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On zeros of solutions to a second-order singular half-linear equation. (English) Zbl 0949.34024

The authors investigate oscillatory properties of the half-linear second-order differential equation \[ u''=p(t)|u|^\alpha|u'|^{1-\alpha}\text{ sgn }u,\quad t\in (a,b), \tag{*} \] with \(\alpha\in (0,1]\), \(p\in L_{\text{loc}}(a,b)\), \(\int_a^b (t-a)^\alpha(b-t)^\alpha |p(t)|dt<\infty\) and \(p\) is allowed to have singularities both at \(t=a\), \(t=b\). A typical result is the following statement: Let there exist \(\lambda,\mu \in (a,b)\) such that \[ \begin{gathered} -\alpha\int_\lambda ^\mu p(t) dt\geq {1\over (\lambda -a)^\alpha}+ {1\over (b-\mu)^\alpha}+{\alpha\over (\lambda -a)^{\alpha+1}} \int_a^\lambda (t-a)^{\alpha+1}p(t) dt\\ + {\alpha\over (b-\mu)^{\alpha+1}}\int_\mu^b(b-t)^{\alpha+1}p(t) dt. \end{gathered} \] Then there exist \(a_1\in [a,b)\), \(b_1\in (a,b]\) and a nontrivial solution \(u\) to (*) such that \(u(a_1+)=0=u(b_1-)\).

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34K11 Oscillation theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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