Stability of the solutions of differential equations (Q1282989)

In order to give stability conditions for a differential equation the author introduces a new norm in the space of polynomial functions in one complex variable, \(P(z)=\sum_{j=0}^{n} a_{j}z^{j}\): \[ ||P||=\left (\sum_{j=0}^{n}j!|a_{j}|^{2}\right)^{1/2} \] This norm is hilbertian, and it can be interpreted as a weighted \(L_{2}\) or \(l_{2}\) norm. In this framework one considers the space \({\mathcal P}_{2}\) of analytic functions \(f(z)=\sum_{j\geq 0}a_{j}z^{j}\) for which \(\sum_{j\geq 0}j!|a_{j}|^{2}<\infty\). Some results concerning the space \({\mathcal P}_{2}\) allow to derive stability conditions for a differential equation \(P(D)u=f\), where \(P\) is any complex polynomial function, \(D={{\partial}\over{\partial {z} }}\), and \(f\in{\mathcal P}_{2}\). For such an equation one looks for \(u\in{\mathcal P}_{2}\), such that \(P(D)u\in{\mathcal P}_{2}\). When conditions that ensure the uniqueness of the solution are fulfilled, then for two \(f,g\) satisfying \(||f-g||<\varepsilon\), the corresponding solutions \(u,v\) are also close, i.e. \(||{u-v}||<\varepsilon/a\), where \(a\) is the leading coefficient of the polynomial \(P\).