Solution branching in linear differential equations
This paper deals with the \(n\)th order differential equation (1) \(L(y)\equiv y^{(n)}+p_{n-1}(t) y^{(n-1)}+\dots+p_ 0(t)y=0\), where \(p_ i\) are assumed to be continuous and real valued on \(\mathbb{R}\), together with the boundary conditions (2) \(y^{(r)}(a_ j)=0\), \(r=0,\dots,i_ j-1\), \(j=1,\dots,k\), where \(a_ 1,\dots,a_ k\) are distinct points and \(\sum_{j=1}^ k i_ j=n\). The principal motivating result of this paper is due to \textit{P. Hartman} [Rend. Circ. Mat. Palermo, II. Ser. 7, 123-142 (1958; Zbl 0085.045)]: If (1) has a nontrivial solution \(y\) with \(n\) zeros counting multiplicities on an open interval \(I\), then there exists a solution \(z\) with \(n\) distinct zeros on \(I\). Hartman's proof does not establish any relation between the solution \(z\) and origin solution \(y\). That is, \(z\) is not necessarily close (in the sense of continuous functions) to \(y\) and the distinct zeros of \(z\) are not necessarily close to the zeros of \(y\). The authors explain why the solutions \(y\) and \(z\) in Hartman's theorem are unrelated, in general, and give assumptions under which we may write \(z=y+\sum_{i=1}^ m c_ i w_ i\), where constants \(c_ 1,\dots,c_ m\) are near zero, \(w_ 1,\dots,w_ m\) are solutions of \(L(y)=0\), \(\| w_ i\|=1\), and zeros of \(z\) are simple and close to zeros of \(y\). Key assumptions are 1- dimensionality of the solution space of BVP (1), (2), an exactness condition on the zeros of \(y\), and a restriction on multiple zero counts.
scientific article; zbMATH DE number 166904