a, b] (X(a−) = X(a), X(b+) = X(b)), d_{1}X(t) = X(t) − X(t−), d_{2}X(t) = X(t+) − X(t).

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

∫_{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 interval ]s, t[ with

respect to the measure μ_{g} corresponding to the function g (if s = t, then ∫_{s}^{t} x(τ)dg(τ) = 0).

1991 Mathematics Subject Classification. 34B05.

Key words and phrases. Differential and integral inequalities, system of linear generalized ordinary differential equations.

If g_{j} : [a, b] → R (j = 1, 2) are nondeacreasing functions, g = g_{1} − g_{2} and x : [a, b] → R, then

∫_{s}^{t} x(τ)dg(τ) = ∫_{s}^{t} x(τ) dg_{1}(τ) − ∫_{s}^{t} x(τ)dg_{2}(τ) for a ≤ s ≤ t ≤ b.

If G = (g_{ik})_{i,k=1}^{n} ∈ BV([a, b], R^{n×n}), x = (x_{k})_{k=1}^{n} ∈ BV([a, b], R^{n}), then

∫_{s}^{t} dG(τ) · x(τ) = ( ∑_{k=1}^{n} ∫_{s}^{t} x_{k}(τ)dg_{ik}(τ))_{i=1}^{n} for a ≤ s ≤ t ≤ b.

Let I ⊂ [a, b] be an interval and A ∈ BV(I, R^{n×n}). A vector-function is said to be a solution of the system of the linear generalized ordinary differential equations dx(t) = dA(t) · x(t) + dq(t) (inequalities dx(t) ≤ dA(t) · x(t) + dq(t)) on I if

x(t) − x(s) − ∫_{s}^{t} dA(τ) · x(τ) − q(t) + q(s) = 0 (≤ 0) for s ≤ t (s, t ∈ I).

Theorem 1. Let c_{ik} (i ≠ k; i, k = 1, …, n) be functions nondecreasing on [a, b], C(t) = (c_{ik}(t))_{i,k=1}^{n},

det (I_{n} + (−1)^{j}d_{j}C(t)) ≠ 0 for (−1)^{j}(t − t_{0}) ≥ 0 (j = 1, 2),  (4)
1 + d_{j}c_{ii}(t) > 0 for (−1)^{j}(t − t_{0}) ≥ 0 (j = 1, 2; i = 1, …, n),  (5)

and

∑_{i=1}^{n} d_{j}c_{ik}(t) < 1 for (−1)^{j}(t − t_{0}) < 0 (j = 1, 2; k = 1, …, n).  (6)

Let, moreover, x ∈ BV([a, t_{0}[, R^{n}) ∩ BV(]t_{0}, b], R^{n}) be a solution of the system (1) satisfying the condition (2). Then

x(t) ≤ y(t) for t ∈ [a, b] \ {t_{0}},  (7)

where y ∈ BV([a, b], R^{n}) is a solution of the problem

dy(t) = [dC(t) · y(t) + dq(t)] sign(t − t_{0}) for t ∈ [a, b] \ {t_{0}},  (8)
(−1)^{j}d_{j}y(t_{0}) = d_{j}C(t_{0}) · y(t_{0}) + d_{j}q(t_{0}) (j = 1, 2),  (9)
y(t_{0}) = c_{0}.  (10)

Theorem 2. Let c_{ik} (i, k = 1, …, n) be functions nondecreasing on [a, b] and (4) and (6) hold, where C(t) = (c_{ik}(t))_{i,k=1}^{n}. Then for every solution x ∈ BV([a, t_{0}[, R^{n}) ∩ BV(]t_{0}, b], R^{n}) of the system (3) satisfying the condition (2), the estimate (7) holds, where y_{0} ∈ BV([a, b], R^{n}) is a solution of the problem (8)–(10).

Remark. The condition

max_{k=1,...,n} ∑_{i=1}^{n} |d_{j}c_{ik}(t)| < 1 for t ∈ [a, b] (j = 1, 2)

guarantees the conditions (4)–(6). Moreover, in view of (4) the problem (8)–(10) has a unique solution (see [1,Theorem III.1.4]).

References

1. Š. Schwabik, M. Tvrdý, and O. Vejvoda, Differential and integral equations: boundary value problems and adjoints. Academia, Praha, 1979.

Author’s address: Sukhumi Branch of Tbilisi State University 19, Al. Chavchavadze St., Tbilisi 380049 Georgia