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On the structure of some set of solutions of a second order ordinary differential equation. (Russian) Zbl 0614.34005

The Cauchy problem \[ (1)\quad x''=f(t,x,x'),\quad x(t_ 0)=x_ 0,\quad x'(t_ 0)=x_ 0' \] is considered, where \(f: I\times R^ 2\to R\) satisfies Caratheodory’s condition, \(t_ 0\in I=[a,b]\), \(-\infty <a<b<+\infty\), \(x_ 0,x_ 0'\in R\). Some properties of the set of solutions to (1) satisfying the condition \[ \min \{\alpha (t),\beta (t)\}\leq x(t)\leq \max \{\alpha (t),\beta (t)\}\quad on\quad I \] are established, where \(\alpha\),\(\beta\) are lower and upper functions of solutions to (1).
Reviewer: I.Foltyńska

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34C99 Qualitative theory for ordinary differential equations
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