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Oscillation theorems for non-canonical self-adjoint differential equations of second order. (English) Zbl 1036.34037

The paper presents two oscillation theorems for the second-order linear differential equation \[ (r(t)u'(t))'+p(t)u(t)=0,\tag{1} \] where \(r,p\in C[t_0,\infty)\) and \(\int^\infty \frac{dt}{r(t)}<\infty\). The results are proved by using the so-called \(v\)-derivative of a function. They can be achived also by the standard change of variable \(s=1/\rho (t)\), \(z(s)=su(t)\), where \(\rho(t)=\int_t^\infty \frac{ds}{r(s)}\) transforming equation (1) into the binomial equation \(z''+r(t)p(t)\rho^3 (t)\frac{z(s)}{s}=0\), and applying the classical Kneser result.

MSC:

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

Keywords:

Oscillation
Full Text: DOI

References:

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