zbMATH — the first resource for mathematics

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.

MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text:
References:
 [1] BORŮVKA O.: Lineare Differentialtransformationen 2. Ordnung, VEB, Deutscher Verlag der Wissenschaften, Berlin, 1971. [2] COURANT R., HILBERT D.: Methods of Mathematical Physics. Vol. I, Interscience, New York, 1953. · Zbl 0053.02805 [3] DOŠLÝ O.: Riccati matrix differential equation and classification of disconjugate differential systems. Arch. Math. (Brno) 23 (1988), 231-242. · Zbl 0637.34025 [4] DOŠLÝ O.: On traits formation of self-adjoint differential systems and their reciprocals. Ann. Polon. Math. 50 (1990), 223-234. [5] DOŠLÝ O.: On some problems in oscillation theory of self-adjoint linear differential equations. Math. Slovaca 41 (1991), 101-111. · Zbl 0753.34018 [6] DOŠLÝ O.: Conjugacy criteria for second order differential equations. Rocky Mountain J. Math.) · Zbl 0794.34025 [7] HAWKING S. W., PENROSE R.: The singularities of gravity collapse and cosmology. Proc. Roy. Soc. London Ser. A 314 ; 1970, 543-548. · Zbl 0954.83012 [8] MACHÁT J.: Phase matrix of self-adjoint linear differential equations. (Czech.), Thesis, Brno, 1989. [9] MÜLLER-PFEIFFER E.: Existence of conjugate points for second and fourth order differential equations. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 281-291. · Zbl 0481.34019 [10] MÜLLER-PFEIFFER E.: On the existence of nodal domains for elliptic differential operators. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), 287-299. · Zbl 0537.35027 [11] MÜLLER-PFEIFFER E.: Nodal domains of one- or two-dimensional elliptic differential equations. Z. Anal. Anwendungen 7 (1988), 135-139. · Zbl 0657.35042 [12] TIPLER F. J.: General relativity and conjugate ordinary differential equations. J. Differential Equations 30 (1978), 165-174. · Zbl 0362.34023 [13] WEIDMAN J.: Linear Operators in Hilbert Spaces. Springer-Verlag, New York-Berlin, 1982.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.