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Some properties of solutions of the homogeneous nonlinear second order differential equations. (English) Zbl 1389.34044

Summary: In this paper we consider the following nonlinear homogeneous second order differential equations \[ F (x, y, y', y'') = 0. \] We present for the solutions, \(y \in C^2 [a, b]\), of this equation, extremal principle, Sturm-type, Nicolescu-type and Butlewski-type separation theorems. Some applications and examples are given. Open problems are also presented.

MSC:

34A09 Implicit ordinary differential equations, differential-algebraic equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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