S. Mukhigulashvili

On Two-Point Boundary Value Problems for Two-Dimensional Differential Systems with Singularities

abstract:
For a differential system
$$ \frac{du_1}{dt}=h_0(t,u_1,u_2)u_2,\quad \frac{du_2}{dt}=-h_1(t,u_1,u_2)u_1^{-\lb}-h_2(t,u_1,u_2), $$
where $\lb\in]0,1[$ and $h_i:]a,b[\tm]0,+\iy[\tm \bR\to[0,+\iy[$ $(i=0,1,2) $ are continuous functions, we have established sufficient conditions for the existence of at least one solution satisfying one of the two boundary conditions
$$\lim_{t\to a}u_1(t)=0,\;\;\;\lim_{t\to b}u_1(t)=0 $$
and
$$\lim_{t\to a}u_1(t)=0,\;\;\;\lim_{t\to b}u_2(t)=0. $$