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 1132.34032The 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].

\[ \frac{1}{\pi}\int_{t_{1}}^{t_{2}}\left[ \left| \frac{a^{\prime}(t)} {a(t)}\right| +\left| \frac{b^{\prime}(t)}{b(t)}\right| \right] dt\leq H<\infty. \] 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].

Reviewer: Svitlana P. Rogovchenko (Famagusta)