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The multiplicity criteria for zero points of second order differential equations. (English) Zbl 0754.34026
The paper is dedicated to the problem of conjugacy and disconjugacy of the second order equation (1) \(y''+p(x)y=0\). If \(p(x)-(k^ 2-1)(1+x^ 2)^{-2}\not\equiv 0\) on \(\mathbb{R}\) and \[ \lim_{t_ 1\searrow- \infty,}\inf_{t^ 2\nearrow\infty}\int^{t_ 2}_{t_ 1}[p(x)- (k^ 2-1 )(1+x^ 2)^{-2}](1+x^ 2)\sin^ 2k\left(\text{arctg} x+{\pi\over 2}\right)dx\geq 0 \] then (1) possesses a nontrivial solution having at least \((n+1)\) zeros on \(\mathbb{R}\). If there exist \(\varepsilon_ 1\) and \(\varepsilon_ 2\) strictly positive and \(c\in(a,b)\) such that \[ \varepsilon_ 1\int^ b_ c\exp\left\{2\int^ x_ c\left[\int^ t_ cp(s)ds-\varepsilon_ 1 \right]dt\right\}dx>A, \varepsilon_ 2\int^ c_ a\exp\left\{2\int^ x_ c\left[\int ^ t_ cp(s)ds+\varepsilon_ 2\right]dt\right\}dx>B \] and \(\varepsilon_ 1+\varepsilon_ 2-\pi(\varepsilon_ 1B+\varepsilon_ 2A)/2AB>0\) then (1) is conjugate on \((a,b)\). Interesting extensions to partial differential equations are suggested.

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