Mem. Differential Equations Math. Phys. 10(1997), 111–112

M. Ashordia

ON SUCCESSIVE APPROXIMATIONS FOR SOLVING THE CAUCHY PROBLEM FOR A SYSTEM OF LINEAR GENERALIZED ORDINARY

DIFFERENTIAL EQUATIONS

(Reported on February 12–19, 1996)

In the present note, we consider a succesive approximations method of construction of the solution of the Cauchy problem

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

where t_{0} ∈ [a, b], c_{0} ∈ R^{n}, A : [a, b] → R^{n×n} and q : [a, b] → R^{n} are a matrix-function and a vector-function with bounded variation components, respectively.

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]; 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).

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

1991 Mathematics Subject Classification. 34B05.

Key words and phrases. Cauchy problem, system of linear generalized ordinary differential equations, successive approximations.

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.

A vector-function x ∈ BV([a, b], R^{n}) is said to be a solution of the problem (1),(2) if it satisfies the condition (2) and

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

Theorem. Let

det (I_{n} + (−1)^{j}d_{j}A(t)) ≠ 0 for (−1)^{j}(t − t_{0}) < 0 (j = 1, 2).

Then the problem (1),(2) has a unique solution x and

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

where

x_{k}(t_{0}) = c_{0} (k = 0, 1, …),
x_{0}(t) = (I_{n} + (−1)^{j}d_{j}A(t))^{−1}c_{0} for (−1)^{j}(t − t_{0}) < 0 (j = 1, 2)

and

x_{k}(t) = (I_{n} + (−1)^{j}d_{j}A(t))^{−1}[c_{0} + ∫_{t_{0}}^{t} dA(τ) · x_{k−1}(τ) +

+ (−1)^{j}d_{j}A(t) · x_{k−1}(t) + q(t) − q(t_{0})]
for (−1)^{j}(t − t_{0}) < 0 (j = 1, 2; k = 1, 2, …).

Note that the unique solvability of the problem (1),(2) is proved in [1].

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