Resolvent Estimates for Ordinary Differential Operators of Mixed Type

A. V. Chueshov

Matematičeskie trudy, Tome 3 (2000) no. 1, pp. 144-196.

Résumé

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 $u^{(j)}(t)=\frac{d^{j}u(t)}{d{t}^j}$, 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 $(0,1)$.