Periodic and Bounded Solutions of Second-Order Nonlinear Differential Equations

Ahmedov, J. T. , Mukhamadiev, E. M. , Nurov, I. J.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 2.

Abstract:
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.

Keywords: differential equation, periodic solutions, boundedness, homogeneous equation, homotopy, completely continuous vector fields, phase portraits.

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