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On a Nevanlinna type result for solutions of nonautonomous equation $$y'= a(t)F_{1}(x,y),$$ $$x'=b(t)F_{2}(x,y)$$. (English) Zbl 1132.34032
The authors are concerned with the oscillation of solutions of the system of differential equations $x^{\prime}=a(t)F_{1}(x,y),\qquad y^{\prime}=b(t)F_{2}(x,y),\tag{1}$ where the functions $$F_{1}$$ and $$F_{2}$$ are smooth in the closure $$\overline{D}$$ of a given domain $$D$$ in the plane, whereas $$a$$ and $$b$$ are smooth bounded functions satisfying
$\frac{1}{\pi}\int_{t_{1}}^{t_{2}}\left[ \left| \frac{a^{\prime}(t)} {a(t)}\right| +\left| \frac{b^{\prime}(t)}{b(t)}\right| \right] dt\leq H<\infty.$ Taking into account that in this case $$a,b,F_{1},F_{2}$$ and the first-order partial derivatives of $$F_{1}$$ and $$F_{2}$$ are bounded in $$D,$$ the problem looks quite unusual for the classical theory of oscillation of ordinary differential equations. In fact, the authors have chosen a different approach to the problem related to the ideas and techniques exploited in the complex analysis. The concept of $$a$$-points of solutions is introduced as the points $$\tau_{i}$$ such that for some values $$(a^{\prime},a^{\prime\prime})$$ in the $$xy$$-plane, solutions of the system (1) satisfy $$x(\tau_{i})=a^{\prime}$$ and $$y(\tau_{i})=a^{\prime\prime}.$$ Main results of the paper provide upper bounds on the number $$n(t_{1},t_{2},a)$$ of $$a$$-points of solutions of the system (1) in the interval $$(t_{1},t_{2}).$$
Since the authors mention that they are not aware of oscillation results for the systems of differential equations, the reviewer would like to mention a few monographs where related theorems and further references can be found. Oscillation results for two-dimensional systems of ordinary differential, delay differential and neutral differential equations are reported in R. P. Agarwal, S. R. Grace and D. O’Regan [Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Dordrecht: Kluwer Academic Publishers (2002; Zbl 1073.34002), Chapter 7], R. P. Agarwal, M. Bohner and W.-T. Li [Nonoscillation and oscillation: theory for functional differential equations, New York: Marcel Dekker (2004; Zbl 1068.34002), Chapter 7], L. H. Erbe, Q. Kong and B. G. Zhang [Oscillation theory for functional-differential equations, New York: Marcel Dekker (1995; Zbl 0821.34067), Chapter 6], G. S. Ladde, V. Lakshmikantham and B. G. Zhang [Oscillation theory of differential equations with deviating arguments, New York: Marcel Dekker (1987; Zbl 0832.34071), Chapter 5], K. Gopalsamy [Stability and oscillations in delay differential equations of population dynamics, Dordrecht: Kluwer Academic Publishers (1992; Zbl 0752.34039), Chapter 5].
##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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