I. Kiguradze

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

ON A MULTI-POINT BOUNDARY VALUE PROBLEM FOR HIGHER ORDER SINGULAR DIFFERENTIAL EQUATIONS

Optimal sufficient conditions are found for the solvability and unique solvability of the boundary value problem

u^{(n)} = f(t, u, ⋯, u^{(n−1)}),  (1)
u^{(i−1)}(t_{0})=0 (i=1, …, n − 1), ∑_{k=1}^{n}α_{k}u^{(k−1)}(a_{k})+∑_{k=1}^{n}β_{k}u^{(k−1)}(b_{k})=0. (2)

Here n ≥ 2, a ≤ a_{k} < t_{0} < b_{k} ≤ b (k = 1, …, n), (−1)^{n−k}α_{k} ≥ 0,

β_{k} ≥ 0 (k = 1, …, n), ∑_{k=1}^{n}(|α_{k}| + β_{k}) > 0, and f : I_{t_{0}} × R^{n} → R is a

measurable in the first argument and continuous in the last n arguments function, I_{t_{0}} = [a, b] \ {t_{0}}. Moreover, the function f, generally speaking, is nonintegrable in the first argument on [a, b] as it has the singularity at the point t_{0}.

In particular, the following theorem is proved.

Theorem. Let there exist measurable functions g_{k}, \overline{g}_{k} : I_{t_{0}} → [0, +∞[ and an integrable function g : [a, b] → [0, +∞[ such that

∫_{a}^{b} |t − t_{0}|^{n−k}\overline{g}_{k}(t) dt < +∞ (k = 1, …, n),

\begin{matrix}∑_{k=1}^{n} \frac{1}{(n − k)!} ∫_{a}^{t_{0}}(t_{0} − t)^{n−k}g_{k}(t) dt ≤ 1, \\ ∑_{k=1}^{n} \frac{1}{(n − k)!} ∫_{t_{0}}^{b}(t − t_{0})^{n−k}g_{k}(t) dt ≤ 1, \end{matrix}  (3)

and on the set I_{t_{0}} × R^{n} the following inequalities are satisfied:

f(t, x_{1}, ⋯, x_{n}) sgn[(t − t_{0})^{n−1}x_{1}] ≥ − ∑_{k=1}^{n} g_{k}(t)|x_{k}| − g(t),

|f(t, x_{1}, ⋯, x_{n})| ≤ ∑_{k=1}^{n} \overline{g}_{k}(t)|x_{k}| + g(t).

Then problem (1), (2) has at least one solution.

Note that in the right-hand sides of inequalities (3) we cannot write down 1 + ε instead of 1 no matter how small ε > 0 would be.

The above results are obtained jointly with Prof. R. P. Agarwal.

Supported by GRDF (Grant No. 3318).