Mem. Differential Equations Math. Phys. 6(1995), 121–123

M. Ashordia

ON THE QUESTION OF SOLVABILITY OF THE PERIODIC BOUNDARY VALUE PROBLEM FOR A SYSTEM OF LINEAR GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS

(Reported on October 30 and November 6, 1995)

Let ω be a positive number, A = (a_{ik})_{i,k=1}^{n} : R → R^{n×m} and g = (g_{i})_{i=1}^{n} : R → R^{n} be a matrix function and a vector function from BV_{ω}^{n×m} and BV_{ω}^{n}, respectively.

We consider the ω-periodic boundary value problem

dx(t) = dA(t) · x(t) + dg(t), x(0) = x(ω).  (1)

The use will be made of the following notation and definitions: R = ]−∞, +∞[ ; R^{n×m} is a set of all real n × m-matrices; I is the identity n × n-matrix; R^{n} = R^{n×1}. BV_{ω}^{n×m} is the set of all matrix functions X : R → R^{n×m} such that X(t + ω) = X(t) + X(ω) for t ∈ R, and the restriction on [0, ω] of every its components has bounded total variation; X(t−) and X(t+) are the left and the right limits of X at the point t ∈ R; d_{1}X(t) = X(t) − X(t−), d_{2}X(t) = X(t+) − X(t).

If g : R → R is nondecreasing, x : R → R and s < t, 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 Lebesque–Stieltjes integral over the open interval ]s, t[ with

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

L_{ω}(g) = {p ∈ BV_{ω} : ∫_{0}^{ω} |p(t)|dg(t) < ∞}.

A vector function x = (x_{i})_{i=1}^{n} ∈ BV_{ω}^{n} is a solution of the problem (1) if it is ω-periodic and

x_{i}(t) = x_{i}(s) + ∑_{k=1}^{n} ∫_{s}^{t} x_{k}(τ)da_{ik}(τ) for s ≤ t (i = 1, …, n).

Let natural numbers m and n_{1}, …, n_{m} (0 = n_{0} < n_{1} < ⋯ < n_{m} = n), nondecreasing functions c_{lj} : [0, ω] → R (l = 1, 2; j = 1, …, m), functions α_{lj} ∈ L_{ω}(c_{lj}) (l = 1, 2; j = 1, …, m) and matrix functions P_{lj} = (p_{ljik})_{i,k=1}^{n} (l = 1, 2; j = 1, …, m), p_{ljik} ∈ L_{ω}(c_{lj})

1991 Mathematics Subject Classification. 34B05.

Key words and phrases. Periodic boundary value problem, system of linear generalized ordinary differential equtions, Lebesque–Stieltjes integral, uniqueness solvability.

(i, k = n_{j−1} + 1, …, n_{j}) be such that a_{ik}(t) ≡ 0 (i = n_{j−1} + 1, …, n_{j}; k = n_{j} + 1, …, n; j = 1, …, m − 1),

a_{ik}(t) − \frac{1}{2}( ∑_{0<τ≤t} ∑_{σ=n_{j−1}+1}^{n_{j}} d_{1}a_{σi}(τ) · d_{1}a_{σk}(τ) −

− ∑_{0≤τ<t} ∑_{σ=n_{j−1}+1}^{n_{j}} d_{2}a_{σi}(τ) · d_{2}a_{σk}(τ)) =

= b_{1jik}(t) − b_{2jik}(t) for t ∈ [0, ω] (i, k = n_{j−1} + 1, …, n_{j}; j = 1, …, m),

(−1)^{l+1}σ_{j} ∑_{i,k=n_{j−1}+1}^{n_{j}} p_{ljik}(t)x_{i}x_{k} ≥ α_{lj}(t) ∑_{i=n_{j−1}+1}^{n_{j}} x_{i}^{2}

for μ_{c_{lj}} almost everywhere t ∈ [0, ω], (x_{i})_{i=1}^{n} ∈ R^{n} (l = 1, 2; j = 1, …, m),

where σ_{j} ∈ {−1, 1}, b_{ljik}(t) ≡ ∫_{0}^{t} p_{ljik}(τ)dc_{lj}(τ) (i ≠ k) and b_{ljii} is such that

(−1)^{l+1}σ_{j}(b_{ljii}(t) − b_{ljii}(s) − ∫_{s}^{t} p_{ljii}(τ)dc_{lj}(τ)) ≥ 0 for 0 ≤ s ≤ t ≤ ω.

Then we shall say that

A ∈ Q_{ω}(m, (n_{j}; c_{1j}, c_{2j}; α_{1j}, α_{2j}; P_{1j}, P_{2j})_{j=1}^{m}).  (2)

Theorem. Let there exist natural numbers m and n_{1}, …, n_{m} (0 = n_{0} < n_{1} < ⋯ < n_{m} = n), functions c_{lj} and α_{lj} (l = 1, 2; j = 1, …, m) and matrix functions P_{lj} = (p_{ljik})_{i,k=1}^{n} such that (2) holds. Let, moreover,

det(I + (−1)^{k}d_{k}A(t)) ≠ 0, (1 + σ_{j})d_{1}c_{j}(t) + (1 − σ_{j})d_{2}c_{j}(t) < 2,
(1 − σ_{j})d_{1}c_{j}(t) + (1 + σ_{j})d_{2}c_{j}(t) ≠ −2

and

exp (c_{j}(ω) − ∑_{0<τ≤ω} d_{1}c_{j}(τ) − ∑_{0≤τ<ω} d_{2}c_{j}(τ)) >

> \frac{1}{2}[(1 + σ_{j}) ∏_{0<τ≤ω} (1 − d_{1}c_{j}(τ)) ∏_{0≤τ<ω} (1 + d_{2}c_{j}(τ))^{−1} +

+(1 − σ_{j}) ∏_{0<τ≤ω} (1 + d_{1}c_{j}(τ))^{−1} ∏_{0≤τ<ω} (1 − d_{2}c_{j}(τ))],

for every t ∈ [0, ω] and j ∈ {1, …, m}, where

c_{j}(t) ≡ 2 ∑_{l=1}^{2} ∫_{0}^{t} α_{lj}(τ)dc_{lj}(τ).

Then the problem (1) has one and only one solution.

The analogous question has been considered in [1] for a system of linear ordinary differential equations.

References

1. I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations. (Russian) Current problems in mathematics. Newest results, vol. 30, 3–103, Itogi Nauki i Tekhniki, Akad, Nauk SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987.

Author’s address: Ministry of Defence of Georgia Department of Automatic-Control Systems 1, University St., Tbilisi 380114 Georgia