## 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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