I. Kiguradze

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

ON SINGULAR BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

Suppose m and n are natural numbers, −∞ < a < b < +∞, α : [0, n−1], β ∈ [0, n − 1],

α_{i} = \frac{α + i − n + |α + i − n|}{2} , β_{i} = \frac{β + i − n + |β + i − n|}{2} (i = 1, …, n);

ℝ^{m} is the m-dimensional real Euclidean space with norm ∥ · ∥_{ℝ^{⋗}}; C_{α,β}^{n−1}(]a, b[ ; ℝ^{⋗}) is the Banach space of (n − 1)–times continuously dif-

ferentiable vector functions x : ]a, b[ → ℝ^{⋗} having the limits

lim_{t→a} (t − a)^{α_{i}}x^{(i−1)}(t), lim_{t→b} (b − t)^{β_{i}}x^{(i−1)}(t) (i = 1, …, n),

where the norm is defined by the equality

∥x∥_{C_{α,β}^{n−1}} = sup { ∑_{k=1}^{n}(t − a)^{α_{i}}(b − t)^{β_{i}}∥x^{(i−1)}(t)∥_{ℝ^{⋗}} : a < t < b};

L_{α,β}(]a, b[ ; ℝ^{⋗}) is the Banach space of summable with weight (t − a)^{α} × (b − t)^{β} vector functions y :]a, b[ → ℝ^{⋗} with the norm

∥y∥_{L_{α,β}} = ∫_{a}^{b}(t − a)^{α}(b − t)^{β}∥y(t)∥_{ℝ^{⋗}} dt.

Sufficient conditions for the solvability and unique solvability are established for the singular boundary value problem

x^{(n)}(t) = f(x)(t), h_{i}(x) = 0 (i = 1, …, n),  (1)

where f : C_{α,β}^{n−1}(]a, b[ ; ℝ^{⋗}) → 𝕃_{α,β}(]⅁, [ ; ℝ^{⋗}) and h_{i} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{⋗}) → ℝ^{⋗} (i = 1, …, n) are continuous operators.