Mem. Differential Equations Math. Phys. 10(1997), 132–133

N. Kekelia

ON A SUCCESIVE APPROXIMATIONS METHOD OF SOLVING THE

CAUCHY PROBLEM FOR THE SYSTEM OF GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

(Reported on May 27 and June 3, 1996)

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

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

where t_{0} ∈ [a, b], c_{0} ∈ R^{n}, A = (a_{ik})_{i,k=1}^{n} : [a, b] → R^{n×n} is a matrix-function whose components have bounded variation and are continuous from the right on [a, t_{0}[ and continuous from the left on ]t_{0}, b], f = (f_{k})_{k=1}^{n} : [a, b] × R^{n} → R^{n} is a vector-function belonging to the Carathéodory class corresponding to A.

The following notation and definitions will be used: R =] − ∞, +∞[, [a, b] (a, b ∈ R) is a closed segment, R^{n} is the space of all real column n-vectors x = (x_{i})_{i=1}^{n} with the

norm ∥x∥ = ∑_{i=1}^{n} |x_{i}|, R^{n×n} is the set of all real n × n matrices.

BV([a, b], R^{n}) is the set of all vector-functions x = (x_{i}) : [a, b] → R^{n} with components of bounded variation on [a, b], x(t−) = (x_{i}(t−))_{i=1}^{n} and x(t+) = (x_{i}(t+))_{i=1}^{n} 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 finction g (if s = t, then ∫_{s}^{t} x(τ)dg(τ) = 0).

If G_{j} = (g_{jik})_{i,k=1}^{n} : [a, b] → R^{n×n} (j = 1, 2) are nondecreasing matrix-functions, G = G_{1} − G_{2} and x = (x_{k})_{k=1}^{n} : [a, b] → R^{n}, then

∫_{s}^{t} dG(τ) · x(τ) = ( ∑_{k=1}^{n} ( ∫_{s}^{t} x_{k}(τ) dg_{1ik}(τ) − ∫_{s}^{t} x_{k}(τ)dg_{2ik}(τ)))_{i=1}^{n}

for a ≤ s ≤ t ≤ b;

1991 Mathematics Subject Classification. 34B05.

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

K([a, b] × R^{n}, R^{n}; G_{j}) is the Carathéodery class corresponding to G_{j}, i.e., the set of all vector-functions φ = (φ_{k})_{k=1}^{n} : [a, b] × R^{n} → R^{n} such that for each i, k = {1, …, n}:

(a) the function φ_{k}(·, x) : [a, b] → R^{n} is μ_{g_{jik}} -measurable for every x ∈ R^{n}; (b) the function φ_{k}(t, ·) : R^{n} → R^{n} is continuous for μ_{g_{jik}} -almost every t ∈ [a, b], and the function sup{φ_{k}(·, x) : ∥x∥ ≤ r} is μ_{g_{jik}} -integrable on [a, b] for every positive number r,

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

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) · f(τ, x(τ)) for a ≤ s ≤ t ≤ b.

Theorem. Let f ∈ K([a, b] × R^{n}, R^{n}; A) and

∥f(x, t) − f(t, y)∥ ≤ L∥x − y∥ for (t, x, y) ∈ [a, b] × R^{2n},

where L = const. Then the problem (1),(2) has a unique solution x, and

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

where

x_{0}(t) ≡ c_{0},

x_{k}(t) ≡ c_{0} + ∫_{t_{0}}^{t} dA(τ) · f(τ, x_{k−1}(τ)) (k = 1, 2, …).

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