S. Mukhigulashvili

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

ON A PERIODIC BOUNDARY VALUE PROBLEM FOR THE TWO-DIMENSIONAL LINEAR DIFFERENTIAL SYSTEM

The differential system

u′_{i}(t) = p_{ii}(t)u_{i}(t) + p_{i,3−i}u_{3−i}(t) + q_{i}(t) (i = 1, 2)  (1)

is considered with the periodic boundary conditions

u_{i}(w) = u_{i}(0) (i = 1, 2),  (2)

where p_{ik} : [0, w] → R (i, k = 1, 2) are Lebesgue integrable functions, w > 0. Let

p_{i}(t) = p_{i,3−i}(t) exp ( ∫_{0}^{t}(p_{3−i,3−i}(s) − p_{ii}(s)) ds) for 0 ≤ t ≤ w,

λ_{i} = exp ( − ∫_{0}^{w} p_{ii}(s)ds) (i = 1, 2), l = ∫_{0}^{w} |p_{1}(s)| ds ∫_{0}^{w} |p_{2}(s)| ds,

α_{1} = min{1, λ_{1}λ_{2}}, α_{2} = max{1, λ_{1}λ_{2}},
β_{1} = min{λ_{1}, λ_{2}}, β_{2} = max{λ_{1}, λ_{2}}.

Then the following statemants are valid.

Theorem 1. Let

l > 0, l^{−1}(λ_{1} − 1)(λ_{2} − 1) ∉]α_{1}, α_{2}[

and σ ∈ {−1, 1} be such that

σp_{12}(t) ≥ 0, σp_{21}(t) ≤ 0 for 0 ≤ t ≤ w.

Then the problem (1), (2) has one and only one solution.

Theorem 2. Let σ ∈ {−1, 1} be such that

σp_{1}(t) ≥ 0, σ( ∫_{t}^{w} p_{2}(s) ds + \frac{λ_{2}}{λ_{1}} ∫_{0}^{t} p_{2}(s) ds) < 0 for 0 ≤ t ≤ w

and

0 < ∫_{0}^{w} |p_{1}(s)| ds ∫_{0}^{w}[σp_{2}(s)]_{−} ds ≤ 16 \frac{β_{1}}{β_{2}} .

Let, moreover,

∫_{0}^{w}(p_{22}(s) − p_{11}(s)) ds ∫_{0}^{w} p_{11}(s) ds ≥ 0.

Then the problem (1), (2) has one and only one solution.