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On the Kneser-type solutions for two-dimensional linear differential systems with deviating arguments. (English) Zbl 1135.34332

Summary: For the differential system
\[ u_1'(t)=p(t)u_2(\tau(t)), u_2'(t)=q(t)u_1(\sigma(t)),\quad t\in [0,+\infty), \]
where \(p,q\in L_{\text{loc}}(\mathbb R_+;\mathbb R_+)\), \(\tau,\sigma\in C(\mathbb R_+;\mathbb R_+)\), \(\lim_{t\to+\infty}\tau(t)=\lim_{t\to+\infty}\sigma(t)=+\infty\), we get necessary and sufficient conditions that this system does not have solutions satisfying the condition \(u_1(t)u_2(t)<0\) for \(t\in [t_0,+\infty)\). The inequality \((\delta+\Delta)\sqrt{pq}>2/e\) is necessary and sufficient for nonexistence of solutions satisfying this condition.

MSC:

34K06 Linear functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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References:

[1] Chanturiya TA: Specific conditions for the oscillation of solutions of linear differential equations with retarded argument.Ukrainskiĭ Matematicheskiĭ Zhurnal 1986,38(5):662-665, 681.
[2] Koplatadze R: Monotone and oscillating solutions ofth-order differential equations with retarded argument.Mathematica Bohemica 1991,116(3):296-308. · Zbl 0743.34075
[3] Koplatadze R: Specific properties of solutions of differential equations with deviating argument.Ukrainskiĭ Matematicheskiĭ Zhurnal 1991,43(1):60-67.
[4] Koplatadze R: On oscillatory properties of solutions of functional-differential equations.Memoirs on Differential Equations and Mathematical Physics 1994, 3: 179. · Zbl 0843.34070
[5] Ladde GS, Lakshmikantham V, Zhang BG: Oscillation Theory of Differential Equations with Deviating Arguments, Monographs and Textbooks in Pure and Applied Mathematics. Volume 110. Marcel Dekker, New York, NY, USA; 1987:vi+308.
[6] Shmul’yan MG: On the oscillating solutions of a linear second order differential equation with retarding argument.Differentsial’nye Uravneniya 1995, 31: 622-629.
[7] Skubačevskiĭ AL: The oscillatory solutions of a second order linear homogeneous differential equation with retarded argument.Differentsial’nye Uravneniya 1975, 11: 462-469, 587-588. · Zbl 0306.34092
[8] Labovskiĭ SM: A condition for the nonvanishing of the Wronskian of a fundamental system of solutions of a linear differential equation with retarded argument.Differentsial’nye Uravneniya 1974, 10: 426-430, 571.
[9] Azbelev NV, Domoshnitsky A: On the question of linear differential inequalities—I.Differential Equations 1991,27(3):257-263. translation from Differentsial’nye Uravneniya, vol. 27, pp. 376-384, 1991 translation from Differentsial’nye Uravneniya, vol. 27, pp. 376-384, 1991 · Zbl 0806.34010
[10] Azbelev NV, Domoshnitsky A: On the question of linear differential inequalities—II.Differential Equations 1991,27(6):641-647. translation from Differentsial’nye Uravneniya, vol. 27, pp, 923-931, 1991 translation from Differentsial’nye Uravneniya, vol. 27, pp, 923-931, 1991 · Zbl 0811.34051
[11] Domoshnitsky A: Wronskian of fundamental system of delay differential equations.Functional Differential Equations 2002,9(3-4):353-376. · Zbl 1048.34128
[12] Kiguradze I, Partsvania N: On the Kneser problem for two-dimensional differential systems with advanced arguments.Journal of Inequalities and Applications 2002,7(4):453-477. 10.1155/S1025583402000231 · Zbl 1017.34070 · doi:10.1155/S1025583402000231
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