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Focal points and comparison theorems for a class of two point boundary value problems. (English) Zbl 0781.34023

The authors study oscillation properties of the \(n\)th order linear differential equation. Denote by \(T(i_ 1,\dots,i_{n-k}; b)\) the boundary conditions \(y^{(l)}(0)=0\), \(l=0,\dots,k-1\), \(y^{(l)}(b)=0\), \(l=i_ 1,\dots,i_{n-k}\), where \(k\in \{1,\dots,n-1\}\) and \(1<i_ 1\leq i_ 2<\dots <i_{n-k}\leq n-1\). The point \(b_ 0\) is said to be the first \(i_ 1,\dots,i_{n-k}\)-focal point of \(x=0\) relative to the equation (*) \(L(y)= \sum^ \mu_{l=0} p_ l(x)y^{(l)}\), where \(\mu\in\{0,\dots,i_ 1\}\) and \(L(y)= y^{(n)} +q(x)y\), if \(b_ 0=\inf\{b\): (*), \(T(i_ 1,\dots,i_{n-k};b)\) has a nontrivial solution}. Under natural sign restrictions on the functions \(p_ l\), \(l=0,\dots,\mu\), the value \(b_ 0\) is characterized by means of the spectral radius of an integral operator defined via the Green function of (*), \(T(i_ 1,\dots,i_{n-k};b)\), and comparison theorems for \(b_ 0\) relative to two different boundary conditions \(T(i_ 1,\dots,i_{n- k};b)\), \(T(j_ 1,\dots, j_{n-k};b)\) are given.
If \((i_ 1,\dots,i_{n-k})=(0,\dots,n-k-1)\) or \((i_ 1, \dots,i_{n- k})= (k,\dots,n-1)\), the results of the paper reduce to known statements concerning the conjugate and focal points.
Reviewer: O.Došlý (Brno)

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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