Abstract:
In the present article, we consider the problem
\begin{equation}
Hu+\lambda u=f(t), \quad t\in (0,1),
\tag{1}
\end{equation}
where $\lambda$ is a complex parameter and $H$ stands for an ordinary differential operator of order $l\ge 2$ defined by the differential expression
$$
Hu=k(t)u^{(l)}(t)+a(t)u^{(l-1)}(t)+\sum_{j=0}^{l-2}a_j(t)u^{(j)}(t),
$$
with $u^{(j)}(t)=\frac{d^ju(t)}{dt^j}$, and the collection of boundary conditions
$$
l_1u=u^{(p)}(1)+\sum_{\nu=0}^{p-1}\alpha_{\nu}u^{(\nu)}(1)=0, \quad l_0u=u^{(q)}(0)+\sum_{\nu=0}^{q-1}\beta_{\nu}u^{(\nu)}(0)=0.
$$
Using a priori bounds, we prove existence and uniqueness theorems of boundary value problems for linear ordinary differential equations and study dependence of solutions on a parameter. The peculiarity of the problem lies in the fact that the leading coefficient in the equation is of an arbitrary sign on the interval $(0,1)$.
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