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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)

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