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Solutions with prescribed numbers of zeros for nonlinear second order differential equations. (English) Zbl 0820.34019
The authors consider nonlinear second order differential equations \((*)\) \(x''+ p(t) f(x)= 0\), \(a\leq t\leq b\), where \(p\in C^ 1[a, b]\), \(p(t)> 0\) for \(a\leq t\leq b\), and \(f\) satisfies \(f\in C(-\infty, \infty)\), \(f(- u)= f(u)\) for \(u\in (- \infty, \infty)\), \(f(u)> 0\) for \(u\in (0, \infty)\) and some further conditions near 0, and \(\infty\). In particular, they study the Emden-Fowler equation \(x''+ p(t)| x|^ \gamma\text{sign } x= 0\). They consider the solution \(x_ \lambda(t)\) of \((*)\) defined by \(x(a)= 0\), \(x'(a)= \lambda\), where \(\lambda\) is a real parameter. Their results describe mainly the number of zeros of \(x_ \lambda(t)\) in \((a, b]\) with respect to \(\lambda\).
Reviewer: F.Neuman (Brno)

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations