N. Partsvania

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

BOUNDARY VALUE PROBLEMS FOR

TWO-DIMENSIONAL DELAYED DIFFERENTIAL SYSTEMS

The question on the solvability of the boundary value problem

\frac{du_{i}(t)}{dt} = f_{i}(t, u_{1}(τ_{1}(t)), u_{2}(τ_{2}(t))) (i = 1, 2),  (1)
u_{1}(a) = αu_{2}(a), β_{1}u_{1}(b) + β_{2}u_{2}(b) = γ  (2)

is investigated. Here f_{i} : [a, b] × ℝ_{+}^{2} → ℝ_{+} and τ_{i} : [a, b] → [a, b] (i = 1, 2) are continuous functions, ℝ_{+} = [⊬, +∞[ , α ∈ ℝ_{+}, β_{i} ∈ ℝ_{+} (i = 1, 2), β_{1} + β_{2} > 0, γ > 0, and

τ_{i}(t) ≤ t, f_{i}(t, 0, 0) = 0 for a ≤ t ≤ b (i = 1, 2).

In particular, the following theorem is proved.

Theorem. For the solvability of problem (1), (2) it is sufficient one of the following five conditions to be fulfilled:

(i) β_{1} > 0, β_{2} > 0;

(ii) β_{1} = 0, β_{2} = 1, and there exists a positive constant ℓ such that

f_{1}(t, x, y) ≤ ℓ[1 + f_{2}(t, x, y)] for a ≤ t ≤ b, x ≥ 0, 0 ≤ y ≤ γ;

(iii) β_{1} = 0, β_{2} = 1, and

τ_{1}(t) < t for a < t ≤ b;

(iv) α > 0, β_{1} = 0, β_{2} = 0, and there exists a positive constant ℓ such that

f_{2}(t, x, y) ≤ ℓ[1 + f_{1}(t, x, y)] for a ≤ t ≤ b, 0 ≤ x ≤ γ, y ≥ 0;

(v) α > 0, β_{1} = 1, β_{2} = 0, and

τ_{2}(t) < 0 for a < t ≤ b.