Geodesic
Periodic and bounded solutions of second-order nonlinear differential equations
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 2, pp. 35-50 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper investigates questions about the existence of periodic or bounded solutions of nonlinear second-order differential equations of the form $y''+g(y,y')=f(t,y,y')$. Here the function $g(y,z)$ is continuous and positively homogeneous of the first order, and $f(t,y,z)$ is continuous function defined for all values of $t$, $y$, $z$ and satisfying the smallness condition with respect to $|y|+|z|$ at infinity. For this equation, the questions of the existence of a priori estimate, periodic solutions in the case of a function $f(t,y,z)$ periodic in $t$, and bounded solutions in the case only the boundedness in $t$ of the function $f(t,y,z)$ are closely related to the qualitative behavior of the solution of the homogeneous equations $y''+g(y,y')=0$. Therefore, at the first stage, it seems important to study the nature of the behavior of the trajectory equivalent to the homogeneous equation of the system. Passing to polar coordinates, we obtain the representations of the solution of the system, which allow one to describe the complete classification of all possible phase portraits of the solution of the system in terms of the property of the function $g(y,y')$. In particular, the conditions for the the absence of nonzero periodic solutions or solutions bounded on the entire axis were obtained. The challenge of existence periodic solutions to the original equation is equivalent to the existence of solutions to the integral equations in the space $C[0, T]$-functions continuous on the segment $[0,T]$. In turn, the integral equation generates a completely continuous vector field in the space $C[0,T]$, the zeros of which determine solution of an integral equation. Formulas for calculating the rotation of a vector field on spheres of sufficiently large radius of the space $C[0,T]$ are obtained. Based on the results obtained, conditions for the existence of periodic and bounded solutions of an inhomogeneous equation are found. Note that the results obtained have been brought up to calculation formulas. 
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J. T. Ahmedov; E. M. Mukhamadiev; I. J. Nurov. Periodic and bounded solutions of second-order nonlinear differential equations. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 2, pp. 35-50. http://geodesic.mathdoc.fr/item/VMJ_2022_24_2_a3/

[1] Kafarov V. V., Glebov M. B., Mathematical Modeling of the Main Processes Chemical Industries, UNITY, M., 1991, 400 pp. (in Russian)

[2] Riznichenko G. Yu., Lectures on Mathematical Models in Biology, M.–Izhevsk, 2002, 232 pp. (in Russian)

[3] Zang V. B., Synergetic Economy. Time and Change in the Nonlinear Economic Theories, World, M., 1999, 335 pp. (in Russian)

[4] Kolemaev V. A., Mathematical Economics, Higher school, M., 1998, 240 pp. (in Russian) | MR

[5] Krasnoselsky M. A., Zabreik P. P., Geometric Methods of Nonlinear Analysis, Nauka, M., 1975, 512 pp. (in Russian)

[6] Petrovsky I. G., Lectures on the Theory of Ordinary Differential Equations, Moscow State University, M., 1984, 296 pp. (in Russian) | MR

[7] Bobylev N. A., “About Constructing Correct Guiding Functions”, Doklady Math. USSR, 183:2 (1968), 265–266 (in Russian) | Zbl

[8] Muhamadiev E. M., “On the Theory of Periodical Solutions of Systems of Ordinary Differential Equations”, Doklady Math. USSR, 194:3 (1970), 510–513 (in Russian) | Zbl

[9] Muhamadiev E. M., “On the Theory of Bounded Solutions of Ordinary Systems Differential Equations”, Differentsial'nye Uravneniya, 10:5 (1974), 635–646 (in Russian) | Zbl

[10] Ahmedov J. T., Muhamadiev E. M., Nurov I. D., “Periodic and Limited Solutions Second-Order Quasilinear Equations”, Vestnik Voronezh State University, 3 (2019), 59–66 (in Russian) | MR

[11] Gomory R. E., “Critical Points at Infinity and Forced Oscillation”, Annals of Mathematics Studies, 3:36 (1956), 85–126 | MR

[12] Abduvaitov H., “Some Sufficient Conditions for the Existence of Periodic and Bounded Solutions of Second-Order Nonlinear Differential Equations”, Differentsial'nye Uravneniya, 21:12 (1985), 74–84 (in Russian) | MR

[13] Azizov R. E., “A Priori Estimates for Bounded Solutions of a System of Three Differential Equations with Homogeneous Principal Terms”, Reports of the Academy of Sciences of the Tajik SSR, 31:9 (1988), 555–558 (in Russian) | MR | Zbl

[14] Ahmedov J. T., “To Theory of Periodic and Bounded Solutions of Nonlinear Dynamic Second-Order Systems with Phase Portraits”, Reports of the Academy of Sciences of the Republic of Tajikistan, 63:9–10 (2020), 579–585 (in Russian)